Density of Complex Critical Points of a Real Random SO(m+1) Polynomials

TL;DR

This paper investigates how the density of complex critical points of real random SO(m+1) polynomials converges to that of complex random polynomials, using the Kac-Rice formula, with explicit formulas provided for the one-variable case.

Contribution

It introduces a new proof using the Kac-Rice formula for the convergence of critical point densities and provides exact and scaling limit formulas in the one-variable case.

Findings

Density of complex critical points approaches that of complex random polynomials

Explicit formulas for one-variable critical point densities

Convergence results proven using Kac-Rice formula

Abstract

We study the density of complex critical points of a real random SO(m+1) polynomial in m variables. In a previous paper [Mac09], the author used the Poincare- Lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials, the SU(m+1) polynomials. In this paper, we use the Kac- Rice formula to prove an analogous result: the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial. In one variable, we give an exact formula and a scaling limit formula for the density of critical points of the real random SO(2) polynomial as well as for the density of critical points of the corresponding complex random SU(2) polynomial.