The lemniscate tree of a random polynomial

Epstein, Michael; Hanin, Boris; Lundberg, Erik

Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1663-1687.

À chaque polynôme complexe générique $p\left(z\right)$ est associé un arbre binaire étiqueté (appelé dans cet aticle, “arbre de lemniscate”) qui code le type topologique du graphe de $\left|p\right(z\left)\right|$. La structure de ramification de l’arbre de lemniscate est déterminée par la configuration (c’est-à-dire la disposition dans le plan) des composants singuliers de ces ensembles de niveaux $\left|p\right(z\left)\right|=t$ passant par un point critique.

Dans cet article, nous nous intéressons à la question suivante : combien de branches apparaissent typiquement dans un arbre de lemniscate  ? Nous répondons d’abord à cette question pour un arbre de lemniscate échantillonné uniformément dans la classe combinatoire de tous ces arbres associés à un polynôme générique de degré fixé, et ensuite pour un arbre de lemniscate résultant d’un polynôme aléatoire avec des zéros indépendants et identiquement distribués. D’un point de vue plus général, ces résultats constituent un premier pas vers un traitement probabiliste (dans un cadre spécialisé) du programme d’Arnold consistant à énumérer les fonctions algébriques de Morse.

To each generic complex polynomial $p\left(z\right)$ is associated a labeled binary tree (here referred to as a “lemniscate tree”) that encodes the topological type of the graph of $\left|p\right(z\left)\right|$. 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 $\left|p\right(z\left)\right|=t$ 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.

Classification : 30C15, 60G60, 31A15, 14P25, 05A15, 60C05, 60F05
Keywords: random polynomial, binary tree, lemniscate, analytic combinatorics
Mot clés : polynôme aléatoire, arbre binaire, lemniscate, combinatoire analytique

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