Mathematics

We investigate the landscape given by considering the modulus $|p(z)|$ of a complex polynomial.  In the spirit of Arnold's program of enumerating algebraic Morse functions, F. Catanese and M. Paluszny classified (1991) the landscapes generated by complex polynomials, and they enumerated all possible topological equivalence classes (the equivalence being up to diffeomorphism of the domain and range) using a combinatorial scheme based on a certain class of labeled binary trees that we refer to as "lemniscate trees".  The branching structure of the lemniscate tree is determined by the configuration (i.e., arrangement in the plane) of the various singular level sets $|p(z)|=t$.  The level sets are referred to as lemniscates, and a generic polynomial of degree n has $n-1$ singular lemniscates (each passes through a critical point of $p$).  In this talk, we investigate the global structure of the landscape by addressing 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.  This is joint work with Boris Hanin and Michael Epstein.