Research Contribution


Received: 10 March 2024; Accepted 26 September 2024.


On intersection of lemniscates of rational functions

Stepan Orevkov IMT, l'université Paul Sabatier, 119 route de Narbonne, 31062 Toulouse, France;
Steklov Mathematical Institute, ul. Gubkina 8, Moscow 119991, Russia;
email: orevkov@math.ups-tlse.fr× and Fedor Pakovich Department of Mathematics, Ben Gurion University of the Negev,
P.O.B. 653, Beer Sheva, 8410501, Israel;
email: pakovich@math.bgu.ac.il;
This research was supported by ISF Grant No. 1092/22.×



Abstract

For a non-constant complex rational function \(P\), the lemniscate of \(P\) is defined as the set of points \(z\in \C \) such that \(\vert P(z)\vert =1\). The lemniscate of \(P\) coincides with the set of real points of the algebraic curve given by the equation \(L_P(x,y)=0\), where \(L_P(x,y)\) is the numerator of the rational function \(P(x+iy)\overline { P}(x-iy)-1.\) In this paper, we study the following two questions: under what conditions two lemniscates have a common component, and under what conditions the algebraic curve \(L_P(x,y)=0\) is irreducible. In particular, we provide a sharp bound for the number of complex solutions of the system \(\vert P_1(z)\vert =\vert P_2(z)\vert =1\), where \(P_1\) and \(P_2\) are rational functions.

1. Introduction

Let \(P\) be a non-constant complex rational function on the Riemann sphere \(\bC \). The lemniscate of \(P\) is defined as

\begin{equation} \f L_P=\{z\in \C : \, \vert P(z)\vert =1\}. \end{equation}

The paper is devoted to two more or less independent problems concerning the lemniscates. The first problem is to determine the maximal number of intersection points of \(\f L_{P_1}\) and \(\f L_{P_2}\), i.e., the number of solutions of the system of equations

\begin{equation} \label {sys1} \vert P_1(z)\vert =\vert P_2(z)\vert =1, \end{equation}

where \(P_1\) and \(P_2\) are rational functions of given degrees \(n_1\) and \(n_2\). The second problem is to find out when a lemniscate (considered as a real algebraic curve in \(\R ^2\)) is irreducible.

Our result about the intersection of lemniscates is the following statement.

Theorem 1.1.
Let \(P_1\) and \(P_2\) be non-constant complex rational functions of degrees \(n_1\) and \(n_2\). The following three conditions are equivalent:

  1. (i) \({\f L}_{P_1}\) and \({\f L}_{P_2}\) have more than \(2n_1 n_2\) common points.
  2. (ii) \({\f L}_{P_1}\cap {\f L}_{P_2}\) is infinite.
  3. (iii) \(P_1=B_1\circ W\) and \(P_2=B_2\circ W\) for some rational functions \(W\), \(B_1\), \(B_2\) such that each of \(B_1,B_2\) maps the unit circle \(\T \) to itself.

Furthermore, for any natural \(n_1\) and \(n_2\) there exist rational functions of degrees \(n_1\) and \(n_2\) such that \(\f L_{P_1}\) and \(\f L_{P_2}\) have exactly \(2n_1n_2\) intersection points.

We remark that for a rational function \(B\), the condition \(B(\T )\subset \T \) can be written in the form

\begin{equation} \label {eq.bla} B(z)\overline {B}(1/z)=1 \qquad \text {for all $z\in \bC $.} \end{equation}

Such functions are known in the complex analysis under the name of quotients of finite Blaschke products.

In a recent paper [13] , the equivalence of conditions \(\rm (i), \rm (ii), \rm (iii)\) was proven with a weaker bound \((n_1+n_2)^2\) instead of the bound \(2n_1n_2\). Notice that the result of [13] implies the main result of the paper [1] by Ailon and Rudnick, which is equivalent to the following statement: if \(P_1\) and \(P_2\) are complex polynomials, then

\begin{equation} \#\bigcup _{k=1}^\infty \{z \in \C ~:~P_1(z)^k = P_2(z)^k =1\} \le C(P_1,P_2), \end{equation}

for some constant \(C(P_1,P_2)\) that depends only on \(P_1\) and \(P_2\), unless for some non-zero integers \(m_1\) and \(m_2\) the equality

\begin{equation} \label {PP} P_1^{m_1}(z)P_2^{m_2}(z)=1 \end{equation}

holds (see Section 2.4 below).

In addition, the result of [13] answers the question of Corvaja, Masser, and Zannier ( [5] ) about the intersection of an irreducible curve \(\cC \) in \(\C ^*\times \C ^*\) with \(\T \times \T \), in case \(\cC \) has genus zero and is parametrized by rational functions \(P_1,\) \(P_2\) (see [13] for more details). These applications of [13] stem from the fact that the numbers \(C(P_1,P_2)\) and \(\#(\cC \cap \T \times \T )\) obviously are bounded from above by the number of solutions of ( 2 ). Thus, a sharp bound for the last number is of great interest, and our Theorem  1.1 provides it.

