| Editor-in-Chief: Askold Khovanskii |
| Managing Editor: Vladlen Timorin |
| A Journal of the IMS, |
| Stony Brook University |
| Published by |
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Contents
Research Papers
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Two-Sided Fundamental Theorem of Affine Geometry
Alexey Gorinov
Arnold Mathematical Journal (2022)
Published: 24 March 2022
Abstract
Download PDF of the paper .The fundamental theorem of affine geometry says that if a self-bijection $f$ of an affine space of dimenion n over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then $f$ of the expected type, namely $f$ is a composition of an affine map and an automorphism of the field. We prove a two-sided analogue of this: namely, we consider self-bijections as above which take affine subspaces to affine subspaces but which are allowed to take left subspaces to right ones and vice versa. We show that under some conditions these maps again are of the expected type. -
Renormalization of Bicritical Circle Maps
Gabriela Estevez, Pablo Guarino
Arnold Mathematical Journal (2022)
Published: 03 March 2022
Abstract
Download PDF of the paper .A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper, we establish this principle for a large class of bicritical circle maps, which are $C^3$ circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo in (J Eur Math Soc 1:339–392, 1999) for the case of a single critical point. When combined with the recent papers (Estevez et al. in Complex bounds for multicritical circle maps with bounded type rotation number, arXiv:2005.02377 , 2020; Yampolsky in C R Math Rep Acad Sci Can 41:57–83, 2019), our main theorem implies $C^{1+\alpha }$ rigidity for real-analytic bicritical circle maps with rotation number of bounded type (Corollary 1.1). -
Dynamical Moduli Spaces and Polynomial Endomorphisms of Configurations
Talia Blum, John R. Doyle, Trevor Hyde, Colby Kelln, Henry Talbott, Max Weinreich
Arnold Mathematical Journal 2022
Published: 22 February 2022
Abstract
Download PDF of the paper .A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree. -
Catastrophe in Elastic Tensegrity Frameworks
Alexander Heaton, Sascha Timme
Arnold Mathematical Journal (2022)
Published: 18 February 2022
Abstract
Download PDF of the paper .We discuss elastic tensegrity frameworks made from rigid bars and elastic cables, depending on many parameters. For any fixed parameter values, the stable equilibrium position of the framework is determined by minimizing an energy function subject to algebraic constraints. As parameters smoothly change, it can happen that a stable equilibrium disappears. This loss of equilibrium is called catastrophe, since the framework will experience large-scale shape changes despite small changes of parameters. Using nonlinear algebra, we characterize a semialgebraic subset of the parameter space, the catastrophe set, which detects the merging of local extrema from this parametrized family of constrained optimization problems, and hence detects possible catastrophe. Tools from numerical nonlinear algebra allow reliable and efficient computation of all stable equilibrium positions as well as the catastrophe set itself. -
The ${{\mathbb {F}}}_p$-Selberg Integral
Rimányi, Richárd, Alexander Varchenko
Arnold Mathematical Journal (2022) 8:1
Published: 24 January 2022
Abstract
Download PDF of the paper .We prove an ${{\mathbb {F}}}_p$-Selberg integral formula, in which the ${{\mathbb {F}}}_p$-Selberg integral is an element of the finite field ${{\mathbb {F}}}_p$ with odd prime number p of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo p. -
Partial Duality of Hypermaps
S. Chmutov, F. Vignes-Tourneret
Arnold Mathematical Journal (2022)
Published: 03 January 2022
Abstract
Download PDF of the paper .We introduce partial duality of hypermaps, which include the classical Euler–Poincaré duality as a particular case. Combinatorially, hypermaps may be described in one of three ways: as three involutions on the set of flags (bi-rotation system or $\tau $-model), or as three permutations on the set of half-edges (rotation system or $\sigma $-model in orientable case), or as edge 3-coloured graphs. We express partial duality in each of these models. We give a formula for the genus change under partial duality. -
A Symplectic Dynamics Proof of the Degree–Genus Formula
Peter Albers, Hansjörg Geiges, Kai Zehmisch
Arnold Mathematical Journal (2021)
Published: 20 December 2021
Abstract
Download PDF of the paper .We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere (defining the Hopf fibration), with boundaries oriented positively by the flow. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory. -
Equivalence of Neighborhoods of Embedded Compact Complex Manifolds and Higher Codimension Foliations
Xianghong Gong, Laurent Stolovitch
Arnold Mathematical Journal (2021)
Published: 15 October 2021
Abstract
Download PDF of the paper .We consider an embedded n-dimensional compact complex manifold in $n+d$ dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert’s formal principle program. We will give conditions ensuring that a neighborhood of $C_n$ in $M_{n+d}$ is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold’s result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in $M_{n+d}$ having $C_n$ as a compact leaf, extending Ueda’s theory to the high codimension case. Both problems appear as a kind of linearization problems involving small divisors condition arising from solutions to their cohomological equations. -
On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field
A. Haddley, R. Nair
Arnold Mathematical Journal (2021)
Published: 15 October 2021
Abstract
Download PDF of the paper .Let ${\mathcal {M}}$ denote the maximal ideal of the ring of integers of a non-Archimedean field K with residue class field k whose invertible elements, we denote $k^{\times }$, and a uniformizer we denote $\pi $. In this paper, we consider the map $T_{v}: {\mathcal {M}} \rightarrow {\mathcal {M}}$ defined by $\begin{aligned} T_v(x) = \frac{\pi ^{v(x)}}{x} - b(x), \end{aligned}$ where $b(x)$ denotes the equivalence class to which $\frac{\pi ^{v(x)}}{x}$ belongs in $k^{\times }$. We show that $T_v$ preserves Haar measure $\mu $ on the compact abelian topological group ${\mathcal {M}}$. Let ${\mathcal {B}}$ denote the Haar $\sigma $-algebra on ${\mathcal {M}}$. We show the natural extension of the dynamical system $({\mathcal {M}}, {\mathcal {B}}, \mu , T_v)$ is Bernoulli and has entropy $\frac{\#( k)}{\#( k^{\times })}\log (\#( k))$. The first of these two properties is used to study the average behaviour of the convergents arising from $T_v$. Here for a finite set A its cardinality has been denoted by $\# (A)$. In the case $K = {\mathbb {Q}}_p$, i.e. the field of p-adic numbers, the map $T_v$ reduces to the well-studied continued fraction map due to Schneider. -
Varieties in Cages: A Little Zoo of Algebraic Geometry
Gabriel Katz
Arnold Mathematical Journal (2021)
Published: 30 September 2021
Abstract
Download PDF of the paper .A $d^{\{n\}}$-cage $\mathsf K$ is the union of n groups of hyperplanes in $\mathbb P^n$, each group containing d members. The hyperplanes from the distinct groups are in general position, thus producing $d^n$ points where hyperplanes from all groups intersect. These points are called the nodes of $\mathsf K$. We study the combinatorics of nodes that impose independent conditions on the varieties $X \subset \mathbb P^n$ containing them. We prove that if X, given by homogeneous polynomials of degrees $\le d$, contains the points from such a special set $\mathsf A$ of nodes, then it contains all the nodes of $\mathsf K$. Such a variety X is very special: in particular, X is a complete intersection. -
An Extension of the $\mathfrak {sl}_2$ Weight System to Graphs with $n \le 8$ Vertices
Evgeny Krasilnikov
Arnold Mathematical Journal (2021)
Published: 06 September 2021
Abstract
Download PDF of the paper .Chord diagrams and 4-term relations were introduced by Vassiliev in the late 1980. Various constructions of weight systems are known, and each of such constructions gives rise to a knot invariant. In particular, weight systems may be constructed from Lie algebras as well as from the so-called 4-invariants of graphs. A Chmutov–Lando theorem states that the value of the weight system constructed from the Lie algebra $\mathfrak {sl}_2$ on a chord diagram depends on the intersection graph of the diagram, rather than the diagram itself. This inspired a question due to Lando about whether it is possible to extend the weight system $\mathfrak {sl}_2$ to a graph invariant satisfying the four term relations for graphs. We show that for all graphs with up to 8 vertices such an extension exists and is unique, thus answering in affirmative to Lando’s question for small graphs. -
New Invariants of Poncelet–Jacobi Bicentric Polygons
Pedro Roitman, Ronaldo Garcia, Dan Reznik
Arnold Mathematical Journal (2021)
Published: 18 August 2021
Abstract
Download PDF of the paper .The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville’s theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that the pedal polygons with respect to the family’s limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard N-periodics with respect to a focus-centered circle, an important corollary is that (iii) elliptic billiard focus-inversive N-gons have constant perimeter. Interestingly, these also conserve their sum of cosines (except for the $N=4$ case). -
Element-Building Games on $\mathbb {Z}_n$
Bret Benesh, Robert Campbell
Arnold Mathematical Journal (2021)
Published: 18 August 2021
Abstract
Download PDF of the paper .We consider a pair of games where two players alternately select previously unselected elements of $\mathbb {Z}_n$ given a particular starting element. On each turn, the player either adds or multiplies the element they selected to the result of the previous turn. In one game, the first player wins if the final result is 0; in the other, the second player wins if the final result is 0. We determine which player has the winning strategy for both games except for the latter game with nonzero starting element when $n \in \{2p,4p\}$ for some odd prime p. -
Cutpoints of Invariant Subcontinua of Polynomial Julia Sets
Alexander Blokh, Lex Oversteegen, Vladlen Timorin
Arnold Mathematical Journal (2021)
Published: 16 August 2021
Abstract
Download PDF of the paper .We prove fixed point results for branched covering maps $f$ of the plane. For complex polynomials $P$ with Julia set $J_{P}$ these imply that periodic cutpoints of some invariant subcontinua of $J_{P}$ are also cutpoints of $J_{P}$. We deduce that, under certain assumptions on invariant subcontinua $Q$ of $J_{P}$, every Riemann ray to $Q$ landing at a periodic repelling/parabolic point $x\in Q$ is isotopic to a Riemann ray to $J_{P}$ relative to $Q$. -
Surfaces of Section for Seifert Fibrations
Bernhard Albach, Hansjörg Geiges
Arnold Mathematical Journal (2021)
Published: 05 August 2021
Abstract
Download PDF of the paper .We classify global surfaces of section for flows on 3-manifolds defining Seifert fibrations. We discuss branched coverings—one way or the other—between surfaces of section for the Hopf flow and those for any other Seifert fibration of the 3-sphere, and we relate these surfaces of section to algebraic curves in weighted complex projective planes. -
On a Theorem of Lyapunov-Poincaré in Higher Dimensions
V. León, B. Scárdua
Arnold Mathematical Journal (2021)
Published: 13 July 2021
Abstract
Download PDF of the paper .The classical Lyapunov-Poincaré center theorem assures the existence of a first integral for an analytic 1-form near a center singularity in dimension two, provided that the first jet of the 1-form is nondegenerate. The basic point is the existence of an analytic first integral for the given 1-form. In this paper, we consider generalizations for two main frameworks: (1) real analytic foliations of codimension one in higher dimension and (2) singular holomorphic foliations in dimension two. All this is related to the problem of finding criteria assuring the existence of analytic first integrals for a given codimension one germ with a suitable first jet. Our approach consists in giving an interpretation of the center theorem in terms of holomorphic foliations and, following an idea of Moussu, apply the holomorphic foliations arsenal to obtain the required first integral. As a consequence we are able to revisit some of Reeb's classical results on integrable perturbations of exact homogeneous 1-forms, and prove versions of these in the framework of non-isolated (perturbations of transversely Morse type) singularities. -
Real Lines on Random Cubic Surfaces
Rida Ait El Manssour, Mara Belotti, Chiara Meroni
Arnold Mathematical Journal (2021)
Published: 02 July 2021
Abstract
Download PDF of the paper .We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e., a surface $$Z\subset {\mathbb {R}}{\mathrm {P}}^3$$ defined by a random gaussian polynomial whose probability distribution is invariant under the action of the orthogonal group O(4) by change of variables. Such invariant distributions are completely described by one parameter $$\lambda \in [0,1]$$ and as a function of this parameter the expected number of real lines equals: $$\begin{aligned} E_\lambda =\frac{9(8\lambda ^2+(1-\lambda )^2)}{2\lambda ^2+(1-\lambda )^2}\left( \frac{2\lambda ^2}{8\lambda ^2+(1-\lambda )^2}-\frac{1}{3}+\frac{2}{3}\sqrt{\frac{8\lambda ^2+(1-\lambda )^2}{20\lambda ^2+(1-\lambda )^2}}\right) . \end{aligned}$$ This result generalizes previous results by Basu et al. (Math Ann 374(3–4):1773–1810, 2019) for the case of a Kostlan polynomial, which corresponds to $$\lambda =\frac{1}{3}$$ and for which $$E_{\frac{1}{3}}=6\sqrt{2}-3.$$ Moreover, we show that the expectation of the number of real lines is maximized by random purely harmonic cubic polynomials, which corresponds to the case $$\lambda =1$$ and for which $$E_1=24\sqrt{\frac{2}{5}}-3$$. -
Strange Duality Between the Quadrangle Complete Intersection Singularities
Wolfgang Ebeling, Atsushi Takahashi
Arnold Mathematical Journal (2021)
Published: 22 June 2021
Abstract
Download PDF of the paper .There is a strange duality between the quadrangle isolated complete intersection singularities discovered by the first author and Wall. We derive this duality from a variation of the Berglund–Hübsch transposition of invertible polynomials introduced in our previous work about the strange duality between hypersurface and complete intersection singularities using matrix factorizations of size two. -
Remarks on Joachimsthal Integral and Poritsky Property
Maxim Arnold, Serge Tabachnikov
Arnold Mathematical Journal (2021) 7:3, 483-491
Published: 01 June 2021
Abstract
Download PDF of the paper .The billiard in an ellipse has a conserved quantity, the Joachimsthal integral. We show that the existence of such an integral characterizes conics. We extend this result to the spherical and hyperbolic geometries and to higher dimensions. We connect the existence of Joachimsthal integral with the Poritsky property, a property of billiard curves, called so after H. Poritsky whose important paper Poritsky (Ann Math 51:446--470, 1950) was one of the early studies of the billiard problem. -
A Galois–Dynamics Correspondence for Unicritical Polynomials
Robin Zhang
Arnold Mathematical Journal (2021) 7:3, 467-481
Published: 09 June 2021
Abstract
Download PDF of the paper .In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in the proof of the Mordell–Lang conjecture, we describe a correspondence between the action of a Galois group and the dynamical action of a rational map. For nonlinear polynomials with rational coefficients, the irreducibility of the associated dynatomic polynomial serves as a convenient criterion, although we also verify that the correspondence occurs in several cases when the dynatomic polynomial is reducible. The work of Morton, Morton–Patel, and Vivaldi–Hatjispyros in the early 1990s connected the irreducibility and Galois-theoretic properties of dynatomic polynomials to rational periodic points; from the Galois–dynamics correspondence, we derive similar consequences for quadratic periodic points of unicritical polynomials. This is sufficient to deduce the non-existence of quadratic periodic points of quadratic polynomials with exact period 5 and 6, outside of a specified finite set from Morton and Krumm’s work in explicit Hilbert irreducibility. -
Conjectures on Stably Newton Degenerate Singularities
Jan Stevens
Arnold Mathematical Journal (2021) 7:3, 441-465
Published: 07 June 2021
Abstract
Download PDF of the paper .We discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions. We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $x^p+x^q$ in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate. -
Inflation of Poorly Conditioned Zeros of Systems of Analytic Functions
Michael Burr, Anton Leykin
Arnold Mathematical Journal (2021) 7:3, 431-440
Published: 27 May 2021
Abstract
Download PDF of the paper .Given a system of analytic functions and an approximate zero, we introduce inflation to transform this system into one with a regular quadratic zero. This leads to a method for isolating a cluster of zeros of the given system. -
The Yomdin–Gromov Algebraic Lemma Revisited
Gal Binyamini, Dmitry Novikov
Arnold Mathematical Journal (2021) 7:3, 419-430
Published: 03 May 2021
Abstract
Download PDF of the paper .In 1987, Yomdin proved a lemma on smooth parametrizations of semialgebraic sets as part of his solution of Shub’s entropy conjecture for $C^\infty $ maps. The statement was further refined by Gromov, producing what is now known as the Yomdin–Gromov algebraic lemma. Several complete proofs based on Gromov’s sketch have appeared in the literature, but these have been considerably more complicated than Gromov’s original presentation due to some technical issues. In this note, we give a proof that closely follows Gromov’s original presentation. We prove a somewhat stronger statement, where the parametrizing maps are guaranteed to be cellular. It turns out that this additional restriction, along with some elementary lemmas on differentiable functions in o-minimal structures, allows the induction to be carried out without technical difficulties. -
Interpolation of Weighted Extremal Functions
Alexander Rashkovskii
Arnold Mathematical Journal 2021 7:3, 407-417
Published: 16 March 2021
Abstract
Download PDF of the paper .An approach to interpolation of compact subsets of ${{\mathbb {C}}}^n$, including Brunn–Minkowski type inequalities for the capacities of the interpolating sets, was developed in [8] by means of plurisubharmonic geodesics between relative extremal functions of the given sets. Here we show that a much better control can be achieved by means of the geodesics between weighted relative extremal functions. In particular, we establish convexity properties of the capacities that are stronger than those given by the Brunn–Minkowski inequalities. -
On the Density of Dispersing Billiard Systems with Singular Periodic Orbits
Otto Vaughn Osterman
Arnold Mathematical Journal (2021) 7:3, 387-406
Published: 21 April 2021
Abstract
Download PDF of the paper .Dynamical billiards, or the behavior of a particle traveling in a planar region D undergoing elastic collisions with the boundary has been extensively studied and is used to model many physical phenomena such as a Boltzmann gas. Of particular interest are the dispersing billiards, where D consists of a union of finitely many open convex regions. These billiard flows are known to be ergodic and to possess the K-property. However, Turaev and Rom-Kedar proved that for dispersing systems permitting singular periodic orbits, there exists a family of smooth Hamiltonian flows with regions of stability near such orbits, converging to the billiard flow. They conjecture that systems possessing such singular periodic orbits are dense in the space of all dispersing billiard systems and remark that if this conjecture is true then every dispersing billiard system is arbitrarily close to a non-ergodic smooth Hamiltonian flow with regions of stability [6]. We present a partial solution to this conjecture by showing that any system with a near-singular periodic orbit satisfying certain conditions can be perturbed to a system that permits a singular periodic orbit. We comment on the assumptions of our theorem that must be removed to prove the conjecture of Turaev and Rom-Kedar. -
Near Parabolic Renormalization for Unicritical Holomorphic Maps
Arnaud Chéritat
Arnold Mathematical Journal (2021)
Published: 10 February 2021
Abstract
Download PDF of the paper .Inou and Shishikura provided a class of maps that is invariant by near-parabolic renormalization, and that has proved extremely useful in the study of the dynamics of quadratic polynomials. We provide here another construction, using more general arguments. This will allow to extend the range of applications to unicritical polynomials of all degrees. -
Simplicity of Spectra for Bethe Subalgebras in ${\mathrm {Y}}({\mathfrak {gl}}_2)$
Inna Mashanova-Golikova
Arnold Mathematical Journal (2021) 7:2, 313-339
Published: 07 January 2021
Abstract
Download PDF of the paper .We consider Bethe subalgebras B(C) in the Yangian ${\mathrm {Y}}({\mathfrak {gl}}_2)$ with C regular $2\times 2$ matrix. We study the action of Bethe subalgebras of ${\mathrm {Y}}({\mathfrak {gl}}_2)$ on finite-dimensional representations of ${\mathrm {Y}}({\mathfrak {gl}}_2)$. We prove that B(C) with real diagonal C has simple spectrum on any irreducible ${\mathrm {Y}}({\mathfrak {gl}}_2)$-module corresponding to a disjoint union of real strings. We extend this result to limits of Bethe algebras. Our main tool is the computation of Shapovalov-type determinant for the nilpotent degeneration of B(C). -
