Density of Complex Zeros of a System of Real Random Polynomials

TL;DR

This paper investigates how the density of complex zeros in systems of real random polynomials behaves, revealing rapid convergence to the complex coefficient case and differing near real space depending on the number of variables.

Contribution

It demonstrates the asymptotic behavior of zero densities for real random polynomial systems and highlights differences between one-variable and multi-variable cases.

Findings

Density approaches the complex coefficient case rapidly

Near real space, density diverges for multiple variables

For one variable, density tends linearly to zero

Abstract

We study the density of complex zeros of a system of real random SO($m+1$) polynomials in several variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case. We also show that the behavior the scaled density of complex zeros near $\mathbb{R}^m$ of the system of real random polynomials is different in the $m\geq 2$ case than in the $m=1$ case: the density goes to infinity instead of tending linearly to zero.