In brief, our proof of Theorem  1.1 given in Section 2 goes as follows. If \(P\) is a complex rational function of degree \(n\), then under the standard identification of \(\C \) with \(\R ^2\) the lemniscate \(\f L_P\) coincides with the set of real points of the affine algebraic curve of degree \(2n\) given by the equation \(L_P(x,y)=0\), where \(L_P(x,y)\) is the numerator of the rational function

\begin{equation} \label {eq.L} P(x+iy)\overline { P}(x-iy)-1. \end{equation}

After the linear change of variables \(z=x+iy\), \(w=x-iy\) (in \(\C ^2\)) the Newton polygon of \(L_P\) becomes the square \(n\times n\). Thus, the Bézout Theorem for bihomogeneous polynomials ( [14], §4.2.1 ), which is also a simplest case of the Bernstein-Kushnirenko Theorem [2] , implies that if (i) holds, then \(L_{P_1}\) and \(L_{P_2}\) have a common factor, i.e., the system \(L_{P_1}=L_{P_2}=0\) has infinitely many complex solutions. This is not yet (ii), but we prove a kind of “real” version of the Bézout theorem (Proposition 2.3 ), which implies in particular that if (i) holds, then the system \(L_{P_1}=L_{P_2}=0\) has infinitely many real solutions (Corollary 2.4 (c)). This gives us the implication (i) \(\Rightarrow \) (ii). In turn, the implication (ii) \(\Rightarrow \) (iii) is deduced from a “rational” version of the Cartwrite Theorem [3] (Corollary  2.2 ). Finally, since (iii) implies that \(\f L_W\subset \f L_{P_1}\cap \f L_{P_2}\), the implication (iii) \(\Rightarrow \) (i) is obvious.

Notice that the equivalence of conditions \(\rm (i), \rm (ii), \rm (iii)\) can be proven also by modifying the approach of the paper [13] (see Section 2.3 ).

Another problem considered in this paper (in Section  3 ) is the following. Given a rational function \(P\), under what conditions the curve \(L_{P}(x,y)=0\) is irreducible over \(\C \)? It is not hard to see that the following “composition condition” is sufficient for the reducibility: there exists a quotient of finite Blaschke products \(B\) of degree at least 2 and a rational function \(W\) such that \(P=B\circ W\). Notice that if \(P\) is a polynomial, the composition condition reduces to the condition that \(P=W^k\) for some polynomial \(W\) and \(k\ge 2\). Its necessity for the reducibility of \(L_P(x,y)\) in the polynomial case was established by the first author in [12] (notice that this result has found applications to complex dynamics, see [11] ). In Section  3.2 , we show however that for rational \(P\) the reducibility of \(L_P(x,y)=0\) does not imply in general the composition condition.

Our approach to the problem of irreducibility of algebraic curves \(L_P(x,y)=0\) is based on the observation that a change of variables allows us to consider this problem in the context of the more general problem of irreducibility of “separated variables" curves \(P(x)-Q(y)=0\), where \(P(z)\) and \(Q(z)\) are rational functions. In particular, we deduce our examples of reducible \(L_P(x,y)\) for \(P\) not satisfying the composition condition from examples of reducible separated variables curves found in [4] . On the other hand, modifying the arguments from [12] , we provide a handy sufficient condition for the irreducibility of separated variables curves in case one of rational functions \(P(z)\) and \(Q(z)\) is a polynomial (Theorem  3.5 ).

2. Intersection of lemniscates

2.1. The Cartwright theorem.

For a simple closed curve \(\Gamma \) in \(\C \), we denote by \(\fM _{\Gamma }\) the set of all non-constant functions meromorphic on \(\C \) having modulus one on \(\Gamma \). The following result was proved by Cartwright in [3] .

Theorem 2.1.
Let \(\Gamma \) be a simple closed curve in \(\C \) such that \(\fM _{\Gamma }\) is nonempty. Then there exists a function \(\varphi \in \fM _{\Gamma }\) such that each function in \(\fM _{\Gamma }\) may be written in the form \(f=B\circ \varphi \), where \(B\) is a quotient of finite Blaschke products.

A detailed discussion and generalizations of the Cartwright theorem can be found in the papers [15] , [16] . Below we need the following specialization of the Cartwright theorem, where the notation \(\fR _{\Gamma }\) stands for the set of all non-constant complex rational functions on \(\C \) having modulus one on \(\Gamma \).

Corollary 2.2.
Let \(\Gamma \) be a simple closed curve in \(\C \) such that \(\fR _{\Gamma }\) is nonempty. Then there exists a rational function \(W\in \fR _{\Gamma }\) such that each function in \(\fR _{\Gamma }\) may be written in the form \(P=B\circ W\), where \(B\) is a quotient of finite Blaschke products.

Proof. Applying Theorem 2.1 , we conclude that there exists a meromorphic function \(W\in \fM _{\Gamma }\) such that if \(P\in \fR _{\Gamma }\), then \(P=B\circ W\) for some quotient of finite Blaschke products \(B\). However, since the great Picard theorem implies that any non-rational function \(W\) meromorphic on \(\C \) takes all but at most two values in \(\mathbb C\mathbb P^1\) infinitely often, this equality implies that \(W\) is rational.

2.2. A real version of the Bézout Theorem.

Let us recall that the classical Bézout theorem about intersections of curves in \(\C \P ^2\) implies that the number of intersection points of two affine algebraic curves \(F(x,y)=0\) and \(G(x,y)=0\) of degrees \(m\) and \(n\) in \(\C \times \C \) does not exceed \(mn\), unless the polynomials \(F(x,y)\) and \(G(x,y)\) have a non-constant common factor in \(\C [x,y]\). In case the bidegrees \((m_1,m_2)\) and \((n_1,n_2)\) of \(F(x,y)\) and \(G(x,y)\) are relatively small with respect to \(m\) and \(n\), a better bound can be obtained from the bihomogeneous Bézout theorem, which implies that the number of intersection points of \(F(x,y)=0\) and \(G(x,y)=0\) does not exceed \(n_1m_2+n_2m_1\), unless \(F(x,y)\) and \(G(x,y)\) have a non-constant common factor (see [14], §4.2.1 ). In the proof of Theorem 1.1 , we will use a real version of this last bound provided by Corollary  2.4 (c) below.