Accessible Boundary Points in the Shift Locus of a Family of Meromorphic Functions with Two Finite Asymptotic Values
Tao Chen, Yunping Jiang, Linda Keen
Arnold Mathematical Journal (2021)
Published: 07 January 2021
Abstract
Download PDF of the paper .In this paper, we continue the study, began in Chen et al. (Slices of parameter space for meromorphic maps with two asymptotic values, arXiv:1908.06028, 2019), of the bifurcation locus of a family of meromorphic functions with two asymptotic values, no critical values, and an attracting fixed point. If we fix the multiplier of the fixed point, either of the two asymptotic values determines a one-dimensional parameter slice for this family. We proved that the bifurcation locus divides this parameter slice into three regions, two of them analogous to the Mandelbrot set and one, the shift locus, analogous to the complement of the Mandelbrot set. In Fagella and Keen (Stable components in the parameter plane of meromorphic functions of finite type, arXiv:1702.06563, 2017) and Chen and Keen (Discrete and Continuous Dynamical Systems 39(10):5659–5681, 2019), it was proved that the points in the bifurcation locus corresponding to functions with a parabolic cycle, or those for which some iterate of one of the asymptotic values lands on a pole are accessible boundary points of the hyperbolic components of the Mandelbrot-like sets. Here, we prove these points, as well as the points where some iterate of the asymptotic value lands on a repelling periodic cycle are also accessible from the shift locus. -
Hypergeometric Integrals Modulo p and Hasse–Witt Matrices
Alexey Slinkin, Alexander Varchenko
Arnold Mathematical Journal (2020) 7:2, 267-311
Published: 21 November 2020
Abstract
Download PDF of the paper .We consider the KZ differential equations over ${\mathbb {C}}$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field ${\mathbb {F}}_p$. We study the space of polynomial solutions of these differential equations over ${\mathbb {F}}_p$, constructed in a previous work by Schechtman and the second author. Using Hasse–Witt matrices, we identify the space of these polynomial solutions over ${\mathbb {F}}_p$ with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these polynomial solutions over ${\mathbb {F}}_p$ and the hypergeometric solutions over ${\mathbb {C}}$. -
Herman Rings of Elliptic Functions
Mónica Moreno Rocha
Arnold Mathematical Journal (2020) 6:3, 551-570
Published: 11 November 2020
Abstract
Download PDF of the paper .It has been shown by Hawkins and Koss that over any given lattice, the Weierstrass $\wp $ function does not exhibit cycles of Herman rings. We show that, regardless of the lattice, any elliptic function of order two cannot have cycles of Herman rings. Through quasiconformal surgery, we obtain the existence of elliptic functions of order at least three with an invariant Herman ring. Finally, if an elliptic function has order $o\ge 2$, then we show there can be at most $o-2$ invariant Herman rings. -
Torus Action on Quaternionic Projective Plane and Related Spaces
Anton Ayzenberg
Arnold Mathematical Journal (2020) 7:2, 243-266
Published: 18 November 2020
Abstract
Download PDF of the paper .For an effective action of a compact torus T on a smooth compact manifold X with nonempty finite set of fixed points, the number $\frac{1}{2}\dim X-\dim T$ is called the complexity of the action. In this paper, we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that ${\mathbb {H}}P^2/T^3\cong S^5$ and $S^6/T^2\cong S^4$, for the homogeneous spaces ${\mathbb {H}}P^2={{\,\mathrm{Sp}\,}}(3)/({{\,\mathrm{Sp}\,}}(2)\times {{\,\mathrm{Sp}\,}}(1))$ and $S^6=G_2/{{\,\mathrm{SU}\,}}(3)$. Here, the maximal tori of the corresponding Lie groups ${{\,\mathrm{Sp}\,}}(3)$ and $G_2$ act on the homogeneous spaces from the left. Next we consider the quaternionic analogues of smooth toric surfaces: they give a class of eight-dimensional manifolds with the action of $T^3$. This class generalizes ${\mathbb {H}}P^2$. We prove that their orbit spaces are homeomorphic to $S^5$ as well. We link this result to Kuiper–Massey theorem and its generalizations studied by Arnold. -
A Boothby–Wang Theorem for Besse Contact Manifolds
Marc Kegel, Christian Lange
Arnold Mathematical Journal (2020) 7:2, 225-241
Published: 19 November 2020
Abstract
Download PDF of the paper .A Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly with different periods. We characterize contact manifolds whose Reeb flows are Besse as principal $S^1$-orbibundles over integral symplectic orbifolds satisfying some cohomological condition. Apart from the cohomological condition, this statement appears in the work of Boyer and Galicki in the language of Sasakian geometry (Boyer and Galicki in Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008). We illustrate some non-commonly dealt with perspective on orbifolds in a proof of the above result. More precisely, we work with orbifolds as quotients of manifolds by smooth Lie group actions with finite stabilizer groups. By introducing all relevant orbifold notions in this equivariant way, we avoid patching constructions with orbifold charts. As an application, and building on work by Cristofaro-Gardiner–Mazzucchelli, we deduce a complete classification of closed Besse contact 3-manifolds up to strict contactomorphism. -
A Topological Bound on the Cantor–Bendixson Rank of Meromorphic Differentials
Guillaume Tahar
Arnold Mathematical Journal (2020) 7:2, 213-223
Published: 20 October 2020
Abstract
Download PDF of the paper .In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles (corresponding to meromorphic differentials). The Cantor–Bendixson rank of their set of directions is a measure of descriptive set-theoretic complexity. Drawing on a previous work of David Aulicino, we prove a sharp upper bound that depends only on the genus of the underlying topological surface. The proof uses a new geometric lemma stating that in a sequence of three nested invariant subsurfaces the genus of the third one is always bigger than the genus of the first one. -
On Lagrangian and Legendrian Singularities
Vyacheslav D. Sedykh
Arnold Mathematical Journal (2020) 7:2, 195-212
Published: 20 October 2020
Abstract
Download PDF of the paper .We describe the topology of stable simple multisingularities of Lagrangian and Legendrian maps. In particular, the tables of adjacency indices of monosingularities to multisingularities are given for generic caustics and wave fronts in spaces of small dimensions. The paper is an extended version of the author’s talk in the International Conference “Contemporary mathematics” in honor of the 80th birthday of V. I. Arnold (Moscow, Russia, 2017). -
Probabilistic Schubert Calculus: Asymptotics
Antonio Lerario, Léo Mathis
Arnold Mathematical Journal (2020) 7:2, 169-194
Published: 18 September 2020
Abstract
Download PDF of the paper .In the recent paper Bürgisser and Lerario (Journal für die reine und angewandte Mathematik (Crelles J), 2016) introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by $\delta _{k,n}$ the average number of projective k-planes in ${\mathbb {R}}\mathrm {P}^n$ that intersect $(k+1)(n-k)$ many random, independent and uniformly distributed linear projective subspaces of dimension $n-k-1$. They called $\delta _{k,n}$ the expected degree of the real Grassmannian ${\mathbb {G}}(k,n)$ and, in the case $k=1$, they proved that: $$\begin{aligned} \delta _{1,n}= \frac{8}{3\pi ^{5/2}} \cdot \left( \frac{\pi ^2}{4}\right) ^n \cdot n^{-1/2} \left( 1+{\mathcal {O}}\left( n^{-1}\right) \right) . \end{aligned}$$ Here we generalize this result and prove that for every fixed integer $k>0$ and as $n\rightarrow \infty $, we have $$\begin{aligned} \delta _{k,n}=a_k \cdot \left( b_k\right) ^n\cdot n^{-\frac{k(k+1)}{4}}\left( 1+{\mathcal {O}}(n^{-1})\right) \end{aligned}$$ where $a_k$ and $b_k$ are some (explicit) constants, and $a_k$ involves an interesting integral over the space of polynomials that have all real roots. For instance: $$\begin{aligned} \delta _{2,n}= \frac{9\sqrt{3}}{2048\sqrt{2\pi }} \cdot 8^n \cdot n^{-3/2} \left( 1+{\mathcal {O}}\left( n^{-1}\right) \right) . \end{aligned}$$ Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and give an explicit formula for $\delta _{1,n}$ involving a one-dimensional integral of certain combination of Elliptic functions. -
On the Non-vanishing of the Powers of the Euler Class for Mapping Class Groups
Solomon Jekel, Rita Jiménez Rolland
Arnold Mathematical Journal (2020) 7:1, 159-168
Published: 07 September 2020
Abstract
Download PDF of the paper .The mapping class group of an orientable closed surface with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation-preserving homeomorphisms of the circle. This inclusion pulls back the “discrete universal Euler class” producing a non-zero class in the second integral cohomology of the mapping class group. In this largely expository note, we determine the non-vanishing behavior of the powers of this class. Our argument relies on restricting the cohomology classes to torsion subgroups of the mapping class group. -
On Nörlund–Voronoi Summability and Instability of Rational Maps
Carlos Cabrera, Peter Makienko, Alfredo Poirier
Arnold Mathematical Journal 6:3, 523-549
Published: 08 September 2020
Abstract
Download PDF of the paper .We investigate the connection between the instability of rational maps and summability methods applied to the spectrum of a critical point belonging to the Julia set of a rational map. -
Newton Polyhedra and Good Compactification Theorem
Askold Khovanskii
Arnold Mathematical Journal (2020) 7:1, 135-157
Published: 03 September 2020
Abstract
Download PDF of the paper .A new transparent proof of the well-known good compactification theorem for the complex torus $({\mathbb {C}}^*)^n$ is presented. This theorem provides a powerful tool in enumerative geometry for subvarieties in the complex torus. The paper also contains an algorithm constructing a good compactification for a subvariety in $({\mathbb {C}}^*)^n$ explicitly defined by a system of equations. A new theorem on a toroidal-like compactification is stated. A transparent proof of this generalization of the good compactification theorem which is similar to proofs and constructions from this paper will be presented in a forthcoming publication. -
Quotients of Torus Endomorphisms and Lattès-Type Maps
Mario Bonk, Daniel Meyer
Arnold Mathematical Journal (2020) 6:3, 495-521
Published: 14 September 2020
Abstract
Download PDF of the paper .We show that if an expanding Thurston map is the quotient of a torus endomorphism, then it has a parabolic orbifold and is a Lattès-type map. -
Twisted Forms of Differential Lie Algebras over ${\mathbb {C}}(t)$ Associated with Complex Simple Lie Algebras
Akira Masuoka, Yuta Shimada
Arnold Mathematical Journal (2020) 7:1, 107-134
Published: 18 August 2020
Abstract
Download PDF of the paper .Discussed here is descent theory in the differential context where everything is equipped with a differential operator. To answer a question personally posed by A. Pianzola, we determine all twisted forms of the differential Lie algebras over ${\mathbb {C}}(t)$ associated with complex simple Lie algebras. Hopf–Galois Theory, a ring-theoretic counterpart of theory of torsors for group schemes, plays a role when we grasp the above-mentioned twisted forms from torsors. -
More About Areas and Centers of Poncelet Polygons
Ana C. Chavez-Caliz
Arnold Mathematical Journal (2020) 7:1, 91-106
Published: 14 August 2020
Abstract
Download PDF of the paper .We study the locus of the Circumcenter of Mass of Poncelet polygons, and the limit of the Center of Mass (when we consider the polygon as a “homogeneous lamina”) for degenerate Poncelet polygons. We also provide a proof for one of Dan Reznik invariants for billiard trajectories. Plus, we take a look at how the scene looks like when we shift to spherical geometry. -
The Structure of the Conjugate Locus of a General Point on Ellipsoids and Certain Liouville Manifolds
Jin-ichi Itoh, Kazuyoshi Kiyohara
Arnold Mathematical Journal (2020) 7:1, 31-90
Published: 28 July 2020
Abstract
Download PDF of the paper .It is well known since Jacobi that the geodesic flow of the ellipsoid is “completely integrable”, which means that the geodesic orbits are described in a certain explicit way. However, it does not directly indicate that any global behavior of the geodesics becomes easy to see. In fact, it happened quite recently that a proof for the statement “The conjugate locus of a general point in two-dimensional ellipsoid has just four cusps” in Jacobi’s Vorlesungen über dynamik appeared in the literature. In this paper, we consider Liouville manifolds, a certain class of Riemannian manifolds which contains ellipsoids. We solve the geodesic equations; investigate the behavior of the Jacobi fields, especially the positions of the zeros; and clarify the structure of the conjugate locus of a general point. In particular, we show that the singularities arising in the conjugate loci are only cuspidal edges and $D_4^+$ Lagrangian singularities, which would be the higher dimensional counterpart of Jacobi’s statement. -
Complex Caustics of the Elliptic Billiard
Corentin Fierobe
Arnold Mathematical Journal (2020) 7:1, 1-30
Published: 06 August 2020
Abstract
Download PDF of the paper .The article studies a generalization of the elliptic billiard to the complex domain. We show that the billiard orbits also have caustics, and that the number of such caustics is bigger than for the real case. For example, for a given ellipse E, there exist exactly two confocal ellipses such that the triangular orbits of E are circumscribed about one of them, and each tangent line to one of those ellipses is a side of a triangular orbit. In the case of 4-periodic orbits, we get generically three caustics. We also give an upper bound on the number of caustics for orbits with a fixed number of sides, and explain how to compute its exact value. -
Algebroidally Integrable Bodies
V. A. Vassiliev
Arnold Mathematical Journal (2020) 6:2, 291-309
Published: 07 August 2020
Abstract
Download PDF of the paper .V. Arnold’s problem 1987–14 from his Problems book asks whether there exist bodies with smooth boundaries in ${{\mathbb {R}}}^N$ (other than the ellipsoids in odd-dimensional spaces) for which the volume of the segment cut by any hyperplane from the body depends algebraically on the hyperplane. We present a series of very realistic candidates for the role of such bodies, and prove that the corresponding volume functions are at least algebroid, in particular their analytic continuations are finitely valued; to prove their algebraicity it remains to check the condition of finite growth. -
Centralizers in Mapping Class Groups and Decidability of Thurston Equivalence
Kasra Rafi, Nikita Selinger, Michael Yampolsky
Arnold Mathematical Journal (2020) 6:2, 271-290
Published: 20 July 2020
Abstract
Download PDF of the paper .We find a constructive bound for the word length of a generating set for the centralizer of an element of the Mapping Class Group. As a consequence, we show that it is algorithmically decidable whether two postcritically finite branched coverings of the sphere are Thurston equivalent. -
Double Extensions of Restricted Lie (Super)Algebras
Saïd Benayadi, Sofiane Bouarroudj, Mounir Hajli
Arnold Mathematical Journal (2020) 6:2, 231-269
Published: 27 July 2020
Abstract
Download PDF of the paper .A double extension (${\mathscr {D}}$-extension) of a Lie (super)algebra ${\mathfrak {a}}$ with a non-degenerate invariant symmetric bilinear form ${\mathscr {B}}$, briefly, a NIS-(super)algebra, is an enlargement of ${\mathfrak {a}}$ by means of a central extension and a derivation; the affine Kac–Moody algebras are the best known examples of double extensions of loops algebras. Let ${\mathfrak {a}}$ be a restricted Lie (super)algebra with a NIS ${\mathscr {B}}$. Suppose ${\mathfrak {a}}$ has a restricted derivation ${\mathscr {D}}$ such that ${\mathscr {B}}$ is ${\mathscr {D}}$-invariant. We show that the double extension of ${\mathfrak {a}}$ constructed by means of ${\mathscr {B}}$ and ${\mathscr {D}}$ is restricted. We show that, the other way round, any restricted NIS-(super)algebra with non-trivial center can be obtained as a ${\mathscr {D}}$-extension of another restricted NIS-(super)algebra subject to an extra condition on the central element. We give new examples of ${\mathscr {D}}$-extensions of restricted Lie (super)algebras, and pre-Lie superalgebras indigenous to characteristic 3. -
Fatou’s Associates
Vasiliki Evdoridou, Lasse Rempe, David J. Sixsmith
Arnold Mathematical Journal (2020) 6:3, 459-493
Published: 26 October 2020
Abstract
Download PDF of the paper .Suppose that $f$ is a transcendental entire function, $V \subsetneq {\mathbb {C}}$ $f^{-1}(V)$. Using Riemann maps, we associate the map $f :U \rightarrow V$ to an inner function $g :{\mathbb {D}}\rightarrow {\mathbb {D}}$. It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of $f$ in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of $f$, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of $f$ there are currently very few examples where the relationship between the pair (f, U) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function $f$ with an invariant Fatou component such that g is associated with $f$ in the above sense. Furthermore, there exists a single transcendental entire function $f$ with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of $f$ to a wandering domain. -
Maps That Must Be Affine or Conjugate Affine: A Problem of Vladimir Arnold
Alexey Gorinov, Ralph Howard, Virginia Johnson, George F. McNulty
Arnold Mathematical Journal (2020) 6:2, 213-229
Published: 14 July 2020
Abstract
Download PDF of the paper .A k-flat in a vector space is a k-dimensional affine subspace. Our basic result is that an injection $T:{{\mathbb {C}}}^n\rightarrow {{\mathbb {C}}}^n$ that for some $k\in \{1,2,\ldots ,n-1\}$, T maps all k-flats to flats of ${{\mathbb {C}}}^n$ and is either continuous at a point or Lebesgue measurable, is either an affine map or a conjugate-affine map. An analogous result is proven for injections of the complex projective spaces. In the case of continuity at a point, this is generalized in several directions, the main one being that the complex numbers can be replaced by a finite-dimensional division algebra over an Archimedean ordered field. We also prove injective versions of the Fundamental Theorems of affine and projective geometry and give a counter-example to the surjective version of the latter. This extends work of A. G. Gorinov on a problem of V. I. Arnold. -
Julia Sets of Cubic Rational Maps with Escaping Critical Points
Jun Hu, Arkady Etkin
Arnold Mathematical Journal (2020) 6:3, 431-457
Published: 07 July 2020
Abstract
Download PDF of the paper .It is known that the Julia set of a quadratic rational map is either connected or a Cantor set. In this paper, we explore this dichotomy for the maps in a type of three-dimensional space of cubic rational maps. We show that for a cubic rational map $f$, if $f$ has an attracting fixed point p and all critical points are attracted to p under the iteration of $f$, then the Julia set $J(f)$ is either a Cantor set or a connected set (and locally connected) with one possible exception; we also give a necessary and sufficient condition for $J(f)$ to be a Sierpinski curve when it is connected. -
On a Characterization of Polynomials Among Rational Functions in Non-Archimedean Dynamics
Yûsuke Okuyama, Małgorzata Stawiska
Arnold Mathematical Journal (2020) 6:3, 407-430
Published: 13 August 2020
Abstract
Download PDF of the paper .We study a question on characterizing polynomials among rational functions of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value, from the viewpoint of dynamics and potential theory on the Berkovich projective line. -
Dispersion of the Arnold’s Asymptotic Ergodic Hopf Invariant and a Formula for Its Calculation
P. M. Akhmet’ev, I. V. Vyugin
Arnold Mathematical Journal (2020) 6:2, 199-211
Published: 18 June 2020
Abstract
Download PDF of the paper .The main result of the paper is the formula that calculates the dispersion of the asymptotic Hopf invariant of a magnetic field. The paper contain examples, which describe magnetic fields in a conductive medium. -
A Note on Complex-Hyperbolic Kleinian Groups
Subhadip Dey, Michael Kapovich
Arnold Mathematical Journal (2020) 6:3, 397-406
Published: 08 June 2020
Abstract
Download PDF of the paper .Let $\varGamma $ be a discrete group of isometries acting on the complex hyperbolic n-space $\mathbb {H}^n_\mathbb {C}$. In this note, we prove that if $\varGamma $ is convex-cocompact, torsion-free, and the critical exponent $\delta (\varGamma )$ is strictly lesser than 2, then the complex manifold $\mathbb {H}^n_\mathbb {C}/\varGamma $ is Stein. We also discuss several related conjectures. -
Use of Block Toeplitz Matrix in the Study of Möbius Pairs of Simplexes in Higher-Dimensional Projective Space
Golak Bihari Panda, Saroj Kanta Misra
Arnold Mathematical Journal (2020) 6:2, 189-197
Published: 12 May 2020
Abstract
Download PDF of the paper .A simplex in a projective space of dimension n is expressed by a matrix of order n + 1, where each row represents the homogeneous coordinates of a vertex of the simplex with respect to a reference frame. In the present study, a block Toeplitz matrix is used to express a simplex which forms a Möbius pair along with the reference simplex. A pair of mutually inscribed, circumscribed tetrahedrons is called a Möbius pair. The existence of such pairs of simplexes in higher-dimensional (odd) projective spaces is already established. In the present study an existence of an infinite chain of simplexes in a five-dimensional projective space is established where any two simplexes from the chain form a Möbius pair in some order of their vertices. This is studied with the help of powers of a block Toeplitz matrix. Also, attempt has been made to generalize this result to 2n + 1-dimensional projective space. -
