Distribution of the eigenvalues of a random system of homogeneous polynomials

TL;DR

This paper studies the distribution of eigenvalues of random homogeneous polynomial systems in complex variables, generalizing matrix eigenvalues, and analyzes their asymptotic behavior as the system size grows.

Contribution

It derives the eigenvalue distribution for random polynomial systems and characterizes the limit distribution as the number of variables tends to infinity.

Findings

Eigenvalue distribution derived for random homogeneous polynomial systems

Limit distribution identified for large system size

Eigenvalues follow a specific probabilistic pattern in the asymptotic regime

Abstract

Let $f=(f_1,\ldots,f_n)$ be a system of $n$ complex homogeneous polynomials in $n$ variables of degree $d$. We call $\lambda\in\mathbb{C}$ an eigenvalue of $f$ if there exists $v\in\mathbb{C}^n\backslash\{0\}$ with $f(v)=\lambda v$, generalizing the case of eigenvalues of matrices ($d=1$). We derive the distribution of $\lambda$ when the $f_i$ are independently chosen at random according to the unitary invariant Weyl distribution and determine the limit distribution for $n\to\infty$.