Let \(X\) be a compact non-singular complex algebraic (or analytic) variety. A real structure on \(X\) is an anti-holomorphic involution \(\sigma :X\to X\). We denote the variety \(X\) endowed with the real structure \(\sigma \) by \(X^\sigma \). A point \(p\) of \(X^\sigma \) is called real if \(\sigma (p)=p\). The set of real points of \(X^\sigma \) (the real locus of \(X^\sigma \)) is denoted by \(\R X^\sigma \). A basic example is a projective variety in \(\C \P ^n\) defined by polynomial equations with real coefficients endowed with the involution of complex conjugation. In this case \(\R X^\sigma \) is the subset of \(\R \P ^n\) defined by the same equations.

If \(X=\C \P ^1\times \C \P ^1\), then there are two non-isomorphic real structures on \(X\):

\[ \sigma _h:(x,y)\mapsto (\bar x,\bar y), \qquad \sigma _e:(x,y)\mapsto (\bar y,\bar x). \]

The real loci \(\R X^{\sigma _h}\) and \(\R X^{\sigma _e}\) are, respectively, \(\R \P ^1\times \R \P ^1\) (a torus) and the image of \(\C \P ^1\) in \(X\) under the embedding \(z\mapsto (z,\bar z)\) (a sphere). Note that \(X^{\sigma _h}\) and \(X^{\sigma _e}\) are isomorphic to the complexifications of, respectively, hyperboloid and ellipsoid in \(\C \P ^3\) endowed with the usual complex conjugation:

\[ X^{\sigma _h}\cong \{z\mid z_0^2+z_1^2=z_2^2+z_3^2\}, \qquad X^{\sigma _e}\cong \{z\mid z_0^2=z_1^2+z_2^2+z_3^2\}; \]

here \(z\) stands for \((z_0:z_1:z_2:z_3)\). Indeed, it is straightforward to check that the following mapping \(X\to \C \P ^3\) defines the required isomorphism:

\[ (x,y)\mapsto (x_0y_0+x_1y_1:\alpha (x_0y_1-x_1y_0):x_0y_0-x_1y_1:x_0y_1+x_1y_0), \]

where \(\alpha =1\) for \(\sigma _h\) and \(\alpha =i\) for \(\sigma _e\); here \(x\) and \(y\) stand for \((x_0:x_1)\) and \((y_0:y_1)\).

Let us recall that if \(C\) is an irreducible curve on a smooth compact algebraic surface \(X\), then by the genus formula (see e.g.  [14], §4.4.1 ) the number of singular points of \(C\) is bounded from above by its arithmetic genus \(p_a(C)\), which in this case can be computed by the formula

\[ 2p_a(C) = 2 + C\cdot (C+K_X), \]

where \(K_X\) is the canonical class of \(X\).

Proposition 2.3.
Let \(X\) be a smooth compact complex algebraic surface endowed with a real structure \(\sigma \). Let \(A\) and \(B\) be \(\sigma \)-invariant algebraic curves on \(X\). Suppose that \(\R A\cap \R B\) is finite and \(C^2\ge p_a(C)\) for each irreducible component \(C\) of \(A\cup B\). Then

\begin{equation} \#(\R A\cap \R B) \le A\cdot B. \end{equation}

Proof.

We say that a \(\sigma \)-invariant curve \(C\) on \(X\) is \(\sigma \)-irreducible if \(C=D\cup \sigma (D)\) where \(D\) is irreducible.

Let us first consider the case when \(A\) and \(B\) are \(\sigma \)-irreducible. If \(A\ne B\), then \(A\) and \(B\) have no common components, and hence

\[ \#(\R A\cap \R B) \le \#(A\cap B) \le A\cdot B. \]

Suppose now that \(A=B\). Let us show that \(\R A\subset \Sing (A)\), where \(\Sing (A)\) is the set of singular points of \(A\). Indeed, for any \(p\in \R A\) one can choose local coordinates \((z,w)\) in a neighborhood of \(p\) such that \(\R X=\{\Im z=\Im w=0\}\) and \(A=\{f(z,w)=0\}\), where \(f\) is a polynomial with real coefficients. Thus, if \(A\) were non-singular at \(p\), then \(\R A\) would be infinite by the Implicit Function Theorem, which contradicts our hypothesis that \(\R A\cap \R B\) is finite. Thus, \(\R A\subset \Sing (A)\).

It follows that if \(A=B\) is irreducible (in the usual sense), we have

\[ \#(\R A\cap \R B)=\#\R A \le \#\Sing (A) \le p_a(A) \le A^2 = A\cdot B. \]

On the other hand, if \(A=B\) is reducible, then \(A=C\cup \sigma (C)\) where \(C\) is irreducible and \(\sigma (C)\ne C\). Therefore, we have \(\R A\subset C\cap \sigma (C)\), and hence

\[ \#(\R A\cap \R B) = \#\R A \le C\cdot \sigma (C) \le C^2+\sigma (C)^2+2C\cdot \sigma (C) = A^2 = A\cdot B. \]

This completes the proof in the case where \(A\) and \(B\) are \(\sigma \)-irreducible.

Now we proceed to the general case. Let \(A=A_1\cup \dots \cup A_k\) and \(B=B_1\cup \dots \cup B_l\) where each \(A_i\) and each \(B_j\) is \(\sigma \)-invariant and \(\sigma \)-irreducible. Then

\[ \#(\R A\cap \R B) \le \sum _{i,j}\#(\R A_i\cap \R B_j) \le \sum _{i,j} A_i\cdot B_j = A\cdot B. \]


Corollary 2.4.
(a). Let \(F(x_1,x_2)\) and \(G(x_1,x_2)\) be polynomials with real coefficients of degrees \(m\) and \(n\). Then the number of real solutions of the system \(F=G=0\) is either infinite or bounded above by \(mn\).

