On intersection of lemniscates of rational functions

2023/09/10 by S. Yu. Orevkov, Orevkov, Stepan, Fedor Pakovich +1
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Functional Equations Stability Results #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2309.04983

openalex publication_date 2023/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

For a non-constant complex rational function P, the lemniscate of P is defined as the set of points z∈ \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) 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 P₁(z)\vert =\vert P₂(z)\vert =1, where P₁ and P₂ are rational functions.

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