On Rational Functions Sharing the Measure of Maximal Entropy
F. Pakovich
Arnold Mathematical Journal (2020) 6:3, 387-396
Published: 29 April 2020
Abstract
Download PDF of the paper .We show that describing rational functions $f_1,$ $f_2,$ $\dots ,f_n$ sharing the measure of maximal entropy reduces to describing solutions of the functional equation $A\circ X_1=A\circ X_2=\dots =A\circ X_n$ in rational functions. We also provide some results about solutions of this equation. -
The intrinsic hyperplane arrangement in an arbitrary irreducible representation of the symmetric group
N. V. Tsilevich, A. M. Vershik, S. Yuzvinsky
Arnold Mathematical Journal (2020) 6:2, 173-187
Published: 29 April 2020
Abstract
Download PDF of the paper .For every irreducible complex representation $\pi _\lambda $ of the symmetric group ${{\mathfrak {S}}}_n$, we construct, in a canonical way, a so-called intrinsic hyperplane arrangement ${{\mathcal {A}}}_{\lambda }$ in the space of $\pi _\lambda $. This arrangement is a direct generalization of the classical braid arrangement (which is the special case of our construction corresponding to the natural representation of ${{\mathfrak {S}}}_n$ ), has a natural description in terms of invariant subspaces of Young subgroups, and enjoys a number of remarkable properties. -
Two Parameters bt-Algebra and Invariants for Links and Tied Links
F. Aicardi, J. Juyumaya
Arnold Mathematical Journal (2020) 6:1, 131-148
Published: 02 April 2020
Abstract
Download PDF of the paper .We introduce a two-parameters bt-algebra which, by specialization, becomes the one-parameter bt-algebra, introduced by the authors, as well as another one-parameter presentation of it; the invariant for links and tied links, associated to this two-parameter algebra via Jones recipe, contains as specializations the invariants obtained from these two presentations of the bt-algebra and then is more powerful than each of them. Also, a new non Homflypt polynomial invariant is obtained for links, which is related to the linking matrix. -
Uniqueness of a Three-Dimensional Ellipsoid with Given Intrinsic Volumes
Fedor Petrov, Alexander Tarasov
Arnold Mathematical Journal (2020) 6:2, 163-171
Published: 07 April 2020
Abstract
Download PDF of the paper .Let ${\mathcal {E}}$ be an ellipsoid in ${\mathbb {R}}^n$. Gusakova and Zaporozhets conjectured that ${\mathcal {E}}$ is uniquely (up to rigid motions) determined by its intrinsic volumes. We prove this conjecture for $n = 3$. -
Ramanujan’s Theorem and Highest Abundant Numbers
Oleg R. Musin
Arnold Mathematical Journal (2020) 6:1, 119-130
Published: 14 April 2020
Abstract
Download PDF of the paper .In 1915, Ramanujan proved asymptotic inequalities for the sum of divisors function, assuming the Riemann hypothesis (RH). We consider a strong version of Ramanujan’s theorem and define highest abundant numbers that are extreme with respect to the Ramanujan and Robin inequalities. Properties of these numbers are very different depending on whether the RH is true or false. -
The Roots of Exceptional Modular Lie Superalgebras with Cartan Matrix
Sofiane Bouarroudj, Dimitry Leites, Olexander Lozhechnyk, Jin Shang
Arnold Mathematical Journal (2020) 6:1, 63-118
Published: 20 March 2020
Abstract
Download PDF of the paper .For each of the exceptional (not entering infinite series) finite-dimensional modular Lie superalgebras with indecomposable Cartan matrix, we give the explicit list of its roots, and the corresponding Chevalley basis, for one of its inequivalent Cartan matrices, namely the one corresponding to the greatest number of mutually orthogonal isotropic odd simple roots (this number, called the defect of the Lie superalgebra, is important in the representation theory). Our main tools: Grozman’s Mathematica-based code SuperLie, Python, and A. Lebedev’s help. -
Core Entropy of Quadratic Polynomials
Dzmitry Dudko, Dierk Schleicher
Arnold Mathematical Journal (2020) 6:3, 333-385
Published: 25 March 2020
Abstract
Download PDF of the paper .We give a combinatorial definition of “core entropy” for quadratic polynomials as the growth exponent of the number of certain precritical points in the Julia set (those that separate the $\alpha $ fixed point from its negative). This notion extends known definitions that work in cases when the polynomial is postcritically finite or when the topology of the Julia set has good properties, and it applies to all quadratic polynomials in the Mandelbrot set. We prove that core entropy is continuous as a function of the complex parameter. In fact, we model the Julia set as an invariant quadratic lamination in the sense of Thurston: this depends on the external angle of a parameter in the boundary of the Mandelbrot set, and one can define core entropy directly from the angle in combinatorial terms. As such, core entropy is continuous as a function of the external angle. Moreover, we prove a conjecture of Giulio Tiozzo about local and global maxima of core entropy as a function of external angles: local maxima are exactly dyadic angles, and the unique global maximum within any wake occurs at the dyadic angle of lowest denominator. We also describe where local minima occur. An appendix by Wolf Jung relates different concepts of core entropy and biaccessibility dimension and thus shows that biaccessibility dimension is continuous as well. -
Renormalization in the Golden-Mean Semi-Siegel Hénon Family: Non-Quasisymmetry
Jonguk Yang
Arnold Mathematical Journal (2020) 6:3, 313-331
Published: 03 February 2020
Abstract
Download PDF of the paper .For quadratic polynomials of one complex variable, the boundary of the golden-mean Siegel disk must be a quasicircle. We show that the analogous statement is not true for quadratic Hénon maps of two complex variables. -
On Convex Bodies that are Characterizable by Volume Function
Ákos G. Horváth
Arnold Mathematical Journal (2020) 6:1, 1-28
Published: 23 January 2020
Abstract
Download PDF of the paper .The “old-new” concept of a convex-hull function was investigated by several authors in the last seventy years. A recent research on it led to some other volume functions as the covariogram function, the widthness function or the so-called brightness functions, respectively. A very interesting fact that there are many long-standing open problems connected with these functions whose serious investigation was closed before the “age of computers”. In this survey, we concentrate only on the three-dimensional case; we will mention the most important concepts, statements, and problems. -
Protecting Billiard Balls From Collisions
Jayadev Athreya, Krzysztof Burdzy
Arnold Mathematical Journal (2020) 6:1, 57-62
Published: 10 February 2020
Abstract
Download PDF of the paper .We present a game inspired by research on the possible number of billiard ball collisions in the whole Euclidean space. One player tries to place n static “balls” with zero radius (i.e., points) in a way that will minimize the total number of possible collisions caused by the cue ball. The other player tries to find initial conditions for the cue ball to maximize the number of collisions. The value of the game is $\sqrt{n}$ (up to constants). The lower bound is based on the Erdős-Szekeres Theorem. The upper bound may be considered a generalization of the Erdős-Szekeres Theorem. -
Superforms, supercurrents, minimal manifolds and Riemannian geometry
Bo Berndtsson
Arnold Mathematical Journal (2020) 5:4, 501-532
Published: 14 January 2020
Abstract
Download PDF of the paper .Supercurrents, as introduced by Lagerberg, were mainly motivated as a way to study tropical varieties. Here we will associate a supercurrent to any smooth submanifold of ${\mathbb {R}}^n$. Positive supercurrents resemble positive currents in complex analysis, but depend on a choice of scalar product on ${\mathbb {R}}^n$ and reflect the induced Riemannian structure on the submanifold. In this way we can use techniques from complex analysis to study real submanifolds. We illustrate the idea by giving area estimates of minimal manifolds and a monotonicity property of the mean curvature flow. We also use the formalism to give a relatively short proof of Weyl’s tube formula. -
Viterbo’s Conjecture for Certain Hamiltonians in Classical Mechanics
Roman Karasev, Anastasia Sharipova
Arnold Mathematical Journal (2019) 5:4, 483-500
Published: 11 December 2019
Abstract
Download PDF of the paper .We study some particular cases of Viterbo’s conjecture relating volumes of convex bodies and actions of closed characteristics on their boundaries, focusing on the case of a Hamiltonian of classical mechanical type, splitting into summands depending on the coordinates and the momentum separately. We manage to establish the conjecture for sublevel sets of convex 2-homogeneous Hamiltonians of this kind in several particular cases. We also discuss open cases of this conjecture. -
Invariants of Graph Drawings in the Plane
A. Skopenkov
Arnold Mathematical Journal (2020) 6:1, 21-55
Published: 13 February 2020
Abstract
Download PDF of the paper .We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of geometry, combinatorics and topology. We define a ${\mathbb {Z}}_2$ valued self-intersection invariant (i.e. the van Kampen number) and its generalizations. We present elementary formulations and arguments accessible to mathematicians not specialized in any of the areas discussed. So most part of this survey could be studied before textbooks on algebraic topology, as an introduction to starting ideas of algebraic topology motivated by algorithmic, combinatorial and geometric problems. -
Triangulated Endofunctors of the Derived Category of Coherent Sheaves Which do not Admit DG Liftings
Vadim Vologodsky
Arnold Mathematical Journal (2019) 5:4, 387-391
Published: 04 November 2019
Abstract
Download PDF of the paper .In Rizzardo and Van den Bergh (An example of a non-Fourier–Mukai functor between derived categories of coherent sheaves, 2014), constructed an example of a triangulated functor between the derived categories of coherent sheaves on smooth projective varieties over a field k of characteristic 0 which is not of the Fourier–Mukai type. The purpose of this note is to show that if ${{\,\mathrm{{char}}\,}}k =p$ then there are very simple examples of such functors. Namely, for a smooth projective Y over ${{\mathbb {Z}}}_p$ with the special fiber $i: X\hookrightarrow Y$, we consider the functor $L i^* \circ i_*: D^b(X) \rightarrow D^b(X)$ from the derived categories of coherent sheaves on X to itself. We show that if Y is a flag variety which is not isomorphic to ${{\mathbb {P}}}^1$ then $L i^* \circ i_*$ is not of the Fourier–Mukai type. Note that by a theorem of Toën (Invent Math 167:615–667, 2007: Theorem 8.15) the latter assertion is equivalent to saying that $L i^* \circ i_*$ does not admit a lifting to a ${{\mathbb {F}}}_p$-linear DG quasi-functor $D^b_{dg}(X) \rightarrow D^b_{dg}(X)$, where $D^b_{dg}(X)$ is a (unique) DG enhancement of $D^b(X)$. However, essentially by definition, $L i^* \circ i_*$ lifts to a ${{\mathbb {Z}}}_p$-linear DG quasi-functor. -
0-Cycles on Grassmannians as Representations of Projective Groups
R. Bezrukavnikov, M. Rovinsky
Arnold Mathematical Journal (2019) 5:2, 373-385
Published: 05 November 2019
Abstract
Download PDF of the paper .Let F be an infinite division ring, V be a left F-vector space, $r\ge 1$ be an integer. We study the structure of the representation of the linear group $\mathrm {GL}_F(V)$ in the vector space of formal finite linear combinations of r-dimensional vector subspaces of V with coefficients in a field. This gives a series of natural examples of irreducible infinite-dimensional representations of projective groups. These representations are non-smooth if F is locally compact and non-discrete. -
Newton–Okounkov Polytopes of Flag Varieties for Classical Groups
Valentina Kiritchenko
Arnold Mathematical Journal (2019) 5:2, 355-371
Published: 17 October 2019
Abstract
Download PDF of the paper .For classical groups $SL_n(\mathbb {C})$, $SO_n(\mathbb {C})$ and $Sp_{2n}(\mathbb {C})$, we define uniformly geometric valuations on the corresponding complete flag varieties. The valuation in every type comes from a natural coordinate system on the open Schubert cell, and is combinatorially related to the Gelfand–Zetlin pattern in the same type. In types A and C, we identify the corresponding Newton–Okounkov polytopes with the Feigin–Fourier–Littelmann–Vinberg polytopes. In types B and D, we compute low-dimensional examples and formulate open questions. -
On Mutually Semiconjugate Rational Functions
F. Pakovich
Arnold Mathematical Journal (2019) 5:2, 339-354
Published: 09 October 2019
Abstract
Download PDF of the paper .We characterize pairs of rational functions A, B such that A is semiconjugate to B, and B is semiconjugate to A. -
Invariant Spanning Trees for Quadratic Rational Maps
Anastasia Shepelevtseva, Vladlen Timorin
Arnold Mathematical Journal (2019) 5:4, 435-481
Published: 17 October 2019
Abstract
Download PDF of the paper .We study Thurston equivalence classes of quadratic post-critically finite branched coverings. For these maps, we introduce and study invariant spanning trees. We give a computational procedure for searching for invariant spanning trees. This procedure uses bisets over the fundamental group of a punctured sphere. We also introduce a new combinatorial invariant of Thurston classes—the ivy graph. -
Solutions with Compact Time Spectrum to Nonlinear Klein–Gordon and Schrödinger Equations and the Titchmarsh Theorem for Partial Convolution
Andrew Comech
Arnold Mathematical Journal (2019) 5:2, 315-338
Published: 27 September 2019
Abstract
Download PDF of the paper .We prove that finite energy solutions to the nonlinear Schrödinger equation and nonlinear Klein–Gordon equation which have the compact time spectrum have to be one-frequency solitary waves. The argument is based on the generalization of the Titchmarsh convolution theorem to partial convolutions. -
Faithful tropicalizations of elliptic curves using minimal models and inflection points
Paul Alexander Helminck
Arnold Mathematical Journal (2019) 5:4, 401-434
Published: 27 September 2019
Abstract
Download PDF of the paper .We give an elementary proof of the fact that any elliptic curve E over an algebraically closed non-archimedean field K with residue characteristic $\ne {2,3}$ and with $v(j(E))<0$ admits a tropicalization that contains a cycle of length $-v(j(E))$. We first define an adapted form of minimal models over non-discrete valuation rings and we recover several well-known theorems from the discrete case. Using these, we create an explicit family of marked elliptic curves (E, P), where E has multiplicative reduction and P is an inflection point that reduces to the singular point on the reduction of E. We then follow the strategy as in Baker et al. (Algebraic Geom 3(1):63–105, 2016) and construct an embedding such that its tropicalization contains a cycle of length $-v(j(E))$. We call this a numerically faithful tropicalization. A key difference between this approach and the approach in Baker et al. (2016) is that we do not require any of the analytic theory on Berkovich spaces such as the Poincaré–Lelong formula or (Baker et al. 2016) to establish the numerical faithfulness of this tropicalization. -
On Osculating Framing of Real Algebraic Links
Grigory Mikhalkin, Stepan Orevkov
Arnold Mathematical Journal (2019) 5:4, 393-399
Published: 26 August 2019
Abstract
Download PDF of the paper .For a real algebraic link in ${{\mathbb {RP}}}^3$, we prove that its encomplexed writhe (an invariant introduced by Viro) is maximal for a given degree and genus if and only if its self-linking number with respect to the framing by the osculating planes is maximal for a given degree. -
Gelfand–Tsetlin Degeneration of Shift of Argument Subalgebras in Types B, C and D
Leonid Rybnikov, Mikhail Zavalin
Arnold Mathematical Journal (2019) 5:2, 285-313
Published: 26 August 2019
Abstract
Download PDF of the paper .The universal enveloping algebra of any semisimple Lie algebra $\mathfrak {g}$ contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of $\mathfrak {g}$. For $\mathfrak {g}=\mathfrak {gl}_n$ the Gelfand–Tsetlin commutative subalgebra in $U(\mathfrak {g})$ arises as some limit of subalgebras from this family. We study the analogous limit of shift of argument subalgebras for classical Lie algebras ($\mathfrak {g}=\mathfrak {sp}_{2n}$ or $\mathfrak {so}_{n}$). The limit subalgebra is described explicitly in terms of Bethe subalgebras in twisted Yangians $Y^-(2)$ and $Y^+(2)$, respectively. We index the eigenbasis of such limit subalgebra in any irreducible finite-dimensional representation of $\mathfrak {g}$ by Gelfand–Tsetlin patterns of the corresponding type, and conjecture that this indexing is, in appropriate sense, natural. According to Halacheva et al. (Crystals and monodromy of Bethe vectors. arXiv:1708.05105, 2017) such eigenbasis has a natural $\mathfrak {g}$-crystal structure. We conjecture that this crystal structure coincides with that on Gelfand–Tsetlin patterns defined by Littelmann in Cones, crystals, and patterns (Transform Groups 3(2):145–179, 1998). -
Shifted Quantum Affine Algebras: Integral Forms in Type A
Michael Finkelberg, Alexander Tsymbaliuk
Arnold Mathematical Journal (2019) 5:2, 197-283
Published: 01 August 2019
Abstract
Download PDF of the paper .We define an integral form of shifted quantum affine algebras of type A and construct Poincaré–Birkhoff–Witt–Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these integral forms are closed with respect to the coproduct and shift homomorphisms. We prove that the homomorphism from our integral form to the corresponding quantized K-theoretic Coulomb branch of a quiver gauge theory is always surjective. In one particular case we identify this Coulomb branch with the extended quantum universal enveloping algebra of type A. Finally, we obtain the rational (homological) analogues of the above results [proved earlier in Kamnitzer et al. (Proc Am Math Soc 146(2):861–874, 2018a; On category $\mathcal {O}$ for affine Grassmannian slices and categorified tensor products. arXiv:1806.07519, 2018b) via different techniques]. -
Simplicial Equations for the Moduli Space of Stable Rational Curves
Joaquin Maya, Jacob Mostovoy
Arnold Mathematical Journal (2019) 5:2, 187-196
Published: 19 June 2019
Abstract
Download PDF of the paper .In this, largely expository, note, we show how the simplicial structure of the moduli spaces of stable rational curves with marked points allows to produce explicit equations for these spaces. The key argument is an elementary combinatorial statement about the sets of trees with marked leaves. -
Open WDVV Equations and Virasoro Constraints
Alexey Basalaev, Alexandr Buryak
Arnold Mathematical Journal (2019) 5:2, 145-186
Published: 24 June 2019
Abstract
Download PDF of the paper .In their fundamental work, Dubrovin and Zhang, generalizing the Virasoro equations for the genus 0 Gromov–Witten invariants, proved the Virasoro equations for a descendent potential in genus 0 of an arbitrary conformal Frobenius manifold. More recently, a remarkable system of partial differential equations, called the open WDVV equations, appeared in the work of Horev and Solomon. This system controls the genus 0 open Gromov–Witten invariants. In our paper, for an arbitrary solution of the open WDVV equations, satisfying a certain homogeneity condition, we construct a descendent potential in genus 0 and prove an open analog of the Virasoro equations. We also present conjectural open Virasoro equations in all genera and discuss some examples. -
Triangulated Endofunctors of the Derived Category
of Coherent Sheaves Which Do Not Admit DG Liftings
Vadim Vologodsky
Arnold Math J. (2019) 5:1, 139–143
Received: 29 December 2018 / Revised: 29 March 2019 / Accepted: 1 June 2019 / Published online: 12 June 2019
Abstract
Download PDF of the paper (199KB) .In, Rizzardo and Van den Bergh (An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves. arXiv:1410.4039, 2014) constructed an example of a triangulated functor between the derived categories of coherent sheaves on smooth projective varieties over a field $k$ of characteristic 0 which is not of the Fourier-Mukai type. The purpose of this note is to show that if char $k = p$ then there are very simple examples of such functors. Namely, for a smooth projective $Y$ over $\mathbb Z_p$ with the special fiber $i : X\hookrightarrow Y$, we consider the functor $Li^*\circ i_* : D^b(X)\to D^b(X)$ from the derived categories of coherent sheaves on $X$ to itself. We show that if $Y$ is a flag variety which is not isomorphic to $\mathbb P^1$ then $Li^*\circ i_*$ is not of the Fourier-Mukai type. Note that by a theorem of Toen (Invent Math 167:615-667, 2007, Theorem 8.15) the latter assertion is equivalent to saying that $Li^*\circ i_*$ does not admit a lifting to a $\mathbb F_p$-linear DG quasi-functor $D^b_{dg}(X)\to D^b_{dg}(X)$, where $D^b_ {dg}(X)$ is a (unique) DG enhancement of $D^b(X)$. However, essentially by definition, $Li^*\circ i_*$ lifts to a $\mathbb Z_p$-linear DG quasi-functor. -
Laughlin states and gauge theory
Nikita Nekrasov
Arnold Math J. (2019) 5:1, 123–138
Received: 19 November 2018 / Revised: 11 April 2019 / Accepted: 28 May 2019 / Published online: 6 June 2019
Abstract