(b). Let \(F(x_1,x_2)\) and \(G(x_1,x_2)\) be polynomials with real coefficients such that \(\deg _{x_k} F=m_k\) and \(\deg _{x_k} G =n_k\), \(k=1,2\). Then the number of real solutions of the system \(F=G=0\) is either infinite or bounded above by \(m_1n_2+m_2n_1\).

(c). Let \(F(z,w)\) and \(G(z,w)\) be polynomials with complex coefficients such that \(F(z,w) = \overline {F}(w,z)\) and \(G(z,w) = \overline {G}(w,z)\). Let \(\deg _z F=m\) and \(\deg _z G=n\). Then the number of solutions of the system \(F=G=0\) belonging to the set \(\{(z,w)\mid w=\bar z\}\) is either infinite or bounded above by \(2mn\).

Proof. (a). We apply Proposition  2.3 to the curves in \(\C \P ^2\) (with the standard real structure) defined by the homogeneous equations

\[ x_0^m F(x_1/x_0,x_2/x_0) = 0, \qquad x_0^n G(x_1/x_0,x_2/x_0) = 0, \]

and observe that, for a curve \(C\) of degree \(d\) in \(\C \P ^2\), we have (see [14], §4.2.3 )

\[ p_a(C) = (d-1)(d-2)/2 \le d^2 = C^2. \]

(b). We apply Proposition  2.3 to \(X=\C \P ^1\times \C \P ^1\) endowed with the real structure \(\sigma _h\) (see above) and to the curves defined by the bihomogeneous equations

\[ u_1^{m_1}u_2^{m_2} F(x_1/u_1,x_2/u_2) = 0, \qquad u_1^{n_1}u_2^{n_2} G(x_1/u_1,x_2/u_2) = 0. \]

In this case, for a curve \(C\) of bidegree \((d_1,d_2)\) in \(X\), we have (see [14], §4.2.3 )

\[ p_a(C) = (d_1-1)(d_2-1) \le 2d_1d_2 = C^2. \]

(c). The same proof as in (b), but with \(\sigma _e\) instead of \(\sigma _h\). (Note that our hypothesis about \(F\) and \(G\) implies that their bidegrees are \((m,m)\) and \((n,n)\) respectively.)

Notice that higher-dimensional analogs of Corollary  2.4 do not hold. Indeed, let \(F_1=P(x)+P(y)\), \(F_2=F_3=z\), where \(P(x)=\prod _{k=1}^n(x-k)^2\), \(n\ge 3\) (see [8], Example 13.6 ). Then the number of real solutions of the system of equations \(F_1=F_2=F_3=0\) is finite but greater than \(\prod \deg F_i\).

2.3. Proof of Theorem 1.1.

The implication (i) \(\Rightarrow \) (ii) immediately follows from Corollary  2.4 (c), because the embedding \(\C \to \C ^2\), \(z\mapsto (z,\bar z)\) identifies \(\f L_{P_k}\) with

\[ \{(z,w)\,:\,P_k(z)\overline P_k(w)=1\}\,\cap \, \{(z,w)\,:\,w=\bar z\}, \quad k=1,2. \]

The implication (ii) \(\Rightarrow \) (iii) follows from Corollary  2.2 . Indeed, suppose that the intersection \(\f L_{P_1}\cap \f L_{P_2}\) is infinite. Then the complexifications of \(\f L_{P_1}\) and \(\f L_{P_2}\) have a common real component \(A\) whose real locus \(\R A\) is also infinite. By composing \(P_1\) and \(P_2\) with a Möbius transformation, we may assume that \(|P_j(\infty )|\ne 1\), \(j=1,2\), so that \(\R A\) is compact. Recall that the real locus of a real algebraic curve in neighborhood of every its non-isolated point is homeomorphic to a “star” with an even number of branches (see e.g.  [9] , p. 104). Thus, the set of non-isolated points of \(\R A\) is homeomorphic to a graph in \(\R ^2\), all of whose vertices have even valency. Such a graph necessarily have a cycle \(D\), and, by construction, the rational functions \(P_1\) and \(P_2\) have modulus one on \(D\). Applying now Corollary  2.2 to \(D\), we obtain (iii).

To prove (iii) \(\Rightarrow \) (i), it is enough to observe that if \(B\) is a quotient of finite Blaschke products, then \(B^{-1}(\T )\) contains \(\T \). Thus, (iii) implies that both \(\f L_{P_1}\) and \(\f L_{P_2}\) contain the infinite set \(\f L_W\) as a subset.

To prove the last part of the theorem, let us fix a Möbius transformation \(\nu \) that maps the real line to the unit circle, and observe that the lemniscate of \(P_1=\nu \circ z^{n_1}\) is a union of \(n_1\) lines on \(\C \) passing through the origin. Under the identification of \(\bC \) with \(S^2\) via the stereographic projection, \(\f L_{P_1}\) becomes a union of \(n_1\) big circles passing through two antipodal points \(a_1,b_1\). Let \(\delta \) be a Möbius transformation of \(\bC \) corresponding to an isometry of \(S^2\), and \(P_2=\nu \circ z^{n_2}\circ \delta \). Then \(\f L_{P_2}\) is a union of \(n_2\) big circles passing through two antipodal points \(a_2,b_2\). Any two distinct big circles intersect at two points. Hence, if \(\delta \) is chosen generically (so that \(\{a_1,b_1\}\cap \f L_{P_2}=\{a_2,b_2\}\cap \f L_{P_1}=\varnothing \)), then \(\#(\f L_{P_1}\cap \f L_{P_2})=2n_1 n_2\).

