The lemniscate tree of a random polynomial

2018/06/01 by Epstein, Michael, Hanin, Boris, Lundberg, Erik
#05A15 #14P25 #30C15 #31A15 #60C05 #60F05 #60G60 #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1806.00521

Abstract

To each generic complex polynomial p(z) there 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 address the question "How many branches appear in a typical lemniscate tree?" We answer this question first for a lemniscate tree sampled uniformly from the combinatorial class and second for the lemniscate tree arising from a random polynomial generated by 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.

Related

• On the number of components of random polynomial lemniscates
• On the number of unary-binary tree-like structures with restrictions on the unary height
• Asymptotics for the expected number of nodal components for random\n lemniscates
• A review of some recent results on random polynomials over R and over C
• From Trees to Barcodes and Back Again II: Combinatorial and Probabilistic Aspects of a Topological Inverse Problem