Download PDF of the paper (378KB) .Genus one Laughlin wavefunctions, describing the gas of interacting electrons on a two dimensional torus in the presence of a strong magnetic field, analytically continued in the filling fraction, are related to the partition functions of half-BPS surface defects in four dimensional $\mathcal N = 2$ supersymmetric gauge theory. -
Solutions of Polynomial Equations in Subgroups of $\mathbb F^*_p$
Sergei Makarychev, Ilya Vyugin
Arnold Math J. (2019) 5:1, 105–121
Received: 4 December 2018 / Revised: 16 April 2019 / Accepted: 20 May 2019 / Published online: 5 June 2019
Abstract
Download PDF of the paper (342KB) .We present an upper bound on the number of solutions of an algebraic equation $P(x,y) = 0$ where $x$ and $y$ belong to the union of cosets of some subgroup of the multiplicative group $\kappa^*$ of some field of positive characteristic. This bound generalizes the bound of Corvaja and Zannier (J Eur Math Soc 15(5):1927-1942, 2013) to the case of union of cosets. We also obtain the upper bounds on the generalization of additive energy. -
EA-Matrix Integrals of Associative Algebras and Equivariant
Localization
Serguei Barannikov
Arnold Math J. (2019) 5:1, 97–104
Received: 1 November 2018 / Revised: 21 April 2019 / Accepted: 8 May 2019 / Published online: 27 May 2019
Abstract
Download PDF of the paper (262KB) .The theory of periods of noncommutative varieties, depending on commutative parameters, was introduced in Barannikov (2000). The analogue of top-degree holomorphic form in this setting was shown in loc.cit. to be certain element of semi-infinite subspace of negative cyclic homology. The integrals of this element satisfy the second order equation with respect to the parameters of deformations of the varieties. It was proven in loc.cit. that the generating function of genus zero Gromov-Witten invariants of complete intersection in $\mathbb{CP}^d$ with trivial canonical class coincides with the coefficient of this second order equation for the family of mirror varieties. This approach had singled out the $A_\infty$-algebras/categories, satisfying cyclic homology analogue of degeneration of Hodge to de Rham spectral sequence, as the proper definition of (smooth and compact) noncommutative varieties. -
The Index of a Local Boundary Value Problem for Strongly
Callias-Type Operators
Maxim Braverman, Pengshuai Shi
Arnold Math J. (2019) 5:1, 79–96
Received: 21 October 2018 / Revised: 21 March 2019 / Accepted: 27 April 2019 / Published online: 14 May 2019
Abstract
Download PDF of the paper (357KB) .We consider a complete Riemannian manifold $M$ whose boundary is a disjoint union of finitely many complete connected Riemannian manifolds. We compute the index of a local boundary value problem for a strongly Callias-type operator on $M$. Our result extends an index theorem of D. Freed to non-compact manifolds, thus providing a new insight on the Hořava-Witten anomaly. -
On Generic Semi-simple Decomposition of Dimension
Vector for an Arbitrary Quiver
D. A. Shmelkin
Arnold Math J. (2019) 5:1, 69–78
Received: 20 October 2018 / Revised: 15 April 2019 / Accepted: 23 April 2019 / Published online: 14 May 2019
Abstract
Download PDF of the paper (287KB) .Generic (canonical) decomposition of dimension vector for a quiver was introduced by Victor Kac as characterizing the generic module indecomposable summands dimensions, hence, the generic orbit. Derksen and Weyman proposed an elegant algorithm to compute that decomposition, extensively using Schofield's results. We consider generic semi-simple decomposition, which corresponds to generic closed orbit and provide a simple and fast algorithm to compute this decomposition. Generic semi-simple decomposition has two useful application. First, it reduces the computation of generic decomposition to the case of quiver without oriented cycles in a geometric way. Second, it provides a nice novel presentation of the algebra of invariants of quiver representations as a tensor product of similar algebras for the summands. -
Tropical Limits of Decimated Polynomials
Elizaveta Arzhakova, Evgeny Verbitskiy
Arnold Math J. (2019) 5:1, 57–67
Received: 8 November 2018 / Revised: 26 March 2019 / Accepted: 28 March 2019 / Published online: 30 April 2019
Abstract
Download PDF of the paper (462KB) .Motivated by some problems that originate in Statistical Physics and Algebraic Dynamics, we discuss a particular renormalization mechanism of multivariate Laurent polynomials which is called a decimation, and the corresponding tropical limiting shape result obtained in Arzhakova et al. (Decimation of principal actions. Preprint, 2018). -
Geometry of Mutation Classes of Rank 3 Quivers
Anna Felikson, Pavel Tumarkin
Arnold Math J. (2019) 5:1, 37–55
Received: 18 September 2018 / Revised: 16 February 2019 / Accepted: 21 February 2019 / Published online: 4 March 2019
Abstract
Download PDF of the paper (500KB) .We present a geometric realization for all mutation classes of quivers of rank 3 with real weights. This realization is via linear reflection groups for acyclic mutation classes and via groups generated by $\pi$-rotations for the cyclic ones. The geometric behavior of the model turns out to be controlled by the Markov constant $p^2 + q^2 + r^2 -pqr$, where $p$, $q$, $r$ are the weights of arrows in a quiver. We also classify skew-symmetric mutation-finite real $3\times 3$ matrices and explore the structure of acyclic representatives in finite and infinite mutation classes. -
Fundamental Matrix Factorization in the FJRW-Theory
Revisited
Alexander Polishchuk
Arnold Math J. (2019) 5:1, 23–35
Received: 2 October 2018 / Revised: 1 February 2019 / Accepted: 13 February 2019 / Published online: 25 February 2019
Abstract
Download PDF of the paper (300KB) .We present an improved construction of the fundamental matrix factorization in the FJRW-theory given in Polishchuk and Vaintrob (J Reine Angew Math 714:1-22, 2016). The revised construction makes the independence on choices more apparent and works for a possibly nonabelian finite group of symmetries. One of the new ingredients is the category of dg-matrix factorizations over a dg-scheme. -
Localization Properties of Chern Insulators
Roman Bezrukavnikov, Anton Kapustin
Arnold Math J. (2019) 5:1, 15–21
Received: 20 November 2018 / Accepted: 21 January 2019 / Published online: 8 March 2019
Abstract
Download PDF of the paper (200KB) .We study the localization properties of the equal-time electron Green's function in a Chern insulator in an arbitrary dimension and with an arbitrary number of bands. We prove that the Green's function cannot decay super-exponentially if the Hamiltonian is finite-range and the quantum Hall response is nonzero. For a general band Hamiltonian (possibly infinite-range), we prove that the Green's function cannot be finite-range if the quantum Hall response is nonzero. The proofs use methods of algebraic geometry. -
Approximate Identities and Lagrangian Poincaré
Recurrence
Viktor L. Ginzburg, Başak Z. Gürel
Arnold Math J. (2019) 5:1, 5–14
Received: 8 November 2018 / Revised: 19 December 2018 / Accepted: 18 January 2019 / Published online: 4 March 2019
Abstract
Download PDF of the paper (300KB) .In this note we discuss three interconnected problems about dynamics of Hamiltonian or, more generally, just smooth diffeomorphisms. The first two concern the existence and properties of the maps whose iterations approximate the identity map with respect to some norm, e.g., $C^1$- or $C^0$-norm for general diffeomorphisms and the $\gamma$-norm in the Hamiltonian case, and the third problem is the Lagrangian Poincaré recurrence conjecture. -
Effective Birational Rigidity of Fano Double Hypersurfaces
Thomas Eckl, Aleksandr Pukhlikov
Arnold Math J. (2018) 4:3-4, 505–521
Received: 31 December 2018 / Revised: 20 February 2019 / Accepted: 11 March 2019 / Published online: 19 March 2019
Abstract
Download PDF of the paper (307KB) .We prove birational superrigidity of Fano double hypersurfaces of index one with quadratic and multi-quadratic singularities, satisfying certain regularity conditions, and give an effective explicit lower bound for the codimension of the set of non-rigid varieties in the natural parameter space of the family. The lower bound is quadratic in the dimension of the variety. The proof is based on the techniques of hypertangent divisors combined with the recently discovered $4n^2$-inequality for complete intersection singularities. -
On Algorithms that Effectively Distinguish Gradient-Like
Dynamics on Surfaces
Vladislav E. Kruglov, Dmitry S. Malyshev, Olga V. Pochinka
Arnold Math J. (2018) 4:3-4, 483–504
Received: 22 November 2018 / Revised: 4 February 2019 / Accepted: 6 March 2019 / Published online: 15 March 2019
Abstract
Download PDF of the paper (887KB) .In the present paper we survey existing graph invariants for gradient-like flows on surfaces up to the topological equivalence and develop effective algorithms for their distinction (let us recall that a flow given on a surface is called a gradient-like flow if its non-wandering set consists of a finite set of hyperbolic fixed points, and there is no trajectories connecting saddle points). Additionally, we construct a parametrized algorithm for the Fleitas's invariant, which will be of linear time, when the number of sources is fixed. Finally, we prove that the classes of topological equivalence and topological conjugacy are coincide for gradient-like flows, so, all the proposed invariants and distinguishing algorithms works also for topological classification, taking in sense time of moving along trajectories. So, as the main result of this paper we have got multiple ways to recognize equivalence and conjugacy class of arbitrary gradient-like flow on a closed surface in a polynomial time. -
Chamber Structure of Modular Curves $X_1(N)$
Guillaume Tahar
Arnold Math J. (2018) 4:3-4, 459–481
Received: 7 October 2018 / Revised: 4 February 2019 / Accepted: 7 February 2019 / Published online: 7 March 2019
Abstract
Download PDF of the paper (748KB) .Modular curves $X_1(N)$ parametrize elliptic curves with a point of order $N$. They can be identified with connected components of projectivized strata $\mathbb P\mathcal H(a,-a)$ of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the topological changes in the flat structure defined by the meromorphic differentials. We provide formulas for the number of chambers and an effective means for drawing the incidence graph of the chamber structure of any modular curve $X_1(N)$. This defines a family of graphs with specific combinatorial properties. This approach provides a geometrico-combinatorial computation of the genus and the number of punctures of modular curves $X_1(N)$. Although the dimension of a stratum of meromorphic differentials depends only on the genus and the numbers of the singularities, the topological complexity of the stratum crucially depends on the order of the singularities. -
On Centro-Affine Curves and Bäcklund Transformations
of the KdV Equation
Serge Tabachnikov
Arnold Math J. (2018) 4:3-4, 445–458
Received: 20 September 2018 / Revised: 1 February 2019 / Accepted: 11 March 2019 / Published online: 1 April 2019
Abstract
Download PDF of the paper (301KB) .We continue the study of the Korteweg-de Vries equation in terms of cento-affine curves, initiated by Pinkall. A centro-affine curve is a closed parametric curve in the affine plane such that the determinant made by the position and the velocity vectors is identically one. The space of centro-affine curves is acted upon by the special linear group, and the quotient is identified with the space of Hill's equations with periodic solutions. It is known that the space of centro-affine curves carries two pre-symplectic structures, and the KdV flow is identified with is a bi-Hamiltonian dynamical system therein. We introduce a one-parameter family of transformations on centro-affine curves, prove that they preserve both presymplectic structures, commute with the KdV flow, and share the integrals with it. Furthermore, the transformation commute with each other (Bianchi permutability). We also describe integrals of the KdV equation as arising from the monodromy of Riccati equations associated with centro-affine curves. We are motivated by our work (joint with M. Arnold, D. Fuchs, and I. Izmenstiev), concerning the cross-ratio dynamics on ideal polygons in the hyperbolic plane and hyperbolic space, whose continuous version is studied in the present paper. -
On the Geometry of the Set of Symmetric Matrices with
Repeated Eigenvalues
Paul Breiding, Khazhgali Kozhasov, Antonio Lerario
Arnold Math J. (2018) 4:3-4, 423–443
Received: 20 August 2018 / Revised: 15 November 2018 / Accepted: 11 December 2018 / Published online: 2 January 2019
Abstract
Download PDF of the paper (394KB) .We investigate some geometric properties of the real algebraic variety $\Delta$ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart-Young-Mirsky-type theorem for the distance function from a generic matrix to points in $\Delta$. We exhibit connections of our study to real algebraic geometry (computing the Euclidean distance degree of $\Delta$) and random matrix theory. -
Integrable Hamiltonian Systems with a Periodic Orbit or
Invariant Torus Unique in the Whole Phase Space
Mikhail B. Sevryuk
Arnold Math J. (2018) 4:3-4, 415–422
Received: 9 August 2018 / Accepted: 7 November 2018 / Published online: 19 November 2018
Abstract
Download PDF of the paper (370KB) .It is very well known that periodic orbits of autonomous Hamiltonian systems are generically organized into smooth one-parameter families (the parameter being just the energy value). We present a simple example of an integrable Hamiltonian system (with an arbitrary number of degrees of freedom greater than one) with a unique periodic orbit in the phase space (which is not compact). Similar examples are given for Hamiltonian systems with a unique invariant torus (of any prescribed dimension) carrying conditionally periodic motions. Parallel examples for Hamiltonian systems with a compact phase space and with uniqueness replaced by isolatedness are also constructed. Finally, reversible analogues of all the examples are described. -
Deformation Classification of Real Non-singular Cubic
Threefolds with a Marked Line
S. Finashin, V. Kharlamov
Arnold Math J. (2018) 4:3-4, 345–414
Received: 7 August 2018 / Revised: 16 November 2018 / Accepted: 5 January 2019 / Published online: 4 March 2019
Abstract
Download PDF of the paper (1131KB) .We prove that the space of pairs $(X,l)$ formed by a real non-singular cubic hypersurface $X\subset P^4$ with a real line $l\subset X$ has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surface $F_{\mathbb R}(X)$ formed by real lines on $X$. For another interpretation we associate with each of the 18 components a well defined real deformation class of real non-singular plane quintic curves and show that this deformation class together with the real deformation class of $X$ characterizes completely the component. -
On the Reducibility and the Lenticular Sets of Zeroes of
Almost Newman Lacunary Polynomials
Denys Dutykh, Jean-Louis Verger-Gaugry
Arnold Math J. (2018) 4:3-4, 315–344
Received: 15 June 2018 / Revised: 14 January 2019 / Accepted: 22 February 2019 / Published online: 4 March 2019
Abstract
Download PDF of the paper (1150KB) .The class $\mathcal B$ of lacunary polynomials $f(x) := -1 + x + x^n +x^{m_1} + x^{m_2} + \dots + x^{m_s}$, where $s\ge0$, $m_1- n\ge n- 1$, $m_{q+1}-m_q\ge n- 1$ for $1\le q < s$, $n\ge 3$ is studied. A polynomial having its coefficients in $\{0, 1 \}$ except its constant coefficient equal to $-1$ is called an almost Newman polynomial. A general theorem of factorization of the almost Newman polynomials of the class $\mathcal B$ is obtained. Such polynomials possess lenticular roots in the open unit disk off the unit circle in the small angular sector $-\pi/18\le \arg z\le\pi/18$ and their nonreciprocal parts are always irreducible. The existence of lenticuli of roots is a peculiarity of the class $\mathcal B$. By comparison with the Odlyzko-Poonen Conjecture and its variant Conjecture, an Asymptotic Reducibility Conjecture is formulated aiming at establishing the proportion of irreducible polynomials in this class. This proportion is conjectured to be 3/4 and estimated using Monte-Carlo methods. The numerical approximate value $\approx0.756$ is obtained. The results extend those on trinomials (Selmer) and quadrinomials (Ljunggren, Mills, Finch and Jones). -
Modular Cauchy Kernel Corresponding to the Hecke Curve
Nina Sakharova
Arnold Math J. (2018) 4:3-4, 301–313
Received: 8 June 2018 / Revised: 30 August 2018 / Accepted: 22 November 2018 / Published online: 11 February 2019
Download PDF of the paper (370KB) . -
Formal Factorization of Higher Order Irregular Linear
Differential Operators
Leanne Mezuman, Sergei Yakovenko
Arnold Math J. (2018) 4:3-4, 279–299
Received: 10 May 2018 / Revised: 5 March 2019 / Accepted: 11 March 2019 / Published online: 18 March 2019
Abstract
Download PDF of the paper (479KB) .We study the problem of formal decomposition (non-commutative factorization) of linear ordinary differential operators over the field $\mathbb C((t))$ of formal Laurent series at an irregular singular point corresponding to $t = 0$. The solution (given in terms of the Newton diagram and the respective characteristic numbers) is known for quite some time, though the proofs are rather involved. We suggest a process of reduction of the non-commutative problem to its commutative analog, the problem of factorization of pseudopolynomials, which is known since Newton invented his method of rotating ruler. It turns out that there is an ``automatic translation'' which allows to obtain the results for formal factorization in the Weyl algebra from well known results in local analytic geometry. In addition, we draw some (apparently unnoticed) parallels between the formal factorization of linear operators and formal diagonalization of systems of linear first order differential equations. -
Epicycles in the Hyperbolic Sky
Olga Paris-Romaskevich
Arnold Math J. (2018) 4:3-4, 251–277
Received: 9 March 2018 / Revised: 2 November 2018 / Accepted: 27 March 2019 / Published online: 8 April 2019
Abstract
Download PDF of the paper (1918KB) .Consider a swiveling arm on an oriented complete riemannian surface composed of three geodesic intervals, attached one to another in a chain. Each interval of the arm rotates with constant angular velocity around its extremity contributing to a common motion of the arm. Does the extremity of such a chain have an asymptotic velocity? This question for the motion in the euclidian plane, formulated by J.-L. Lagrange, was solved by P. Hartman, E. R. Van Kampen, A. Wintner. We generalize their result to motions on any complete orientable surface of non-zero (and even non-constant) curvature. In particular, we give the answer to Lagrange's question for the movement of a swiveling arm on the hyperbolic plane. The question we study here can be seen as a dream about celestial mechanics on any riemannian surface: how many turns around the Sun a satellite of a planet in the geliocentric epicycle model would make in 1 billion years? -
Counting Borel Orbits in Symmetric Spaces of Types
$BI$ and $CII$
Mahir Bilen Can, Özlem Uǧurlu
Arnold Math J. (2018) 4:2, 213–250
Received: 1 February 2018 / Revised: 29 August 2018 / Accepted: 4 September 2018 / Published online: 10 September 2018
Abstract
Download PDF of the paper (800KB) .This is a continuation of our combinatorial program on the enumeration of Borel orbits in symmetric spaces of classical types. Here, we determine the generating series the numbers of Borel orbits in $\mathbf{SO}_{2n+1}/\mathbf{S}(\mathbf{O}_{2p}\times\mathbf{O}_{2q+1})$ (type $BI$) and in $\mathbf{Sp}_n /\mathbf{Sp}_p\times\mathbf{Sp}_q$ (type $CII$). In addition, we explore relations to lattice path enumeration. -
Solvability of Equations by Quadratures and Newton's
Theorem
Askold Khovanskii
Arnold Math J. (2018) 4:2, 193–211
Received: 25 July 2018 / Revised: 21 August 2018 / Accepted: 27 August 2018 / Published online: 14 September 2018
Abstract
Download PDF of the paper (500KB) .Picard–Vessiot theorem (1910) provides a necessary and sufficient condition for solvability of linear differential equations of order $n$ by quadratures in terms of its Galois group. It is based on the differential Galois theory and is rather involved. Liouville in 1839 found an elementary criterium for such solvability for $n = 2$. Ritt simplified Liouville's theorem (1948). In 1973 Rosenlicht proved a similar criterium for arbitrary $n$. Rosenlicht work relies on the valuation theory and is not elementary. In these notes we show that the elementary Liouville–Ritt method based on developing solutions in Puiseux series as functions of a parameter works smoothly for arbitrary $n$ and proves the same criterium. -
Renormalization for Unimodal Maps with Non-integer
Exponents
Igors Gorbovickis, Michael Yampolsky
Arnold Math J. (2018) 4:2, 179–191
Received: 19 November 2017 / Revsed: 8 August 2018 / Accepted: 12 August 2018 / Published online: 23 August 2018
Abstract
Download PDF of the paper (436KB) .We define an analytic setting for renormalization of unimodal maps with an arbitrary critical exponent. We prove the global hyperbolicity of renormalization conjecture for unimodal maps of bounded type with a critical exponent which is sufficiently close to an even integer. Furthermore, we prove the global $C^{1+\beta}$-rigidity conjecture for such maps, giving the first example of a smooth rigidity theorem for unimodal maps whose critical exponent is not an even integer. -
Conway River and Arnold Sail
K. Spalding, A. P. Veselov