Note that Theorem  1.1 can be proved somewhat shorter using the approach of the paper [13] , where instead of the intersection \(\f L_{P_1}\cap \f L_{P_2}\) an algebraic curve \(\cC \) parametrized by \(P_1\) and \(P_2\) is considered. Under this approach, the proof of Theorem  1.1 reduces to proving that if an algebraic curve \(\cC \) parametrized by rational functions \(P_1\) and \(P_2\) of degrees \(n_1\) and \(n_2\) has more than \(2n_1n_2\) points whose coordinates have modulus one, then it can be parametrized by some quotients of finite Blaschke products. On the other hand, considering instead of the functions \(P_1\), \(P_2\) the functions

\[\widehat P_1=T\circ P_1\circ T^{-1}, \ \ \ \ \widehat P_2=T\circ P_2\circ T^{-1},\]

where

\[T(z) = i\frac {1+z}{1-z},\]

one can easily see that the last statement is equivalent to the statement that if an algebraic curve \(\cC \) parametrized by rational functions \(P_1\) and \(P_2\) of degrees \(n_1\) and \(n_2\) has more than \(2n_1n_2\) real points, then it can be parametrized by some rational functions with real coefficients. Finally, the last statement follows from the bihomogeneous Bézout Theorem since if \(\cC \) is not defined over \(\R \), then the set

\[\cC \cap \R ^2=\cC \cap \overline \cC \]

is finite and contains at most \(\cC \cdot \overline \cC =2n_1n_2\) points (in [13] the usual version of the Bézout Theorem was used).

Our proof of Theorem  1.1 , while longer, considers the lemniscates directly and relates the initial problem with a real version of the Bézout Theorem, which might have an independent interest.

2.4. Intersection of polynomial lemniscates.

The polynomial version of Theorem 1.1 is the following statement.

Theorem 2.5.

Let \(P_1\) and \(P_2\) be non-constant complex polynomials of degrees \(n_1\) and \(n_2\). The following three conditions are equivalent:


  1. (i) \({\f L}_{P_1}\) and \({\f L}_{P_2}\) have more than \(2n_1 n_2\) common points.
  2. (ii) \({\f L}_{P_1}\cap {\f L}_{P_2}\) is infinite.
  3. (iii)* \(P_1=P^{n_1},\) \(P_2=c P^{n_2}\) for some polynomial \(P\), natural \(n_1,n_2\), and \(c\in \C \) with \(\vert c\vert =1\).

Proof. It is clear that Condition (iii)* is a particular case of Condition (iii) for \(B_1=z^{n_1},\) \(B_2=cz^{n_2},\) and \(W=P.\) Thus, in view of Theorem  1.1 , it is enough to show that if \(P_1\) and \(P_2\) are polynomials, then (iii) reduces to (iii)*.

To prove the last statement, let us observe that if \(P_1\) and \(P_2\) are polynomials, then each of the functions \(B_1\) and \(B_2\) has a unique pole, and this pole is common for \(B_1\) and \(B_2\). By condition ( 3 ), each pole of \(B_k\), \(k=1,2\), is symmetric with respect to \(\T \) to a zero of the same multiplicity and vice versa. Therefore, there exists \(a\in \bC \setminus \T \) such that

\[ B_1=c_1\Big ( \frac {z-a}{1-\overline {a}z}\Big )^{n_1}, \qquad B_2=c_2\Big ( \frac {z-a}{1-\overline {a}z}\Big )^{n_2} \]

for some \(n_1,n_2\geq 1\) and \(c_1,c_2\in \C \) with \(\vert c_1 \vert =\vert c_2 \vert =1\), implying that

\begin{equation} \label {b1} P_1=c_1 \widetilde P^{n_1}, \qquad P_2=c_2 \widetilde P^{n_2}, \end{equation}

where

\[ \widetilde P=\frac {z-a}{1-\overline {a}z}\circ W. \]

Finally, it is easy to see that \(\widetilde P\) is a polynomial and (iii)* holds for \(P=c_1^{\frac {1}{n_1}}\widetilde P\) and \(c=c_2c_1^{-{n_2}/{n_1}}\).

Notice that in the paper [13] it is erroneously claimed that in the polynomial case Condition (iii) of Theorem  1.1 simply reduces to the condition that \(P_1 = P^{m_1}\), \(P_2 = P^{m_2}\) for some polynomial \(P\). This inaccuracy however does not affect the main application of results of [13] in the polynomial case: the result of Ailon and Rudnick mentioned in the introduction. Indeed, if \(c\) in (iii)* is not a root of unity, then the system

\begin{equation} \label {sim} P_1(z)^k =1, \ \ \ P_2(z)^k =1 \end{equation}

has no solutions for any \(k\geq 1\), since (iii)* and ( 9 ) imply that

\[ P^{n_1n_2k}(z)=c^{n_1k}P^{n_1n_2k}(z)=1. \]

On the other hand, if \(c^l=1\), then ( 5 ) holds for \(m_1=n_2l\) and \(m_2=-n_1l\).

2.5. On sharpness of the bound \(2n_1n_2\) in the polynomial case.

The last statement of Theorem  1.1 states that the bound \(2n_1n_2\) is sharp if we speak of all rational functions. However, it does not seem so when we restrict ourselves to polynomials only. The maximal number of intersection points of two polynomial lemniscates that we succeeded to realize, is given in the following statement.

