On intersection of lemniscates of rational functions

TL;DR

This paper investigates the geometric and algebraic properties of lemniscates of rational functions, focusing on conditions for common components and irreducibility, and provides bounds on solutions to related systems.

Contribution

It offers new criteria for when lemniscates share components and establishes a sharp bound on the number of solutions for systems involving two rational functions.

Findings

Conditions for common components of lemniscates identified

Irreducibility criteria for algebraic curves of lemniscates established

Sharp bounds on solutions to systems of lemniscates provided

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.