On intersection of lemniscates of rational functions

TL;DR

The paper investigates two central questions about lemniscates of rational functions: when two lemniscates share a common component and when the associated real algebraic curve $L_P(x,y)=0$ is irreducible. It proves a sharp bound: for rational functions of degrees $n_1$ and $n_2$, two lemniscates have more than $2n_1n_2$ complex intersection points only if they arise from a common compositional structure via a map $W$ and Blaschke-type quotients $B_1,B_2$, with each $B_j$ mapping the unit circle to itself; the bound is sharp in general. The argument blends a real version of Bézout’s theorem for bihomogeneous polynomials, a Cartwright-type reduction to Blaschke quotients, and a study of separated-variable curves, yielding a polynomial-special case and a counterexample showing that reducibility does not force a Composition Condition for rational $P$. The results have implications for the intersection theory of lemniscates and for irreducibility criteria of algebraic curves parameterized by rational functions. Overall, the work connects complex analysis, algebraic geometry, and diophantine-style bounds in the setting of lemniscates.

Abstract

For a non-constant complex rational function $P$, the lemniscate of $P$ is defined as the set of points $z\in \mathbb C$ 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.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 2.1
• Corollary 2.2
• Proposition 2.3
• Corollary 2.4
• Theorem 2.5
• Proposition 2.6
• Lemma 3.1
• Theorem 3.2
• Theorem 3.3