Arnold Math J. (2018) 4:2, 169–177
Received: 21 February 2018 / Revised: 21 June 2018 / Accepted: 5 July 2018 / Published online: 18 July 2018
Abstract
Download PDF of the paper (436KB) .We establish a simple relation between two geometric constructions in number theory: the Conway river of a real indefinite binary quadratic form and the Arnold sail of the corresponding pair of lines. -
Upper Bounds on Betti Numbers of Tropical
Prevarieties
Dima Grigoriev, Nicolai Vorobjov
Arnold Math J. (2018) 4:1, 127-136
Received: 5 October 2017 / Revised: 9 March 2018 / Accepted: 12 March 2018
Abstract
Download PDF of the paper (436KB) .We prove upper bounds on the sum of Betti numbers of tropical prevarieties in dense and sparse settings. In the dense setting the bound is in terms of the volume of Minkowski sum of Newton polytopes of defining tropical polynomials, or, alternatively, via the maximal degree of these polynomials. In sparse setting, the bound involves the number of the monomials. -
Trace Test
Anton Leykin, Jose Israel Rodriguez, Frank Sottile
Arnold Math J. (2018) 4:1, 113-125
Received: 16 December 2016 / Revised: 6 February 2018 / Accepted: 10 March 2018
Abstract
Download PDF of the paper (436KB) .The trace test in numerical algebraic geometry verifies the completeness of a witness set of an irreducible variety in affine or projective space. We give a brief derivation of the trace test and then consider it for subvarieties of products of projective spaces using multihomogeneous witness sets. We show how a dimension reduction leads to a practical trace test in this case involving a curve in a low-dimensional affine space. -
Cyclohedron and
Kantorovich–Rubinstein Polytopes
Filip D. Jevtić, Marija Jelić, Rade T. Živaljević
Arnold Math J. (2018) 4:1, 87-112
Received: 14 July 2017 / Revised: 13 December 2017 / Accepted: 10 March 2018/ Published online: 9 April 2018
Abstract
Download PDF of the paper (436KB) .We show that the cyclohedron (Bott–Taubes polytope) $ W_n$ arises as the polar dual of a Kantorovich–Rubinstein polytope $ KR(\rho)$, where $ \rho$ is an explicitly described quasi-metric (asymmetric distance function) satisfying strict triangle inequality. From a broader perspective, this phenomenon illustrates the relationship between a nestohedron $ \Delta_{{\widehat{\mathcal{F}}}}$ (associated to a building set $ {\widehat{\mathcal{F}}}$) and its non-simple deformation $ \Delta_{\mathcal{F}}$, where $ \mathcal{F}$ is an irredundant or tight basis of $ {\widehat{\mathcal{F}}}$. Among the consequences are a new proof of a recent result of Gordon and Petrov (Arnold Math. J. 3(2):205–218, 2017gp) about $ f$-vectors of generic Kantorovich–Rubinstein polytopes and an extension of a theorem of Gelfand, Graev, and Postnikov, about triangulations of the type A, positive root polytopes. -
Affine Hecke Algebras via DAHA
Ivan Cherednik
Arnold Math J. (2018) 4:1, 69-85
Received: 12 October 2017 / Revised: 22 January 2018 / Accepted: 9 March 2018
Abstract
Download PDF of the paper (436KB) . ViewA method is suggested for obtaining the Plancherel measure for Affine Hecke Algebras as a limit of integral-type formulas for inner products in the polynomial and related modules of Double Affine Hecke Algebras. The analytic continuation necessary here is a generalization of "picking up residues" due to Arthur, Heckman, Opdam and others, which can be traced back to Hermann Weyl. Generally, it is a finite sum of integrals over double affine residual subtori; a complete formula is presented for $ A_1$ in the spherical case. -
Semiconjugate Rational Functions: A Dynamical Approach
F. Pakovich
Arnold Math J. (2018) 4:1, 59-68
Received: 7 January 2018 / Accepted: 29 January 2018
Abstract
Download PDF of the paper (436KB) . ViewUsing dynamical methods we give a new proof of the theorem saying that if $A,B,X$ are rational functions of complex variable $z$ of degree at least two such that $A\circ X=X\circ B$ and $\mathbb C(B,X)=\mathbb C(z)$, then the Galois closure of the field extension $\mathbb C(z)/\mathbb C(X)$ has genus zero or one. -
Secant Degeneracy
Index of the Standard Strata in The Space of Binary Forms
Gleb Nenashev, Boris Shapiro, Michael Shapiro
Arnold Math J. (2018) 3:4, 499-510
Received: 30 January 2017 / Revised: 3 November 2017 / Accepted: 7 November 2017
Abstract
Download PDF of the paper (436KB) . ViewThe space $ Pol_d\simeq \mathbb{C} P^d$ of all complex-valued binary forms of degree $ d$ (considered up to a constant factor) has a standard stratification, each stratum of which contains all forms whose set of multiplicities of their distinct roots is given by a fixed partition $ \mu \vdash d$. For each such stratum $ S_\mu,$ we introduce its secant degeneracy index $ \ell_\mu$ which is the minimal number of projectively dependent pairwise distinct points on $ S_\mu$, i.e., points whose projective span has dimension smaller than $ \ell_\mu-1$. In what follows, we discuss the secant degeneracy index $ \ell_\mu$ and the secant degeneracy index $ \ell_{{{\bar{\mu}}}}$ of the closure $ {{\bar{S}}}_\mu$. -
Orbifold Jacobian
Algebras for Exceptional Unimodal Singularities
Alexey Basalaev, Atsushi Takahashi, Elisabeth Werner
Arnold Math J. (2018) 3:4, 483-498
Received: 4 March 2017 / Revised: 27 September 2017 / Accepted: 8 October 2017
Abstract
Download PDF of the paper (512KB) . ViewThis note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund–Hübsch transform of an invertible polynomial defining the strange dual singularity in the sense of Arnold. -
On Tangent Cones of Schubert Varieties
Dmitry Fuchs, Alexandre Kirillov, Sophie Morier-Genoud, Valentin Ovsienko
Arnold Math J. (2018) 3:4, 451-498
Received: 23 March 2017 / Revised: 10 June 2017 / Accepted: 2 August 2017 / First Online: 22 August 2017
Abstract
Download PDF of the paper (615KB) . ViewWe consider tangent cones of Schubert varieties in the complete flag variety, and investigate the problem when the tangent cones of two different Schubert varieties coincide. We give a sufficient condition for such coincidence, and formulate a conjecture that provides a necessary condition. In particular, we show that all Schubert varieties corresponding to the Coxeter elements of the Weyl group have the same tangent cone. Our main tool is the notion of pillar entries in the rank matrix counting the dimensions of the intersections of a given flag with the standard one. This notion is a version of Fulton's essential set. We calculate the dimension of a Schubert variety in terms of the pillar entries of the rank matrix. -
On Irreducible Components of Real Exponential
Hypersurfaces
Cordian Riener, Nicolai Vorobjov
Arnold Math J. (2017) 3:3, 423–443
Received: 31 December 2016 / Revised: 28 July 2017 / Accepted: 2 August 2017 / First Online: 09 August 2017
Abstract
Download PDF of the paper (615KB) . ViewFix any real algebraic extension $ \mathbb K$ of the field $ \mathbb Q$ of rationals. Polynomials with coefficients from $ \mathbb K$ in $ n$ variables and in $ n$ exponential functions are called exponential polynomials over $ {\mathbb K}$. We study zero sets in $ \mathbb R^n$ of exponential polynomials over $ \mathbb K$, which we call exponential-algebraic sets . Complements of all exponential-algebraic sets in $ \mathbb R^n$ form a Zariski-type topology on $ \mathbb R^n$. Let $ P \in {\mathbb K}[X_1, \ldots ,X_n,U_1, \ldots ,U_n]$ be a polynomial and denote \begin{eqnarray*} V:=\{ (x_1, \ldots , x_n) \in \mathbb R^n|\> P(x_1, \ldots ,x_n,, e^{x_1}, \ldots ,e^{x_n})=0 \}. \end{eqnarray*} The main result of this paper states that, if the real zero set of a polynomial $ P$ is irreducible over $ \mathbb K$ and the exponential-algebraic set $ V$ has codimension 1, then, under Schanuel's conjecture over the reals, either $ V$ is irreducible (with respect to the Zariski topology) or each of its irreducible components of codimension 1 is a rational hyperplane through the origin. The family of all possible hyperplanes is determined by monomials of $ P$. In the case of a single exponential (i.e., when $ P$ is independent of $ U_2, \ldots , U_n$) stronger statements are shown which are independent of Schanuel's conjecture. -
Integral Geometry of Euler Equations
Nikolai Nadirashvili, Serge Vlăduţ
Arnold Math J. (2017) 3:3, 397–421
Received: 23 May 2017 / Revised: 24 July 2017 / Accepted: 29 July / First Online: 09 August 2017
Abstract
Download PDF of the paper (610KB) . ViewWe develop an integral geometry of stationary Euler equations defining some function $ w$ on the Grassmannian of affine lines in $ \mathbb R^3$ depending on a putative compactly supported solution $ (v;p)$ of the system and deduce some linear differential equations for $ w$. We conjecture that $ w=0$ everywhere and prove that this conjecture implies that $ v=0.$ -
Origami, Affine Maps, and Complex Dynamics
William Floyd, Gregory Kelsey, Sarah Koch, Russell Lodge, Walter Parry, Kevin M. Pilgrim, Edgar Saenz
Arnold Math J. (2017) 3:3, 365–395
Received: Received: 19 December 2016 / Accepted: 27 July 2017/ First Online: 22 August 2017
Abstract
Download PDF of the paper (1410KB) . ViewWe investigate the combinatorial and dynamical properties of so-called nearly Euclidean Thurston maps , or NET maps . These maps are perturbations of many-to-one folding maps of an affine two- sphere to itself. The close relationship between NET maps and affine maps makes computation of many invariants tractable. In addition to this, NET maps are quite diverse, exhibiting many different behaviors. We discuss data, findings, and new phenomena. -
Moduli Space of a Planar Polygonal Linkage: A Combinatorial Description
Gaiane Panina
Arnold Math J. (2017) 3:3, 351–364
Received: 11 December 2016 / Revised: 8 April 2017 / Accepted: 18 May 2017
Abstract
Download PDF of the paper (670KB) . ViewWe describe and study an explicit structure of a regular cell complex $\mathcal{K}(L)$ on the moduli space $M(L)$ of a planar polygonal linkage $L$. The combinatorics is very much related (but not equal) to the combinatorics of the permutohedron. In particular, the cells of maximal dimension are labeled by elements of the symmetric group. For example, if the moduli space $M$ is a sphere, the complex $\mathcal{K}$ is dual to the boundary complex of the permutohedron.The dual complex $\mathcal{K}^*$ is patched of Cartesian products of permutohedra. It can be explicitly realized in the Euclidean space via a surgery on the permutohedron. -
When is the Intersection
of Two Finitely Generated Subalgebras of a Polynomial Ring Also Finitely
Generated?
Pinaki Mondal
Arnold Math J. (2017) 3:3, 333–350
Received: 6 November 2016 / Revised: 23 March 2017 / Accepted: 27 March 2017
Abstract
Download PDF of the paper (670KB) . ViewWe study two variants of the following question: "Given two finitely generated $ \mathbb C$-subalgebras $ R_1, R_2$ of $ \mathbb C[x_1, \ldots, x_n]$, is their intersection also finitely generated?" We show that the smallest value of $ n$ for which there is a counterexample is $ 2$ in the general case, and $ 3$ in the case that $ R_1$ and $ R_2$ are integrally closed. We also explain the relation of this question to the problem of constructing algebraic compactifications of $ \mathbb C^n$ and to the moment problem on semialgebraic subsets of $ \mathbb R^n$. The counterexample for the general case is a simple modification of a construction of Neena Gupta, whereas the counterexample for the case of integrally closed subalgebras uses the theory of normal analytic compactifications of $ \mathbb C^2$ via key forms of valuations centered at infinity. -
On a Triply Periodic Polyhedral Surface
Whose Vertices are Weierstrass Points
Dami Lee
Arnold Math J. (2017) 3:3, 319–331
Received: 3 May 2016 / Revised: 12 March 2017 / Accepted: 23 March 2017
Abstract
Download PDF of the paper (1633KB) . ViewIn this paper, we will construct an example of a closed Riemann surface $ X$ that can be realized as a quotient of a triply periodic polyhedral surface $ \Pi \subset \mathbb R^3$ where the Weierstrass points of $ X$ coincide with the vertices of $ \Pi.$ First we construct $ \Pi$ by attaching Platonic solids in a periodic manner and consider the surface of this solid. Due to periodicity we can find a compact quotient of this surface. The symmetries of $ X$ allow us to construct hyperbolic structures and various translation structures on $ X$ that are compatible with its conformal type. The translation structures are the geometric representations of the holomorphic 1-forms of $ X.$ Via the basis of 1-forms we find an explicit algebraic description of the surface that suggests the Fermat's quartic. Moreover the 1-forms allow us to identify the Weierstrass points. -
Dynamics of
Polynomial Diffeomorphisms of $ \mathbb{C}^2$: Combinatorial and Topological
Aspects
Yutaka Ishii
Arnold Math J. (2017) 3:1, 119–173
Received: 7 October 2016 / Revised: 10 February 2017 / Accepted: 23 March 2017
Abstract
Download PDF of the paper (407KB) . ViewThe Fig. 1 was drawn by Shigehiro Ushiki using his software called HenonExplorer . This complicated object is the Julia set of a complex Hénon map $ f_{c, b}(x, y)=(x^2+c-by, x)$ defined on $ \mathbb{C}^2$ together with its stable and unstable manifolds, hence it is a fractal set in the real $ 4$-dimensional space! The purpose of this paper is to survey some results, questions and problems on the dynamics of polynomial diffeomorphisms of $ \mathbb{C}^2$ including complex Hénon maps with an emphasis on the combinatorial and topological aspects of their Julia sets. -
Vanishing Cycles and Cartan Eigenvectors
Laura Brillon, Revaz Ramazashvili, Vadim Schechtman, Alexander Varchenko
Arnold Math J. (2017) 3:2, 251–280
Received: 19 December 2015 / Revised: 11 July 2016 / Accepted: 20 July 2016
Abstract
Download PDF of the paper (718KB) . ViewUsing the ideas coming from the singularity theory, we study the eigenvectors of the Cartan matrices of finite root systems, and of q-deformations of these matrices. -
Polynomial Splitting Measures and Cohomology of
the Pure Braid Group
Trevor Hyde, Jeffrey C. Lagarias
Arnold Math J. (2017) 3:2, 219–249
Received: 10 August 2016 / Revised: 27 December 2016 / Accepted: 1 February 2017
Abstract
Download PDF of the paper (691KB) . ViewWe study for each $n$ a one-parameter family of complex-valued measures on the symmetric group $S_n$, which interpolate the probability of a monic, degree $n$, square-free polynomial in $\mathbb F_q[x]$ having a given factorization type. For a fixed factorization type, indexed by a partition $\lambda$ of $n$, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to $S_n$-subrepresentations of the cohomology of the pure braid group $H^{\bullet}(P_n, \mathbb Q)$. We deduce that the splitting measures for all parameter values $z= -\frac{1}{m}$ (resp. $z= \frac{1}{m}$), after rescaling, are characters of $S_n$-representations (resp. virtual $S_n$-representations). -
Combinatorics of the Lipschitz Polytope
J. Gordon, F. Petrov
Arnold Math J. (2017) 3:2, 205–218
Received: 18 July 2016 / Revised: 13 November 2016 / Accepted: 17 January 2017
Abstract
Download PDF of the paper (455KB) . ViewLet $ \rho$ be a metric on the set $ X=\{1,2,\dots,n+1\}$. Consider the $ n$- dimensional polytope of functions $ f:X\rightarrow \mathbb{R}$, which satisfy the conditions $ f(n+1)=0$, $ |f(x)-f(y)|\leq \rho(x,y)$. The question on classifying metrics depending on the combinatorics of this polytope have been recently posed by (Vershik, Arnold Math J 1(1):75-81, 2015). We prove that for any "generic" metric the number of $ (n-m)$-dimensional faces, $ 0\leq m\leq n$, equals $ \binom{n+m}{m,m,n-m}=(n+m)!/m!m!(n-m)!$. This fact is intimately related to regular triangulations of the root polytope (convex hull of the roots of $ A_n$ root system). Also we get two-sided estimates for the logarithm of the number of Vershik classes of metrics: $ n^3\log n$ from above and $ n^2$ from below. -
Convex Shapes and Harmonic Caps
Laura DeMarco, Kathryn Lindsey
Arnold Math J. (2017) 3:1, 97–117
Received: 4 February 2016 / Revised: 30 November 2016 / Accepted: 23 December 2016
Abstract
Download PDF of the paper (755KB) . ViewAny planar shape $P\subset{\mathbb{C}}$ can be embedded isometrically as part of the boundary surface $S$ of a convex subset of $\mathbb{R}^{3}$ such that $\partial P$ supports the positive curvature of $S$. The complement $Q=S{\setminus}P$ is the associated cap. We study the cap construction when the curvature is harmonic measure on the boundary of $({\hat{{\mathbb{C}}}}{\setminus}P,\infty)$. Of particular interest is the case when $P$ is a filled polynomial Julia set and the curvature is proportional to the measure of maximal entropy. -
Random Chain Complexes
Viktor L. Ginzburg, Dmitrii V. Pasechnik
Arnold Math J. (2017) 3:2, 197–204
Received: 16 March 2016 / Revised: 9 December 2016 / Accepted: 23 December 2016
Abstract
Download PDF of the paper (389KB) . ViewWe study random, finite-dimensional, ungraded chain complexes over a finite field and show that for a uniformly distributed differential a complex has the smallest possible homology with the highest probability: either zero or one-dimensional homology depending on the parity of the dimension of the complex. We prove that as the order of the field goes to infinity the probability distribution concentrates in the smallest possible dimension of the homology. On the other hand, the limit probability distribution, as the dimension of the complex goes to infinity, is a super-exponentially decreasing, but strictly positive, function of the dimension of the homology. -
The $4n^{2}$-Inequality for Complete Intersection Singularities
Aleksandr V. Pukhlikov
Arnold Math J. (2017) 3:2, 187–196
Received: 11 July 2016 / Revised: 25 October 2016 / Accepted: 17 November 2016
Abstract
Download PDF of the paper (396KB) . ViewThe famous $4n^{2}$-inequality is extended to generic complete intersection singularities: it is shown that the multiplicity of the self-intersection of a mobile linear system with a maximal singularity is greater than $4n^{2}\mu$, where $\mu$ is the multiplicity of the singular point. -
The Geometry of Axisymmetric Ideal Fluid Flows with Swirl
Pearce Washabaugh, Stephen C. Preston
Arnold Math J. (2017) 3:2, 175–185
Received: 3 February 2016 / Revised: 21 August 2016 / Accepted: 15 October 2016
Abstract
Download PDF of the paper (424KB) . ViewThe sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold $M$ can give information about the stability of inviscid, incompressible fluid flows on $M$. We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swirl, denoted by $\mathcal{D}_{\mu,E}(M)$, has positive sectional curvature in every section containing the field $X=u(r)\partial_{\theta}$ iff $\partial_{r}(ru^{2})>0$. This is in sharp contrast to the situation on $\mathcal{D}_{\mu}(M)$, where only Killing fields $X$ have nonnegative sectional curvature in all sections containing it. We also show that this criterion guarantees the existence of conjugate points on $\mathcal{D}_{\mu,E}(M)$ along the geodesic defined by $X$. -
On Postsingularly Finite Exponential Maps
Walter Bergweile
Arnold Math J. (2017) 3:1, 83–95
Received: 5 December 2015 / Revised: 15 August 2016 / Accepted: 6 September 2016
Abstract
Download PDF of the paper (456KB) . ViewWe consider parameters $\lambda$ for which 0 is preperiodic under the map $z\mapsto\lambda e^{z}$. Given $k$ and $l$, let $n(r)$ be the number of $\lambda$ satisfying $0<|\lambda|\leq r$ such that 0 is mapped after $k$ iterations to a periodic point of period $l$. We determine the asymptotic behavior of $n(r)$ as $r$ tends to $\infty$. -
Spherical Rectangles
Alexandre Eremenko, Andrei Gabrielov
Arnold Math J. (2016) 2:4, 463–486
Received: 24 January 2016 / Revised: 9 August 2016 / Accepted: 30 August 2016
Abstract
Download PDF of the paper (483KB) . ViewWe study spherical quadrilaterals whose angles are odd multiples of $\pi/2$, and the equivalent accessory parameter problem for the Heun equation. We obtain a classification of these quadrilaterals up to isometry. For given angles, there are finitely many one-dimensional continuous families which we enumerate. In each family the conformal modulus is either bounded from above or bounded from below, but not both, and the numbers of families of these two types are equal. The results can be translated to classification of Heunâ~@~Ys equations with real parameters, whose exponent differences are odd multiples of $1/2$, with unitary monodromy. -
q-Polynomial Invariant of Rooted Trees
Jözef H. Przytycki
Arnold Math J. (2016) 2:4, 449–461
Received: 7 December 2015 / Revised: 28 July 2016 / Accepted: 2 August 2016
Abstract
Download PDF of the paper (606KB) . ViewWe describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to readers of Arnold Journal of Mathematics is that we deal here with an elementary, interesting, new mathematics, and after reading this essay readers can take part in developing the topic, inventing new results and connections to other disciplines of mathematics, and likely, statistical mechanics, and combinatorial biology. -
On the Roots of a Hyperbolic Polynomial Pencil
Victor Katsnelson
Arnold Math J. (2016) 2:4, 439–448
Received: 03 May 2016 / Accepted: 20 July 2016 / Published Online: 02 August 2016
Abstract
Download PDF of the paper (483KB) . ViewLet $ \nu_0(t),\nu_1(t),\ldots,\nu_n(t)$ be the roots of the equation $ R(z)=t$, where $ R(z)$ is a rational function of the form
$\displaystyle \begin{eqnarray*} R(z)=z-\sum\limits_{k=1}^n\frac{\alpha_k}{z-\mu_k}, \end{eqnarray*}$ $ \mu_k$ are pairwise distinct real numbers, $ \alpha_k> 0,\,1\leq{}k\leq{}n$. Then for each real $ \xi$, the function $ e^{\xi\nu_0(t)}+e^{\xi\nu_1(t)}+\,\cdots\,+e^{\xi\nu_n(t)}$ is exponentially convex on the interval $ -\infty< t< \infty$.
