Polynomial-lemniscates, trees and braids (Q1184328)

Motivated by investigations of the second author on ordinary differential equations integrable by rational functions, the present authors study lemniscate configurations, where a lemniscate is a singular level set \(\Gamma_ c=\{z|\;| P(z)| = c > 0\}\) for the real function \(| P|\) defined by a complex polynomial \(P \in \mathbb{C}[z]\) of degree \((n+1)\) with a genericity condition on the \(n\) roots of the derivative \(P'\). The lemniscate-generic polynomials \({\mathcal L}_ n\) form an open set in the open set of polynomials of degree exactly \((n+1)\). The main purpose of the paper is to prove that two polynomials \(P_ 1\), \(P_ 2\in {\mathcal L}_ n\) have the same lemniscate configuration (homeomorphism class of singular level sets) if and only if they lie in the same connected component of \({\mathcal L}_ n\).