Statistical Topology of Real Polynomials. I: Two Variables

TL;DR

This paper studies the average gradient degree of random polynomials in two variables, providing asymptotic results for large degrees, which enhances understanding of the topological complexity of polynomial zero sets.

Contribution

It introduces asymptotic analysis of the average gradient degree for two-variable polynomials, a novel approach in the statistical topology of real polynomials.

Findings

Asymptotic behavior of average gradient degree for large n

Quantitative insights into the topology of polynomial zero sets

Extension of statistical topology methods to two-variable polynomials

Abstract

We investigate average gradient degree of normal random polynomials of fixed algebraic degree n. In particular, for polynomials of two variables, asymptotics of the average gradient degree for large values of n is determined.