-
A Generalisation of the Cauchy-Kovalevskaïa Theorem
Mauricio Garay
Arnold Math J. (2016) 2:3, 407–438
Received: 1 July 2015 / Revised: 15 May 2016 / Accepted: 23 June 2016 / Published Online: 09 August 2016
Abstract
Download PDF of the paper (2516KB) . ViewWe prove that time evolution of a linear analytic initial value problem leadsto sectorial holomorphic solutions in time. -
A Classification of Spherical Curves Based on Gauss Diagrams
Guy Valette
Arnold Math J. (2016) 2:3, 383–405
Received: 28 August 2015 / Revised: 4 May 2016 / Accepted: 23 June 2016 / Published Online: 11 July 2016
Abstract
Download PDF of the paper (2516KB) and appendices. ViewWe consider generic smooth closed curves on the sphere $S^{2}$. These curves (oriented or not) are classified relatively to the group $\mbox{Diff}(S^{2})$ or its subgroup $\mbox{Diff}^{+}(S^{2})$, with the Gauss diagrams as main tool. V. I. Arnold determined the numbers of orbits of curves with $n$ double points when $n<6$. This paper explains how a preliminary classification of the Gauss diagrams of order 5, 6 and 7 allows to draw up the list of the realizable chord diagrams of these orders. For each such diagram $\Gamma$ and for each Arnold symmetry type $T$, we determine the number of orbits of spherical curves of type $T$ realizing $\Gamma$. As a consequence, we obtain the total numbers of curves (oriented or not) with 6 or 7 double points on the sphere (oriented or not) and also the number of curves with special properties (e.g. having no simple loop). -
On Malfatti's Marble Problem
Uuganbaatar Ninjbat
Arnold Math J. (2016) 2:3, 309–327
Received: 3 April 2015 / Revised: 9 April 2016 / Accepted: 20 June 2016 / Published Online: 11 July 2016
Abstract
Download PDF (1043KB) ViewConsider the problem of finding three non-overlapping circles in a given triangle with the maximum total area. This is Malfatti's marble problem, and it is known that the greedy arrangement is the solution. In this paper, we provide a simpler proof of this result by synthesizing earlier insights with more recent developments. We also discuss some related geometric extremum problems, and show that the greedy arrangement solves the problem of finding two non-overlapping circles in a tangential polygon with the maximum total radii and/or area. In the light of this discussion, we formulate a natural extension of Melissen's conjecture. -
Volume Polynomials and Duality Algebras of Multi-Fans
Anton Ayzenberg, Mikiya Masuda
Arnold Math J. (2016) 2:3, 329–381
Received: 17 October 2015 / Revised: 12 November 2015 / Accepted: 23 June 2016 / Published Online: 11 July 2016
Abstract
Download PDF (965KB)We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan $\Delta$ determines a volume polynomial $V_\Delta$ whose values are the volumes of multi-polytopes based on $\Delta$. This homogeneous polynomial is further used to construct a Poincare duality algebra $\mathcal{A}^*(\Delta)$. We study the structure and properties of $V_\Delta$ and $\mathcal{A}^*(\Delta)$ and give applications and connections to other subjects, such as Macaulay duality, Novik-Swartz theory of face rings of simplicial manifolds, generalizations of Minkowski's theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the $g$-theorem does not hold for multi-polytopes. -
Generalizations of Tucker-Fan-Shashkin Lemmas
Oleg R. Musin
Arnold Math J. (2016) 2:3, 299–308
Received: 26 November 2014 / Revised: 25 April 2016 / Accepted: 27 May 2016 / Published online: 16 June 2016
Abstract
Download PDF (487KB) ViewTucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about the doubling of manifolds with boundaries that are BUT-manifolds. -
Strange Duality Between Hypersurface and Complete Intersection Singularities
Wolfgang Ebeling, Atsushi Takahashi
Arnold Math J. (2016) 2:3, 277–298
Received: 22 September 2015 / Revised: 9 May 2016 / Accepted: 12 May 2016 / Published online: 24 May 2016
Abstract
Download PDF (614KB) ViewW. Ebeling and C. T. C. Wall discovered an extension of Arnold's strange duality embracing on one hand series of bimodal hypersurface singularities and on the other, isolated complete intersection singularities. In this paper, we derive this duality from the mirror symmetry and the Berglund-Hübsch transposition of invertible polynomials. -
The Coadjoint Operator, Conjugate Points, and the Stability of Ideal Fluids
James Benn
Arnold Math J. (2016) 2:2, 249–266
Received: 11 August 2015 / Revised: 25 February 2016 / Accepted: 11 April 2016 / Published online: 18 May 2016
Abstract
Download PDF (492KB) ViewWe give a new description of the coadjoint operator $Ad^*_{\eta^{-1}(t)}$ along a geodesic $\eta(t)$ of the $L^2$ metric in the group of volume-preserving diffeomorphisms, important in hydrodynamics. When the underlying manifold is two dimensional the coadjoint operator is given by the solution operator to the linearized Euler equations modulo a compact operator; when the manifold is three dimensional the coadjoint operator is given by the solution operator to the linearized Euler equations plus a bounded operator. We give two applications of this result when the underlying manifold is two dimensional: conjugate points along geodesics of the $L^2$ metric are characterized in terms of the coadjoint operator and thus determining the conjugate locus is a purely algebraic question. We also prove that Eulerian and Lagrangian stability of the $2D$ Euler equations are equivalent and that instabilities in the $2D$ Euler equations are contained and small. -
Internal Addresses of the Mandelbrot Set and Galois Groups of Polynomials
Dierk Schleicher
Received: 15 October 2015 / Revised: 9 February 2016 / Accepted: 7 April 2016 / Published online: 02 August 2016
Abstract
Download PDF (881KB) ViewWe describe an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. Internal addresses are a convenient and efficient way of describing the combinatorial structure of the Mandelbrot set, and of giving geometric meaning to the ubiquitous kneading sequences in human-readable form (Sects. 3 and 4). A simple extension, angled internal addresses, distinguishes combinatorial classes of the Mandelbrot set and in particular distinguishes hyperbolic components in a concise and dynamically meaningful way. This combinatorial description of the Mandelbrot set makes it possible to derive existence theorems for certain kneading sequences and internal addresses in the Mandelbrot set (Sect. 6) and to give an explicit description of the associated parameters. These in turn help to establish some algebraic results about permutations of periodic points and to determine Galois groups of certain polynomials (Sect. 7). Through internal addresses, various areas of mathematics are thus related in this manuscript, including symbolic dynamics and permutations, combinatorics of the Mandelbrot set, and Galois groups. -
Non-avoided Crossings for $n$-Body Balanced Configurations in $\mathbb R^3$ Near a Central Configuration
Alain Chenciner
Arnold Math J. (2016) 2:2, 213–248
Received: 4 September 2015 / Revised: 14 January 2016 / Accepted: 10 March 2016 / Published online: 8 April 2016
Abstract
Download PDF (1132KB) ViewThe balanced configurations are those $n$-body configurations which admit a relative equilibrium motion in a Euclidean space $E$ of high enough dimension $2 p$. They are characterized by the commutation of two symmetric endomorphisms of the $(n-1)$-dimensional Euclidean space of codispositions, the intrinsic inertia endomorphism $B$ which encodes the shape and the Wintner-Conley endomorphism $A$ which encodes the forces. In general, $p$ is the dimension $d$ of the configuration, which is also the rank of B. Lowering to $2(d-1)$ the dimension of $E$ occurs when the restriction of $A$ to the (invariant) image of $B$ possesses a double eigenvalue. It is shown that, while in the space of all $d\times d$ symmetric endomorphisms, having a double eigenvalue is a condition of codimension 2 (the avoided crossings of physicists), here it becomes of codimension 1 provided some condition $(H)$ is satisfied. As the condition is always satisfied for configurations of the maximal dimension (i.e. if $d = n-1$), this implies in particular the existence, in the neighborhood of the regular tetrahedron configuration of four bodies with no three of the masses equal, of exactly three families of balanced configurations which admit relative equilibrium motion in a four dimensional space. -
Geodesics on Regular Polyhedra with Endpoints at the Vertices
Dmitry Fuchs
Arnold Math J. (2016) 2:2, 201–211
Received: 3 October 2015 / Revised: 23 October 2015 / Accepted: 3 March 2016 / Published online: 23 March 2016
Abstract
Download PDF (1305KB) ViewIn a recent work of Davis et al. (2016), the authors consider geodesics on regular polyhedra which begin and end at vertices (and do not touch other vertices). The cases of regular tetrahedra and cubes are considered. The authors prove that (in these cases) a geodesic as above never begins at ends at the same vertex and compute the probabilities with which a geodesic emanating from a given vertex ends at every other vertex. The main observation of the present article is that there exists a close relation between the problem considered in Davis et al. (2016) and the problem of classification of closed geodesics on regular polyhedra considered in articles (Fuchs and Fuchs, Mosc Math J 7:265-279, 2007; Fuchs, Geom Dedic 170:319-333, 2014). This approach yields different proofs of result of Davis et al. (2016) and permits to obtain similar results for regular octahedra and icosahedra (in particular, such a geodesic never ends where it begins). -
On Foliations in Neighborhoods of Elliptic Curves
M. Mishustin
Arnold Math J. (2016) 2:2, 195–199
Received: 13 April 2015 / Revised: 24 August 2015 / Accepted: 13 January 2016 / Published online: 26 January 2016
Abstract
Download PDF (372KB) ViewA counterexample is given to a conjecture from the comments to Arnold's problem 1989-11 about the existence of a tangent foliation in a zero type neighborhood of an elliptic curve. -
Skewers
Serge Tabachnikov
Arnold Math J. (2016) 2:2, 171–193
Received: 19 September 2015 / Revised: 29 December 2015 / Accepted: 11 January 2016 / Published online: 27 January 2016
Abstract
Download PDF (875KB) ViewThe skewer of a pair of skew lines in space is their common perpendicular. To configuration theorems of plane projective geometry involving points and lines (such as Pappus or Desargues) there correspond configuration theorems in space: points and lines in the plane are replaced by lines is space, the incidence between a line and a point translates as the intersection of two lines at right angle, and the operations of connecting two points by a line or by intersecting two lines at a point translate as taking the skewer of two lines. These configuration theorems hold in elliptic, Euclidean, and hyperbolic geometries. This correspondence principle extends to plane configuration theorems involving polarity. For example, the theorem that the three altitudes of a triangle are concurrent corresponds to the Petersen-Morley theorem that the common normals of the opposite sides of a space right-angled hexagon have a common normal. We define analogs of plane circles (they are 2-parameter families of lines in space) and extend the correspondence principle to plane theorems involving circles. We also discuss the skewer versions of the Sylvester problem: given a finite collection of pairwise skew lines such that the skewer of any pair intersects at least one other line at right angle, do all lines have to share a skewer? The answer is positive in the elliptic and Euclidean geometries, but negative in the hyperbolic one. -
An Invariant of Colored Links via Skein Relation
Francesca Aicardi
Arnold Math J. (2016) 2:2, 159–169
Received: 30 May 2015 / Accepted: 14 December 2015 / Published online: 1 March 2016
Abstract
Download PDF (546KB) ViewIn this note, we define a polynomial invariant for colored links by a skein relation. It specializes to the Jones polynomial for classical links. -
N-Division Points of Hypocycloids
N. Mani, S. Rubinstein-Salzedo
Arnold Math J. (2016) 2:2, 149–158
Received: 4 May 2015 / Revised: 19 October 2015 / Accepted: 6 December 2015 / Published online: 04 January 2016
Abstract
Download PDF (526KB) ViewWe show that the n-division points of all rational hypocycloids are constructible with an unmarked straightedge and compass for all integers n, given a pre-drawn hypocycloid. We also consider the question of constructibility of n-division points of hypocycloids without a pre-drawn hypocycloid in the case of a tricuspoid, concluding that only the 1, 2, 3, and 6-division points of a tricuspoid are constructible in this manner. -
Polynomials Invertible in k-Radicals
Y. Burda, A. Khovanskii
Arnold Math J. (2016) 2:1, 121–138
Received: 18 May 2015 / Revised: 22 December 2015 / Accepted: 25 December 2015 / Published online: 09 February 2016 2015
Abstract
Download PDF (1364KB) ViewA classic result of Ritt describes polynomials invertible in radicals: they are compositions of power polynomials, Chebyshev polynomials and polynomials of degree at most 4. In this paper we prove that a polynomial invertible in radicals and solutions of equations of degree at most k is a composition of power polynomials, Chebyshev polynomials, polynomials of degree at most k and, if \(k\le 14\), certain polynomials with exceptional monodromy groups. A description of these exceptional polynomials is given. The proofs rely on classification of monodromy groups of primitive polynomials obtained by Müller based on group-theoretical results of Feit and on previous work on primitive polynomials with exceptional monodromy groups by many authors. -
Generalized Plumbings and Murasugi Sums
B. Ozbagci, P. Popescu-Pampu
Arnold Math J. (2016) 2:1, 69–119
Received: 6 July 2015 / Revised: 28 October 2015 / Accepted: 23 November 2015 / Published online: 23 December 2015
Abstract
Download PDF (1699KB) ViewWe propose a generalization of the classical notions of plumbing and Murasugi summing operations to smooth manifolds of arbitrary dimensions, so that in this general context Gabai's credo "the Murasugi sum is a natural geometric operation" holds. In particular, we prove that the sum of the pages of two open books is again a page of an open book and that there is an associated summing operation of Morse maps. We conclude with several open questions relating this work with singularity theory and contact topology. -
The Gabrielov-Khovanskii Problem for Polynomials
A. V. Pukhlikov
Arnold Math J. (2016) 2:1, 29–68
Received: 19 June 2015 / Revised: 24 October 2015 / Accepted: 6 November 2015 / Published online: 27 November 2015
Abstract
Download PDF (657KB) ViewWe state and consider the Gabrielov-Khovanskii problem of estimating the multiplicity of a common zero for a tuple of polynomials in a subvariety of a given codimension in the space of tuples of polynomials. For a bounded codimension we obtain estimates of the multiplicity of the common zero, which are close to optimal ones. We consider certain generalizations and open questions. -
Galois Correspondence Theorem for Picard-Vessiot Extensions
T. Crespo, Z. Hajto, E. Sowa-Adamus
Arnold Math J. (2016) 2:1, 21–27
Received: 16 April 2015 / Revised: 23 September 2015 / Accepted: 23 October 2015 / Published online: 03 November 2015
Abstract
Download PDF (374KB) ViewFor a homogeneous linear differential equation defined over a differential field K, a Picard-Vessiot extension is a differential field extension of K differentially generated by a fundamental system of solutions of the equation and not adding constants. When K has characteristic 0 and the field of constants of K is algebraically closed, it is well known that a Picard-Vessiot extension exists and is unique up to K-differential isomorphism. In this case the differential Galois group is defined as the group of K-differential automorphisms of the Picard-Vessiot extension and a Galois correspondence theorem is settled. Recently, Crespo, Hajto and van der Put have proved the existence and unicity of the Picard-Vessiot extension for formally real (resp. formally p-adic) differential fields with a real closed (resp. p-adically closed) field of constants. This result widens the scope of application of Picard-Vessiot theory beyond the complex field. It is then necessary to give an accessible presentation of Picard-Vessiot theory for arbitrary differential fields of characteristic zero which eases its use in physical or arithmetic problems. In this paper, we give such a presentation avoiding both the notions of differential universal extension and specializations used by Kolchin and the theories of schemes and Hopf algebras used by other authors. More precisely, we give an adequate definition of the differential Galois group as a linear algebraic group and a new proof of the Galois correspondence theorem for a Picard-Vessiot extension of a differential field with non algebraically closed field of constants, which is more elementary than the existing ones. -
On Maps Taking Lines to Plane Curves
V. Petrushchenko, V. Timorin
Arnold Math J. (2016) 2:1, 1–20
Received: 24 March 2015 / Accepted: 16 October 2015 / Published online: 03 November 2015
Abstract
Download PDF (985KB) ViewWe study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i.e., maps taking lines to plane curves. -
Solvability of Linear Differential Systems with Small Exponents in the Liouvillian Sense
R. R. Gontsov, I. V. Vyugin
Arnold Math J. (2015) 1:4, 445–471
Received: 25 November 2014 / Revised: 20 August 2015 / Accepted: 11 November 2015 / Published online: 26 November 2015
Abstract
Download PDF (585KB) ViewThe paper is devoted to solvability of linear differential systems by quadratures, one of the classical problems of differential Galois theory. As known, solvability of a system depends entirely on properties of its differential Galois group. However, detecting solvability or non-solvability of a given system is a difficult problem, because the dependence of its differential Galois group on the coefficients of the system remains unknown. We consider systems with regular singular points as well as those with non-resonant irregular ones, whose exponents (respectively, so-called formal exponents in the irregular case) are sufficiently small. It turns out that for systems satisfying such restrictions criteria of solvability can be given in terms of the coefficient matrix. -
Finite and Infinitesimal Flexibility of Semidiscrete Surfaces
