On Intersection of Lemniscates of Rational Functions

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