Distribution of the eigenvalues of a random system of homogeneous polynomials

Abstract

Let $f=(f_1,\dots ,f_n)$ 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 $n\to \infty$.