O. Karpenkov
Arnold Math J. (2015) 1:4, 403–444
Received: 18 April 2015 / Revised: 28 July 2015 / Accepted: 24 August 2015 / Published online: 3 September 2015
Abstract
Download PDF (724KB) ViewIn this paper we study infinitesimal and finite flexibility for regular semidiscrete surfaces. We prove that regular 2-ribbon semidiscrete surfaces have one degree of infinitesimal and finite flexibility. In particular we write down a system of differential equations describing isometric deformations in the case of existence. Further we find a necessary condition of 3-ribbon infinitesimal flexibility. For an arbitrary \(n\ge 3\) we prove that every regular n-ribbon surface has at most one degree of finite/infinitesimal flexibility. Finally, we discuss the relation between general semidiscrete surface flexibility and 3-ribbon subsurface flexibility. We conclude this paper with one surprising property of isometric deformations of developable semidiscrete surfaces. -
Constructive Geometrization of Thurston Maps and Decidability of Thurston Equivalence
N. Selinger, M. Yampolsky
Arnold Math J. (2015) 1:4, 361–402
Received: 14 November 2014 / Revised: 3 June 2015 / Accepted: 4 August 2015 / Published online: 7 September 2015
Abstract
Download PDF (1017KB)The key result in the present paper is a direct analogue of the celebrated Thurston's Theorem Douady and Hubbard (Acta Math 171:263-297, 1993) for marked Thurston maps with parabolic orbifolds. Combining this result with previously developed techniques, we prove that every Thurston map can be constructively geometrized in a canonical fashion. As a consequence, we give a partial resolution of the general problem of decidability of Thurston equivalence of two postcritically finite branched covers of \(S^2\) (cf. Bonnot et al. Moscow Math J 12:747-763, 2012). -
Bollobás – Riordan and Relative Tutte Polynomials
C. Butler, S. Chmutov
Arnold Math J. (2015) 1:3, 283–298
Received: 8 December 2014 / Revised: 29 June 2015 / Accepted: 5 July 2015 / Published online: 28 July 2015
Abstract
Download PDF (1003KB) ViewWe establish a relation between the Bollobás – Riordan polynomial of a ribbon graph with the relative Tutte polynomial of a plane graph obtained from the ribbon graph using its projection to the plane in a nontrivial way. Also we give a duality formula for the relative Tutte polynomial of dual plane graphs and an expression of the Kauffman bracket of a virtual link as a specialization of the relative Tutte polynomial. -
Critical Set of the Master Function and Characteristic Variety
of the Associated Gauss-Manin Differential Equations
A. Varchenko
Arnold Math J. (2015) 1:3, 253–282
Received: 7 November 2014 / Accepted: 15 June 2015 / Published online: 7 July 2015
Abstract
Download PDF (717KB) ViewWe consider a weighted family of n parallelly transported hyperplanes in a k-dimensional affine space and describe the characteristic variety of the Gauss–Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the associated point in the Grassmannian Gr(k, n). The Laurent polynomials are in involution. These statements generalize (Varchenko, Mathematics 2:218-231, 2014), where such a description was obtained for a weighted generic family of parallelly transported hyperplanes. An intermediate object between the differential equations and the characteristic variety is the algebra of functions on the critical set of the associated master function. We construct a linear isomorphism between the vector space of the Gauss–Manin differential equations and the algebra of functions. The isomorphism allows us to describe the characteristic variety. It also allowed us to define an integral structure on the vector space of the algebra and the associated (combinatorial) connection on the family of such algebras. -
The Exponential Map Near Conjugate Points In 2D Hydrodynamics
G. Misiołek
Arnold Math J. (2015) 1:3, 243–251
Received: 14 January 2015 / Accepted: 3 May 2015 / Published online: 5 August 2015
Abstract
Download PDF (408KB) ViewWe prove that the weak-Riemannian exponential map of the \(L^2\) metric on the group of volume-preserving diffeomorphisms of a compact two-dimensional manifold is not injective in any neighbourhood of its conjugate vectors. This can be viewed as a hydrodynamical analogue of the classical result of Morse and Littauer. -
Homology of Spaces of Non-Resultant Homogeneous Polynomial Systems in ${\mathbb R}^2$
and ${\mathbb C}^2$
V. A. Vassiliev
Arnold Math J. (2015) 1:3, 233–242
Received: 7 November 2014 / Accepted: 3 June 2015 / Published online: 11 August 2015
Abstract
Download PDF (526KB) ViewThe resultant variety in the space of systems of homogeneous polynomials of some given degrees consists of such systems having non-trivial solutions. We calculate the integer cohomology groups of all spaces of non-resultant systems of polynomials \({\mathbb R}^2 \rightarrow {\mathbb R}\), and also the rational cohomology rings of spaces of non-resultant systems and non-m-discriminant polynomials in \({\mathbb C}^2\). -
Local Invariants of Framed Fronts in 3-Manifolds
V. Goryunov, S. Alsaeed
Arnold Math J. (2015) 1:3, 211–232
Received: 14 January 2015 / Accepted: 3 May 2015 / Published online: 5 August 2015
Abstract
Download PDF (821KB) ViewThe front invariants under consideration are those whose increments in generic homotopies are determined entirely by diffeomorphism types of local bifurcations of the fronts. Such invariants are dual to trivial codimension 1 cycles supported on the discriminant in the space of corresponding Legendrian maps. We describe the spaces of the discriminantal cycles (possibly non-trivial) for framed fronts in an arbitrary oriented 3-manifold, both for the integer and mod2 coefficients. For the majority of these cycles we find homotopy-independent interpretations which guarantee the triviality required. In particular, we show that all integer local invariants of Legendrian maps without corank 2 points are essentially exhausted by the numbers of points of isolated singularity types of the fronts. -
A Formula for the HOMFLY Polynomial of rational links
Sergei Duzhin, Mikhail Shkolnikov
Arnold Math J. (2015) 1:4, 345–359
Received: 10 November 2014 / Accepted: 7 April 2015 / Published online: 24 April 2015
Abstract
Download PDF (710KB) ViewWe give an explicit formula for the HOMFLY polynomial of a rational link (in particular, knot) in terms of a special continued fraction for the rational number that defines the given link [after this work was accomplished, the authors learned about a paper by Nakabo (J. Knot Theory Ramif 11(4):565-574, 2002) where a similar result was proved. However, Nakabo's formula is different from ours, and his proof is longer and less clear]. -
Abundance of 3-Planes on Real Projective Hypersurfaces
S. Finashin, V. Kharlamov
Arnold Math J. (2015) 1:3, 171–199
Received: 7 November 2014 / Accepted: 2 May 2015 / Published online: 2 June 2015
Abstract
Download PDF (613KB) ViewWe show that a generic real projective $n$-dimensional hypersurface of odd degree $d$, such that $4(n-2) ={{d+3}\choose3}$, contains ``many'' real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3\log d$, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle.
Erratum to: Abundance of 3-Planes on Real Projective Hypersurfaces
Arnold Math J. (2015) 1:3, 343
Published online: 31 July 2015
Abstract
Download PDF (304KB)When we published this article, there was a typo in the first line of Theorem 5.3.1. Please find the corrected text in the pdf. The publisher apologises for this mistake. -
On Local Weyl Equivalence of Higher Order Fuchsian Equations
Shira Tanny, Sergei Yakovenko
Arnold Math J. (2015) 1:2, 141–170
Received: 26 December 2014 / Accepted: 15 April 2015/ Published online: 08 May 2015
Abstract
Download PDF (653KB) ViewWe study the local classification of higher order Fuchsian linear differential equations under various refinements of the classical notion of the "type of differential equation" introduced by Frobenius. The main source of difficulties is the fact that there is no natural group action generating this classification. We establish a number of results on higher order equations which are similar but not completely parallel to the known results on local (holomorphic and meromorphic) gauge equivalence of systems of first order equations. -
On an Equivariant Version of the Zeta Function of a Transformation
S. M. Gusein-Zade, I. Luengo, A. Melle-Hernández
Arnold Math J. (2015) 1:2, 127–140
Received: 17 December 2014 / Accepted: 4 April 2015 / Published online: 28 April 2015
Abstract
Download PDF (473KB) ViewEarlier the authors offered an equivariant version of the classical monodromy zeta function of a G-invariant function germ with a finite group G as a power series with the coefficients from the Burnside ring of the group G tensored by the field of rational numbers. One of the main ingredients of the definition was the definition of the equivariant Lefschetz number of a G-equivariant transformation given by W. Lück and J. Rosenberg. Here we use another approach to a definition of the equivariant Lefschetz number of a transformation and describe the corresponding notions of the equivariant zeta function. This zeta-function is a power series with the coefficients from the Burnside ring itself. We give an A'Campo type formula for the equivariant monodromy zeta function of a function germ in terms of a resolution. Finally we discuss orbifold versions of the Lefschetz number and of the monodromy zeta function corresponding to the two equivariant ones. -
Vortex Dynamics of Oscillating Flows
V. A. Vladimirov, M. R. E. Proctor, D. W. Hughes
Arnold Math J. (2015) 1:2, 113–126
Received: 22 December 2014 / Accepted: 23 March 2015 / Published online: 10 April 2015
Abstract
Download PDF (341KB) ViewWe employ the method of multiple scales (two-timing) to analyse the vortex dynamics of inviscid, incompressible flows that oscillate in time. Consideration of distinguished limits for Euler's equation of hydrodynamics shows the existence of two main asymptotic models for the averaged flows: strong vortex dynamics (SVD) and weak vortex dynamics (WVD). In SVD the averaged vorticity is 'frozen' into the averaged velocity field. By contrast, in WVD the averaged vorticity is 'frozen' into the 'averaged velocity + drift'. The derivation of the WVD recovers the Craik-Leibovich equation in a systematic and quite general manner. We show that the averaged equations and boundary conditions lead to an energy-type integral, with implications for stability. -
Remarks on the Circumcenter of Mass
Serge Tabachnikov, Emmanuel Tsukerman
Arnold Math J. (2015) 1:2, 101–112
Received: 15 December 2014 / Accepted: 23 March 2015 / Published online: 31 March 2015
Abstract
Download PDF (235KB) ViewSuppose that to every non-degenerate simplex $\Delta\subset\mathbb R^n$ a 'center' $C(\Delta)$ is assigned so that the following assumptions hold:- The map $\Delta\to C(\Delta)$ commutes with similarities and is invariant under the permutations of the vertices of the simplex;
- The map $\Delta\to \operatorname{Vol}(\Delta)C(\Delta)$ is polynomial in the coordinates of the vertices of the simplex.
The motivation for this theorem comes from the recent study of the circumcenter of mass of simplicial polytopes by the authors and by A. Akopyan. -
Quadratic Cohomology
A. A. Agrachev
Arnold Math J. (2015) 1:1, 37–58
Received: 10 November 2014 / Accepted: 16 December 2014
Abstract
Download PDF (259KB) ViewWe study homological invariants of smooth families of real quadratic forms as a step towards a "Lagrange multipliers rule in the large" that intends to describe topology of smooth maps in terms of scalar Lagrange functions. -
Riemannian Geometry of the Contactomorphism Group
David G. Ebin, Stephen C. Preston
Arnold Math J. (2015) 1:1, 5–36
Received: 11 November 2014 / Accepted: 8 December 2014
Abstract
Download PDF (353KB) ViewGiven an odd-dimensional compact manifold and a contact form, we consider the group of contact transformations of the manifold (contactomorphisms) and the subgroup of those transformations that precisely preserve the contact form (quantomorphisms). If the manifold also has a Riemannian metric, we can consider the L2 inner product of vector fields on it, which by restriction gives an inner product on the tangent space at the identity of each of the groups that we consider. We then obtain right-invariant metrics on both the contactomorphism and quantomorphism groups. We show that the contactomorphism group has geodesics at least for short time and that the quantomorphism group is a totally geodesic subgroup of it. Furthermore we show that the geodesics in this smaller group exist globally. Our methodology is to use the right invariance to derive an "Euler-Arnold" equation from the geodesic equation and to show using ODE methods that it has solutions which depend smoothly on the initial conditions. For global existence we then derive a "quasi-Lipschitz" estimate on the stream function, which leads to a Beale-Kato-Majda criterion which is automatically satisfied for quantomorphisms. Special cases of these Euler-Arnold equations are the Camassa-Holm equation (when the manifold is one-dimensional) and the quasi-geostrophic equation in geophysics.
Open Problems
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Open Problems on Billiards and Geometric Optics
Misha Bialy, Corentin Fierobe, Alexey Glutsyuk, Mark Levi, Alexander Plakhov, Serge Tabachnikov
Arnold Mathematical Journal (2022)
Published: 17 January 2022
Abstract
Download PDF of the paper .This is a collection of problems composed by some participants of the workshop “Differential Geometry, Billiards, and Geometric Optics” that took place at CIRM on October 4–8, 2021. -
Fifty New Invariants of N-Periodics in the Elliptic Billiard
Dan Reznik, Ronaldo Garcia, Jair Koiller
Arnold Mathematical Journal (2021) 7:341-355
Published: 18 February 2021
Abstract
Download PDF of the paper .We introduce 50+ new invariants manifested by the dynamic geometry of N-periodics in the Elliptic Billiard, detected with an experimental/interactive toolbox. These involve sums, products and ratios of distances, areas, angles, etc. Though curious in their manifestation, said invariants do all depend upon the two fundamental conserved quantities in the Elliptic Billiard: perimeter and Joachimsthal’s constant. Several proofs have already been contributed (references are provided); these have mainly relied on algebraic geometry. We very much welcome new proofs and contributions. -
Billiard Trajectories in Regular Polygons and Geodesics on Regular Polyhedra
Dmitry Fuchs
Arnold Mathematical Journal (2021)
Published: 07 January 2021
Abstract
Download PDF of the paper .This article is devoted to the geometry of billiard trajectories in a regular polygon and geodesics on the surface of a regular polyhedron. Main results are formulated as conjectures based on ample computer experimentation. -
Conjectural Large Genus Asymptotics of Masur–Veech Volumes and of Area Siegel–Veech Constants of Strata of Quadratic Differentials
Amol Aggarwal, Vincent Delecroix, Élise Goujard, Peter Zograf, Anton Zorich
Arnold Mathematical Journal (2020) 6:2, 149-161
Published: 20 May 2020
Abstract
Download PDF of the paper .We state conjectures on the asymptotic behavior of the Masur–Veech volumes of strata in the moduli spaces of meromorphic quadratic differentials and on the asymptotics of their area Siegel–Veech constants as the genus tends to infinity. -
Algebraic Stories from One and from the Other Pockets
Ralf Fröberg, Samuel Lundqvist, Alessandro Oneto, Boris Shapiro
Arnold Mathematical Journal (2018) 4:2, 137–160
Received: 7 January 2018 / Revised: 31 May 2018 / Accepted: 18 July 2018 / Published online: 31 July 2018
Abstract
Download PDF of the paper (690KB) .In what follows, we present a large number of questions which were posed on the problem solving seminar in algebra at Stockholm University during the period Fall 2014 – Spring 2017 along with a number of results related to these problems. Many of the results were obtained by participants of the latter seminar. -
Open Problems on Configuration Spaces of Tensegrities
Oleg Karpenkov
Arnold Mathematical Journal (2018) 4:1, 19–25
Received: 7 January Received: 8 June 2017 / Revised: 13 January 2018 / Accepted: 22 January 2018
Abstract
Download PDF of the paper (399KB) . ViewIn this small paper we bring together some open problems related to the study of the configuration spaces of tensegrities, i.e. graphs with stresses on edges. These problems were announced in Doray et al. (Discrete Comput Geom 43:436-466, 2010), Karpenkov et al. (ARS Math Contemp 6:305-322, 2013), Karpenkov (The combinatorial geometry of stresses in frameworks. arXiv:1512.02563 [math.MG]), and Karpenkov (Geometric Conditions of Rigidity in Nongeneric settings, 2016 (by F. Doray, J. Schepers, B. Servatius, and the author), for more details we refer to the mentioned articles. -
Modular Periodicity of the Euler Numbers and a Sequence by Arnold
Sanjay Ramassamy
Arnold Math J. (2018) 3:4, 519–524
Received: 19 November 2017 / Accepted: 11 January 2018 / Published online: 22 January 2018
Abstract
Download PDF of the paper (395KB) . ViewFor any positive integer $q$, the sequence of the Euler up/down numbers reduced modulo $q$ was proved to be ultimately periodic by Knuth and Buckholtz. Based on computer simulations, we state for each value of $q$ precise conjectures for the minimal period and for the position at which the sequence starts being periodic. When $q$ is a power of $2$, a sequence defined by Arnold appears, and we formulate a conjecture for a simple computation of this sequence. -
The Number
$ \pi$ and a Summation by $ SL(2,{\mathbb{Z}})$
Nikita Kalinin, Mikhail Shkolnikov
Arnold Math J. (2018) 3:4, 511–517
Received: 7 October 2016 / Revised: 30 June 2017 / Accepted: 8 October 2017
Abstract
Download PDF of the paper (532KB) . ViewThe sum (resp. the sum of squares) of the defects in the triangle inequalities for the area one lattice parallelograms in the first quadrant has a surprisingly simple expression. Namely, let $ f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$. Then,\begin{align*}\tag{Ж} \sum f(a,b,c,d)^2 = 2-\pi/2,\label{eqspspi} \end{align*}\begin{align*}\tag{ж} \sum f(a,b,c,d) = 2,\label{eqspstwo} \end{align*}where the sum runs by all $ a,b,c,d\in{\mathbb{Z}}_{\geq 0}$ such that $ ad-bc=1$. We present a proof of these formulae and list several directions for the future studies.