Proposition 2.6.
Let \(1\le n_1\le n_2\) and \(d=\gcd (n_1,n_2)\). Then there exist polynomials \(P_1\) and \(P_2\) such that \(\deg P_k=n_k\), \(k=1,2\), and

\[ \#(\f L_{P_1}\cap \f L_{P_2}) = M(n_1,n_2) := n_1 n_2 + n_2 + de,\qquad e = \begin {cases} 0, &\text {if $n_1/d$ is odd,}\\ 1, &\text {if $n_1/d$ is even.} \end {cases} \]

Proof. Let \(P_k(z) = (z/r_k)^{n_k} - 1\), \(k=1,2\), where \(r_k>0\). The lemniscate \(\f L_{P_k}\) is “flower-shaped”, i.e. it is a union of \(n_k\) loops (“petals”) outcoming from \(0\). It is clear that \(\f L_{P_k}\) tends to a union of \(n_k\) lines (we denote it by \(L_k\)) when \(r_k\to \infty \). Let us fix \(r_1\). It is not difficult to show that for a suitably chosen rotation \(R\) we have

\[ \#\big (L_2\cap R(\f L_{P_1})\setminus \{0\}) = n_2+de. \]

A further small shift of \(R(\f L_{P_1})\) produces \(n_1 n_2\) additional crossings near \(0\). Finally, we approximate \(L_2\) by \(\f L_{P_2}\) by choosing \(r_2\gg r_1\).

Note that the number \(M(n_1,n_2)\) in Proposition  2.6 coincides with the upper bound \(2n_1 n_2\) given by Theorem  1.1 if and only if \(n_1=1\).

3. Irreducibility of lemniscates

3.1. Irreducibility of separated variables curves and lemniscates.

Let us recall that a separated variable curve is an algebraic curve in \(\C ^2\) given by the equation \(E_{P,Q}(x,y) = 0\), where \(E_{P,Q}(x,y)\) is the numerator of \(P(x) - Q(y)\) for some non-constant complex rational functions \(P(z)\) and \(Q(z)\).

The irreducibility problem for separated variable curves is quite old and still widely open (see [7] for an introduction to the subject). One of the most complete results in this direction is a full classification of reducible curves \(E_{P,Q}(x,y)=0\) in the case when \(P\) and \(Q\) are indecomposable polynomials. All possible ramifications of such \(P\) and \(Q\) were described by Fried in [6] , and polynomials themselves were found by Cassou-Noguès and Couveignes in [4] .

For further progress, we refer the reader to the recent paper [10] and the bibliography therein. Here and below, by the irreducibility we always mean the irreducibility over \(\C \).

Let us recall that \(L_P(x,y)\) is the numerator of the rational function ( 6 ), and set

\[ \E _P(x,y) = E_{P,\,1/\,\overline P}(x,y). \]

The irreducibility problem for curves \(L_P(x,y)=0\) defining lemniscates is a particular case of the irreducibility problem for separated variables curves due to the following statement, which is immediate from the identity \(L_P(x,y) = \E _P(x+iy,x-iy)\).

Lemma 3.1.
Let \(P\) be a non-constant complex rational function. Then the curve \(L_P(x,y)=0\) is irreducible if and only if the curve \(\E _P(x,y)=0\) is irreducible.

The following theorem was proved in the paper [12] .

Theorem 3.2.
Let \(P\) and \(Q\) be non-constant complex polynomials. Then the curve \(E_{P,\,1/Q}(z,w)=0\) is reducible if and only if

\[ P(z)=P_1(z)^d \qquad \text {and}\qquad Q(w)=Q_1(w)^d \]

for some \(d>1\) and polynomials \(P_1(z)\) and \(Q_1(w)\).

It is easy to see that combined with Lemma 3.1 , Theorem 3.2 implies the following result also proved in [12] .

Theorem 3.3.
Let \(P\) be a non-constant complex polynomial. Then the polynomial \(L_P(x,y)\) is reducible if and only if

\[ P(z)=P_1(z)^d \]

for some \(d>1\) and polynomial \(P_1(z)\).

Notice that since \(|P_1(z)^d|=1\Leftrightarrow |P_1(z)|=1\), this theorem implies that any polynomial lemniscate is the zero set of an irreducible polynomial in \(x\) and \(y\).

Definition 3.4.
A complex rational function \(P(z)\) satisfies the Composition Condition if there exist a quotient of finite Blaschke products \(B\) of degree at least two and a non-constant rational function \(W\) such that \(P=B\circ W\).

Theorem 3.3 shows that the Composition Condition is necessary and sufficient for the reducibility of \(L_P(x,y)\) in the polynomial case. Moreover, it is easy to see that the Composition Condition is sufficient for the reducibility of \(L_P(x,y)\) for any rational function \(P\). Indeed, it follows from ( 3 ) that for any quotient of finite Blaschke products \(B\) and a rational function \(W\) the equality \(W(x)=1/\,\overline {W}(y)\) for some \(x,y\in \C \) implies the equality

\[ P(x)=B(W(x))=\qfrac [1pt]{1}{\overline {B}(1/W(x))} =\qfrac [1pt]{1}{\overline {B}(\overline {W}(y))} =\qfrac [1pt]{1}{\overline {P}(y)}. \]

Therefore, the curve \(\E _W(x,y)=0\) is a component of the curve \(\E _P(x,y)=0\), implying that the latter curve is reducible since the considered curves have different degrees (in view of the assumption \(\deg B>1\)). Thus, the curve \(L_P(x,y)=0\) is reducible by Lemma  3.1 .

