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 $z\in \mathbb{C}$ such that $\mathrm{\mid }P(z)\mathrm{\mid }=1$. 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 $P(x+iy)\bar{P}(x-iy)-1.$ 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 $\mathrm{\mid }{P}_1(z)\mathrm{\mid }=\mathrm{\mid }{P}_2(z)\mathrm{\mid }=1$, where $P_1$ and $P_2$ are rational functions.