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Open Problems and Conjectures Related to the Theory of
Mathematical Quasicrystals
Faustin Adiceam, David Damanik, Franz Gähler, Uwe Grimm, Alan Haynes, Antoine Julien, Andrés Navas, Lorenzo Sadun, Barak Weiss
Arnold Math J. (2016) 2:4, 579–592
Received: 15 January 2016 / Revised: 21 May 2016 / Accepted: 11 June 2016 / Published Online: 11 July 2016
Abstract
Download PDF (708KB) ViewThis list of problems arose as a collaborative effort among the participants of the Arbeitsgemeinschaft on Mathematical Quasicrystals, which was held at the Mathematisches Forschungsinstitut Oberwolfach in October 2015. The purpose of our meeting was to bring together researchers from a variety of disciplines, with a common goal of understanding different viewpoints and approaches surrounding the theory of mathematical quasicrystals. The problems below reflect this goal and this diversity and we hope that they will motivate further cross-disciplinary research and lead to new advances in our overall vision of this rapidly developing field. -
Betti Posets and the Stanley Depth
L. Katthän
Arnold Math J. (2016) 2:2, 267–276
Received: 9 October 2015 / Revised: 19 December 2015 / Accepted: 4 February 2016 / Published online: 15 January 2016
Abstract
Download PDF (456KB) ViewLet $S$ be a polynomial ring and let $I \subseteq S$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of $I$ determines the Stanley projective dimension of $S/I$ or $I$. Our main result is that this conjecture implies the Stanley conjecture for $I$, and it also implies that ${{\mathrm{sdepth}}}S/I \ge {{\mathrm{depth}}}S/I - 1$. Recently, Duval et al. (A non-partitionable Cohen-Macaulay simplicial complex, arXiv:1504.04279, 2015), found a counterexample to the Stanley conjecture, and their counterexample satisfies ${{\mathrm{sdepth}}}S/I = {{\mathrm{depth}}}S/I - 1$. So if our conjecture is true, then the conclusion is best possible. -
Volumes of Strata of Abelian Differentials and Siegel-Veech Constants in Large Genera
A. Eskin, A. Zorich
Arnold Math J. (2015) 1:4, 481–488
Received: 19 July 2015 / Revised: 16 September 2015 / Accepted: 20 October 2015 / Published online: 05 November 2015
Abstract
Download PDF (727KB) ViewWe state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures. -
Disconjugacy and the Secant Conjecture
A. Eremenko
Arnold Math J. (2015) 1:3, 339–342
Received: 5 July 2015 / Accepted: 28 July 2015 / Published online: 4 August 2015
Abstract
Download PDF (613KB) ViewWe discuss the so-called secant conjecture in real algebraic geometry, and show that it follows from another interesting conjecture, about disconjugacy of vector spaces of real polynomials in one variable. -
A Few Problems on Monodromy and Discriminants
V. A. Vassiliev
Arnold Math J. (2015) 1:2, 201–209
Received: 15 February 2015 / Accepted: 31 March 2015 / Published online: 16 April 2015
Abstract
Download PDF (405KB) ViewThe article contains several problems concerning local monodromy groups of singularities, Lyashko-Looijenga maps, integral geometry, and topology of spaces of real algebraic manifolds. -
Problems Around Polynomials: The Good, The Bad and The Ugly...
Boris Shapiro
Arnold Math J. (2015) 1:1, 91–99
Received: 7 November 2014 / Accepted: 16 March 2015 / Published online: 25 March 2015
Abstract
Download PDF (150KB) ViewThe Russian style of formulating mathematical problems means that nobody will be able to simplify your formulation as opposed to the French style which means that nobody will be able to generalize it, - Vladimir Arnold. -
Space of Smooth 1-Knots in a 4-Manifold: Is Its Algebraic Topology
Sensitive to Smooth Structures?
Oleg Viro
Arnold Math J. (2015) 1:1, 83–89
Received: 12 December 2014 / Accepted: 12 February 2015
Abstract
Download PDF (149KB) ViewWe discuss a possibility to get an invariant of a smooth structure on a closed simply connected 4-manifold from homotopy invariants of the space of loops smoothly embedded into the manifold. -
Classification of Finite Metric Spaces and Combinatorics of Convex Polytopes
A. M. Vershik
Arnold Math J. (2015) 1:1, 75–81
Received: 8 November 2014 / Accepted: 31 December 2014
Abstract
Download PDF (376KB) ViewWe describe the canonical correspondence between finite metric spaces and symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those polytopes. -
Periods of Pseudo-Integrable Billiards
Vladimir Dragović, Milena Radnović
Arnold Math J. (2015) 1:1, 69–73
Received: 10 November 2014 / Accepted: 26 December 2014
Abstract
Download PDF (175KB) ViewConsider billiard desks composed of two concentric half-circles connected with two edges. We examine billiard trajectories having a fixed circle concentric with the boundary semicircles as the caustic, such that the rotation numbers with respect to the half-circles are ρ1 and ρ2 respectively. Are such billiard trajectories periodic, and what are all possible periods for given ρ1 and ρ2? -
A Baker's Dozen of Problems
Serge Tabachnikov
Arnold Math J. (2015) 1:1, 59–67
Received: 16 September 2014 / Revised: 23 September 2014 / Accepted: 14 October 2014
Abstract
Download PDF (599KB) ViewThis article is a collection of open problems, with brief historical and bibliographical comments, somewhat in the spirit of the problem with which V. Arnold opened his famous seminar every semester and that were recently collected and published in a book form.
Research Expositions
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The Equivalence Problem in Analytic Dynamics for 1-Resonance
Christiane Rousseau
Arnold Mathematical Journal (2022)
Published: 20 January 2022
Abstract
Download PDF of the paper .When are two germs of analytic systems conjugate or orbitally equivalent under an analytic change of coordinates in a neighborhood of a singular point? The present paper, of a survey nature, presents a research program around this question. A way to answer is to use normal forms. However, there are large classes of dynamical systems for which the change of coordinates to a normal form diverges. In this paper, we discuss the case of singularities for which the normalizing transformation is k-summable, thus allowing to provide moduli spaces. We explain the common geometric features of these singularities, and show that the study of their unfoldings allows understanding both the singularities themselves, and the geometric obstructions to convergence of the normalizing transformations. We also present some moduli spaces for generic k-parameter families unfolding such singularities. -
Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey
Klas Modin, Milo Viviani
Arnold Mathematical Journal (2020) 7:3, 357-385
Published: 15 October 2020
Abstract
Download PDF of the paper .Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here, we give a unified framework for proving integrability results for $N=2$, 3, or 4 point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any two-dimensional manifold with a symmetry group; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in two-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix. -
Proof of van der Waerden's Theorem in Nine Figures
Ari Blondal, Veselin Jungić
Arnold Math J. (2018) 4:2, 161-168
Received: 15 July 2018 / Revised: 19 August 2018 / Accepted: 24 August 2018 / Published online: 3 September 2018
Abstract
Download PDF of the paper (1200KB) .This note contains a proof of van der Waerden's theorem, "one of the most elegant pieces of mathematics ever produced," in nine figures. The proof follows van der Waerden's original idea to establish the existence of what are now called van der Waerden numbers by using double induction. It also contains ideas and terminology introduced by I. Leader and T. Tao. -
Two-Valued Groups, Kummer Varieties, and Integrable
Billiards
V. M. Buchstaber, V. Dragović
Arnold Math J. (2018) 4:1, 27-57
Received: 10 July 2017 / Revised: 30 November 2017 / Accepted: 10 March 2018 / Published online: 9 April 2018
Abstract
Download PDF of the paper (600KB) .A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related to elliptic curves and integrable systems such as elliptic billiards and celebrated Kowalevski top. The present paper is devoted to the case of genus 2, to the investigation of algebraic two-valued group structures on Kummer varieties. One of our approaches is based on the theory of $\sigma$-functions. It enables us to study the dependence of parameters of the curves, including rational limits. Following this line, we are introducing a notion of $n$-groupoid as natural multivalued analogue of the notion of topological groupoid. Our second approach is geometric. It is based on a geometric approach to addition laws on hyperelliptic Jacobians and on a recent notion of billiard algebra. Especially important is connection with integrable billiard systems within confocal quadrics. The third approach is based on the realization of the Kummer variety in the framework of moduli of semi-stable bundles, after Narasimhan and Ramanan. This construction of the two-valued structure is remarkably similar to the historically first example of topological formal two-valued group from 1971, with a significant difference: the resulting bundles in the 1971 case were "virtual", while in the present case the resulting bundles are effectively realizable. -
Proof of the
Broué - Malle - Rouquier Conjecture in Characteristic Zero (After I.
Losev and I. Marin - G. Pfeiffer)
Pavel Etingof
Arnold Math J. (2017) 3:3, 445–449
Received: 3 March 2017 / Revised: 11 March 2017 / Accepted: 4 April 2017
Abstract
Download PDF of the paper (405KB) . ViewWe explain a proof of the Broué – Malle – Rouquier conjecture on Hecke algebras of complex reflection groups, stating that the Hecke algebra of a finite complex reflection group $ W$ is free of rank $ |W|$ over the algebra of parameters, over a field of characteristic zero. This is based on previous work of Losev, Marin – Pfeiffer, and Rains and the author. -
Flows in Flatland: A Romance of Few Dimensions
Gabriel Katz
Arnold Math J. (2017) 3:2, 281–317
Received: 2 March 2016 / Revised: 15 October 2016 / Accepted: 23 October 2016
Abstract
Download PDF of the paper (1007KB) . ViewThis paper is about gradient-like vector fields and flows they generate on smooth compact surfaces with boundary. We use this particular 2-dimensional setting to present and explain our general results about non-vanishing gradient-like vector fields on $n$-dimensional manifolds with boundary. We take advantage of the relative simplicity of 2-dimensional worlds to popularize our approach to the Morse theory on smooth manifolds with boundary. In this approach, the boundary effects take the central stage. -
Some Recent Generalizations of the Classical Rigid Body
Systems
Vladimir Dragović, Borislav Gajić
Arnold Math J. (2016) 2:4, 511–578
Received: 20 November 2014 / Revised: 13 July 2016 / Accepted: 25 August 2016 / Published online: 19 September 2016
Abstract
Download PDF of the paper (1007KB) . ViewSome recent generalizations of the classical rigid body systems are reviewed. The cases presented include dynamics of a heavy rigid body fixed at a point in three-dimensional space, the Kirchhoff equations of motion of a rigid body in an ideal incompressible fluid as well as their higher-dimensional generalizations. -
Building Thermodynamics for Non-uniformly Hyperbolic Maps
Vaughn Climenhaga, Yakov Pesin
Arnold Math Journal (2017) 3:1, 37-82
Research Exposition, Received: 4 February 2016 / Accepted: 20 July 2016 / Published online: 09 August 2016
Abstract
Download PDF of the paper (955KB) . ViewWe briefly survey the theory of thermodynamic formalism for uniformly hyperbolic systems, and then describe several recent approaches to the problem of extending this theory to non-uniform hyperbolicity. The first of these approaches involves Markov models such as Young towers, countable-state Markov shifts, and inducing schemes. The other two are less fully developed but have seen significant progress in the last few years: these involve coarse-graining techniques (expansivity and specification) and geometric arguments involving push-forward of densities on admissible manifolds. -
Kepler's Laws and Conic Sections
A. Givental
Arnold Math Journal (2016) 2:1, 139–148
Received: 5 July 2015 / Revised: 7 September 2015 / Accepted: 24 October 2015 / Published online: 23 December 2015
Abstract
Download PDF (713KB) ViewThe geometry of Kepler's problem is elucidated by lifting the motion from the (x, y)-plane to the cone \(r^2=x^2+y^2\). -
The Conley Conjecture and Beyond
V. L. Ginzburg, B. Z. Gürel
Arnold Math J. (2015) 1:3, 299–337
Received: 25 November 2014 / Accepted: 19 May 2015 / Published online: 4 June 2015
Abstract
Download PDF (882KB) ViewThis is (mainly) a survey of recent results on the problem of the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb flows. We focus on the Conley conjecture, proved for a broad class of closed symplectic manifolds, asserting that under some natural conditions on the manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic orbits. We discuss in detail the established cases of the conjecture and related results including an analog of the conjecture for Reeb flows, the cases where the conjecture is known to fail, the question of the generic existence of infinitely many periodic orbits, and local geometrical conditions that force the existence of infinitely many periodic orbits. We also show how a recently established variant of the Conley conjecture for Reeb flows can be applied to prove the existence of infinitely many periodic orbits of a low-energy charge in a non-vanishing magnetic field on a surface other than a sphere.
Personalia
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Foreword to the Special Issue Dedicated to Misha Lyubich
Anna Miriam Benini, Tanya Firsova, Scott Sutherland, Michael Yampolsky
Arnold Mathematical Journal (2020) 6:3, 311-312
Published: 11 November 2020
Download PDF of the paper . -
Foreword to the Special Issue Dedicated to Rafail Kalmanovich Gordin
Vladimir Dotsenko, Alexander Shen, Mark Spivakovsky
Arnold Mathematical Journal (2019) 5:1, 1-4
Published: 10 July 2019
Download PDF of the paper .
Journal Description
This journal intends to present mathematics so that it would be understandable and interesting to mathematicians independently on their narrow research fields. We invite articles exercising all formal and informal approaches to "unhide" the process of mathematical discovery.
The name of the journal is not only a dedication to the memory of Vladimir Igorevich Arnold (1937-2010), one of the most influential mathematicians of the twentieth century, but also a declaration that the journal hopes to maintain and promote the style which makes the best mathematical works by Arnold so enjoyable and which Arnold implemented in the journals where he was an editor-in-chief.
The ArMJ is organized jointly by the Institute for Mathematical Sciences (IMS) at Stony Brook, USA, and Springer Verlag, Germany.
1. Objectives
The journal intends to publish interesting and understandable results in all areas of Mathematics. The following are the most desirable features of publications that will serve as selection criteria:
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The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions that are necessary for understanding must be provided but also informal motivations even if they are well-known to the experts in the field. If a general statement is given, then the simplest examples of it are also welcome.
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Interdisciplinary and multidisciplinary mathematics
We would like to have many research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, except for the most popular combinations such as algebraic geometry and mathematical physics, analysis and dynamical systems, algebra and combinatorics, and the like. For this reason, this kind of research is often under-represented in specialized mathematical journals. The ArMJ will try to compensate for this.
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Problems, objectives, work in progress
Most scholarly publications present results of a research project in their "final" form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned but the very process of mathematical discovery remains hidden. Following Arnold, we will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. The journal intends to publish well-motivated research problems on a rather regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold's principle, a general formulation is less desirable than the simplest partial case that is still unknown.
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The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author's responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author's understanding of the overall picture is presented; however, these parts must be clearly indicated. Including motivations, informal parts, descriptions of other lines of research, possibly conducted by other mathematicians, should serve this principal objective: being interesting.
1.1 Types of Journal Articles
- Research contribution.
This is the classical format: a short (usually up to 20 pages) account of a research project containing original results and complete proofs of them. However, all of the above applies. Contributions containing very technical arguments may not be suitable for the ArMJ.
- Research exposition.
This is an exposition of a broad mathematical subject containing a description of recent results (proofs may be included or omitted), historical overview, motivations, open problems. A research exposition may take 60 pages or more.
- Problem contribution.
This is a description of an open problem. The problem must be well-motivated, illustrated by examples, and the importance of the problem must be explained. Alternatively, and closer to the original style of Arnold, a problem contribution may consist of a set of several problems that take very short space to state. Problems do not need to be original, however, the authorship must be carefully acknowledged. A problem contribution is meant to be short (normally, up to 4 pages, but exceptions are possible).
1.2 Comparison with Existing Journals
We feel that the following journals have objectives somewhat similar to those of the ArMJ.
- Functional Analysis and its Applications
- Russian Mathematical Surveys
- American Mathematical Monthly
- Bulletin of the AMS
However, each of these journals complies with only a part of our objectives list.
1.3 Why the Name
There are many great mathematicians of the twentieth century. The choice of the name may look random (why not, say, "Gelfand Mathematical Journal"? - we are often asked) but we have very specific reasons for using the name of Vladimir Arnold.
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The principles, according to which the journal operates, are most accurately associated with Vladimir Arnold. He had been actively promoting these or similar principles.
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For many years, V. Arnold had been the Editor-in-Chief of the journal Functional Analysis and its Applications (FAA). In 2006, V. Arnold launched a new journal, Functional Analysis and Other Mathematics (FAOM). The initial composition of the ArMJ Editorial Board consists mostly of former editors of the FAOM.
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Despite the close connections with the FAA and the FAOM, we decided to avoid mentioning "Functional Analysis" in the name of the journal. These names have appeared historically, and have nothing to do with scientific principles of the journals. More than that, the names are even confusing: not all mathematicians could guess that, say, Functional Analysis and its Applications welcomes papers in all areas of mathematics, including algebra and number theory. On the other hand, we wanted to have an indication of these connections in the name of the journal. The name of Vladimir Arnold serves as this indication.
2. Submissions
The journal is published quarterly, every issue consists of 100-150 pages. Manuscripts should be submitted online at http://www.editorialmanager.com/armj. Accepted file formats are LaTeX source (preferred) and MS Word.
Submission of a manuscript implies: that the work described has not been published before; that it is not under consideration for publication anywhere else; that its publication has been approved by all co-authors, if any, as well as by the responsible authorities - tacitly or explicitly - at the institute where the work has been carried out.
Authors wishing to include figures, tables, or text passages that have already been published elsewhere are required to obtain permission from the copyright owner(s) for both the print and online format and to include evidence that such permission has been granted when submitting their papers. Any material received without such evidence will be assumed to originate from the authors.
Editors
Editor-in-Chief:
Askold Khovanskii,
Toronto
e-mail: askold@math.toronto.edu
Managing Editor:
Vladlen Timorin, Moscow
e-mail: vtimorin@hotmail.com
Andrei Agrachev, Trieste
e-mail: agrachevaa@gmail.com
Edward Bierstone, Toronto
e-mail: bierston@math.toronto.edu
Gal Binyamini, The Weizmann Institute of Science, Israel
e-mail:
gal.binyamini@weizmann.ac.il
Felix Chernous'ko, Moscow
e-mail: chern@ipmnet.ru
David Eisenbud, Berkeley
e-mail: de@msri.org
Uriel Frisch, Nice
e-mail: uriel@oca.eu; uriel@obs-nice.fr
Dmitry Fuchs, UC Davis, CA, USA
e-mail: fuchs@math.ucdavis.edu
Alexander Givental, Berkeley
e-mail: givental@math.berkeley.edu
Victor Goryunov, Liverpool
e-mail: Victor.Goryunov@liverpool.ac.uk
Sandro Graffi, Bologna
e-mail: graffi@dm.unibo.it
Sabir Gusein-Zade, Moscow
e-mail: sabirg@list.ru
Yulij Ilyashenko, Moscow and Cornell
e-mail: yulijs@gmail.com
Oleg Karpenkov, Liverpool
e-mail: O.Karpenkov@liverpool.ac.uk
Sergei Kuksin, Paris
e-mail: kuksin@gmail.com
Anatoly Neishtadt, Loughborough
e-mail: A.Neishtadt@lboro.ac.uk
Michael Shubin, Boston
e-mail: m.shubin@neu.edu
Alexander Varchenko, Chapel Hill
e-mail: anv@email.unc.edu
Oleg Viro, Stony Brook
e-mail: oleg.viro@gmail.com
Eduard Zehnder, Zurich
e-mail: eduard.zehnder@math.ethz.ch