In the next section, we show that Composition Condition is not necessary for the reducibility of \(L_P\), while in this section, modifying the idea of [12] , we establish a sufficient condition for the irreducibility of an algebraic curve \(E_{P,Q}(x,y)=0\) in the case when one of the rational functions \(P\), \(Q\) is a polynomial.

Namely, we prove the following result.

Theorem 3.5.
Let \(P\) be a polynomial of degree \(n\geq 1\), and \(Q\) a rational function. Assume that multiplicities \(q_1,q_2,\dots ,q_l\) of poles of \(Q\) satisfy the condition \(\gcd (q_1,q_2,\dots ,q_l,n)=1\). Then the curve \(E_{P,Q}(x,y)=0\) is irreducible.

Notice that like Theorem 3.2 , Theorem 3.5 easily implies Theorem 3.3 .

First proof of Theorem 3.5 .

(cf. the proof of [12], Theorem 1 ). Let \(C\) be the closure of the curve \(E_{P,Q}=0\) in \(\C \P ^1\times \C \P ^1\), that is, the curve defined by the bihomogeneous equation \(u^n v^m E_{P,Q}(x/u,y/v)=0\), \(m=\deg Q\). We identify \(\C \P ^1\) with \(\bC \) denoting \((x:1)\) by \(x\) and \((1:0)\) by \(\infty \).

Suppose there exists a proper factor \(E'(x,y)\) of \(E_{P,Q}\). Let \(C'\) be the corresponding subset of \(C\). Let \(y_1,\dots ,y_l\in \C \P ^1\) be the poles of \(Q\) of multiplicities \(q_1,\dots ,q_l\) respectively. The germ of \(C\) at \((\infty ,y_i)\) has the form \(U^n=Y^{q_i}\) in some local analytic coordinates \((U,Y)\). Indeed, the equation of \(C\) in the affine coordinates \(\widehat U=u/x\), \(\widehat Y=y/v-y_i\) (\(\widehat Y=v/y\) if \(y_i=\infty \)) has the form

\[ \widehat U^n f_i(\widehat U)=\widehat Y^{q_i}g_i(\widehat Y),\qquad f_i(0)g_i(0)\ne 0, \]

thus it has the desired form in the local coordinates \(U=\widehat U f_i^{1/n}\), \(Y=\widehat Y g_i^{1/q_i}\) for any choice of one-valued branches of the roots of \(f_i\) and \(g_i\) near \(0\).

The binomial \(U^n-Y^{q_i}\) factorizes as \(\prod _j^{k_i}(U^{b_i}-\omega ^j Y^{a_i})\), where \(k_i=\gcd (q_i,n)\), \(a_i=q_i/k_i\), \(b_i=n/k_i\), and \(\omega \) is a primitive \(k_i\)-th root of unity. Thus the germ of \(C\) at \((\infty ,y_i)\) has \(k_i\) local analytic branches, which we denote by \(\gamma _{ij}\), \(j=1,\dots ,k_i\). Let \(k'_i\) be the number of those that belong to \(C'\).

Let \(L_i=\C \P ^1\times \{y_i\}\). For local intersections, we have \((\gamma _{ij}\cdot L_i)_{(\infty ,y_i)}=b_i\) for each \(i,j\). Hence

\[ k'_i b_i = (C'.L_i)_{(\infty ,y_i)}=n' := \deg _x E', \qquad i=1,\dots ,l, \]

and we obtain

\begin{equation} \label {imp} \frac {k'_i a_i}{q_i} = \frac {k'_i}{k_i} = \frac {k'_i b_i}{n} = \frac {n'}{n}, \qquad 1\le i\le l. \end{equation}

Let \(d'/d\) be the reduced form of this fraction, i.e., \(d'/d=n'/n\) and \(\gcd (d',d)=1\). Then \(d>1\) (because \(n'<n\)) and \(d\) divides all the denominators in ( 10 ), which implies that \(\gcd (q_1,\dots ,q_l,n)>1\).

Now we expose more or less the same proof using the language of field extensions. The notations in both proofs are consistent. Of course, the proof of [12], Theorem 1 can be reinterpreted similarly.

Second proof of Theorem 3.5 .

For a compact Riemann surface \(C\), we denote the field of meromorphic functions on \(C\) by \(\f M(C)\). Given a meromorphic function \(\theta : C\rightarrow \C \P ^1\), we denote the local multiplicity of \(\theta \) at the point \(z\) by \(e_{\theta }(z)\).

Assume that the curve \(E_{P,Q}(x,y)=0\) is reducible, and let \(C\) be a desingularization of some of its components. Then there exist meromorphic functions \(\phi :C\rightarrow \C \P ^1\) and \(\psi :C\rightarrow \C \P ^1\) of degrees \(m'<m=\deg Q\) and \(n'<n\) such that

\begin{equation} \label {ps} P\circ \phi =Q\circ \psi \end{equation}

and the compositum of the fields \(\phi ^*\f M(\C \P ^1)\subseteq \f M(C)\) and \(\psi ^*\f M(\C \P ^1)\subseteq \f M(C)\) is the whole field \(\f M(C)\). Furthermore, if \(\theta :C\rightarrow \C \P ^1\) is a meromorphic function defined by any of the sides of equality ( 11 ), then by the Abhyankar Lemma (see e. g.  [17] , Theorem 3.9.1) for every point \(t_0\) of \(C\) the following equality holds:

\begin{equation} \label {abj} e_{\theta }(t_0)={\rm lcm} \big (e_{P}(\phi (t_0)), e_{Q}(\psi (t_0))\big ) \end{equation}

