The lemniscate tree of a random polynomial

To each generic complex polynomial p(z) is associated a labeled binary tree (here referred to as a “lemniscate tree”) that encodes the topological type of the graph of |p(z)|. The branching structure of the lemniscate tree is determined by the configuration (i.e., arrangement in the plane) of the singular components of those level sets |p(z)| = t passing through a critical point. In this paper, we ask: how many branches appear in a typical lemniscate tree? We answer this question first for a lemniscate tree sampled uniformly from the combinatorial class of all such trees associated to a generic polynomial of fixed degree and second for the lemniscate tree arising from a random polynomial with i.i.d. zeros. From a more general perspective, these results take a first step toward a probabilistic treatment (within a specialized setting) of Arnold’s program of enumerating algebraic Morse functions.