Let \(Q^{-1}\{\infty \}=\{y_1,y_2,\dots ,y_l\},\) where \(e_Q(y_i)=q_i\), \(1\leq i \leq l\), and

\[ \psi ^{-1}\{y_i\}=\{z_{i1},z_{12},\dots ,z_{ik'_i}\},\quad 1\leq i \leq l. \]

Let us set

\[ k_i=\gcd (q_i,n), \quad a_i=q_i/k_i, \quad b_i=n/k_i. \]

Since \(P^{-1}\{\infty \}=\infty \), by ( 12 ) we have

\[ e_{\theta }(z_{ij})={\rm lcm}(q_i,n),\quad 1\leq j \leq k'_i,\quad 1\leq i \leq l. \]

Therefore,

\[ e_{\psi }(z_{ij}) = \frac {{\rm lcm}(q_i,n)}{q_i} = b_i, \qquad 1\leq j \leq k'_i,\quad 1\leq i \leq l, \]

implying that, for each \(i=1,\dots ,l\), we have \(k'_ib_i = \deg \psi = n'\), whence we obtain the equation ( 10 ) and conclude in the same way as in the first proof.

3.2. A counterexample.

In this subsection we show that the Composition Condition (see Definition  3.4 ) is not necessary for reducibility of the curve \(L_P(x,y)=0\). Let us set

\begin{equation} \label {if} T(z) = \frac {z-i}{z+i}. \end{equation}

We recall that this Möbius transformation (called the Cayley transform) maps \(\bR :=\R \cup \{\infty \}\) to the unit circle \(\T \).

Lemma 3.6.
Let \(S\) be a non-constant complex rational function. Then the curve \(\E _{T\circ S}(x,y)=0\) is irreducible if and only if the curve \(E_{S,\overline {S}}(x,y)=0\) is irreducible.

Proof. If \(S=S_1/S_2\), where \(S_1\) and \(S_2\) have no common roots, then

\[ (T\circ S)(z)=\frac {S_1(z)-iS_2(z)}{S_1(z)+iS_2(z)}. \]

Moreover, it is easy to see that \(S_1(z)-iS_2(z)\) and \(S_1(z)+iS_2(z)\) have no common roots. Thus,

\begin{align*} \E _{T\circ S}(x,y)&=(S_1(x)-iS_2(x))(\overline {S}_1(y)+i\overline {S}_2(y)) -(S_1(x)+iS_2(x))(\overline {S}_1(y)-i\overline {S}_2(y)) \\ &=2i(S_1(x)\overline {S}_2(y)-S_2(x)\overline {S}_1(y)) =2iE_{S,\overline {S}}(x,y). \hskip 34.8mm{\Box } \end{align*}

To show that the Composition Condition is not necessary for the reducibility of \(L_P(x,y)\), one can use examples of reducible curves of the form \(E_{S,\overline {S}}(x,y)=0\) found in [4] . For instance, we can take

\begin{equation} \label {id} \begin{split} S(z) &=\frac {1}{11} z^{11} - (a+1)z^9 + 2z^8 + (3a-9)z^7 - 16(a+1)z^6 + (21a+36)z^5\\ &\;+ (30a-90)z^4 - 63a z^3 + (100a+120)z^2 + (24a-117)z - 18(a+1), \end {split} \end{equation}

where \(a\) satisfies \(a^2 + a + 3 = 0\). It is shown in [4] that for \(S(z)\) the following conditions hold. First, the curve \(S(x)-\overline {S}(y)=0\) is reducible. Second,

\begin{equation} \label {cont} \overline {S}(z)\neq S(cz+b), \end{equation}

for any \(c\in \C ^*\), \(b\in \C \) (the last condition makes this example non-trivial since every curve of the form \(S(x)-S(cy+b)=0\) obviously has a factor \(x-cy-b=0\)).

Let us consider a rational function \(P=T\circ S\), where \(T\) and \(S\) are defined by ( 13 ) and ( 14 ). By Lemma  3.6 , the curve \(\E _P(x,y)=0\) is reducible, implying by Lemma  3.1 that the curve \(L_P(x,y)=0\) is also reducible. Since the degree of \(P\) is a prime number, if the Composition Condition were satisfied by \(P\), we would have

\begin{equation} \label {iwi} P=T\circ S=B\circ \mu \end{equation}

for some quotient of finite Blaschke products \(B\) and Möbius transformation \(\mu \). Let now \(\nu \) be a Möbius transformation that maps \(\bR \) to \(\mu ^{-1}(\T )\). Then \((\mu \circ \nu )(\bR )=\T \) and it follows from ( 16 ) combined with the equalities \(B(\T )=\T \) and \(T(\bR )=\mathbb T\) that

\begin{equation} (S\circ \nu )(\bR )=(S\circ \mu ^{-1})(\T )=(T^{-1}\circ B)(\T )=T^{-1}(\T )=\bR . \end{equation}

Therefore the rational function \(\overline {(S\circ \nu )}-(S\circ \nu )\) identically vanishes on \(\R \), and hence on \(\C \). Thus,

\[ S\circ \nu =\overline {S\circ \nu }=\overline {S}\circ \overline {\nu }, \]

and denoting the Möbius transformation \(\nu \circ \overline {\nu }^{-1}\) by \(\delta \), we rewrite this equality as

\begin{equation} \label {eq.last} \overline {S}=S\circ \delta . \end{equation}

Since \(S\) is a polynomial, ( 18 ) implies that \(\delta \) also is a polynomial. Moreover, \(\deg \delta =1\) (because \(\delta \) is a Möbius transformation), thus ( 18 ) contradicts ( 15 ). The obtained contradiction shows that \(P\) does not satisfy the Composition Condition.

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