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Distribution of the eigenvalues of a random system of homogeneous polynomials
Citation key BB-Distribution-Of-The-Eigenvalues-Of-A-Random-System-Of-Homogeneous-Polynomials
Author Paul Breiding and Peter Bürgisser
Pages 88-107
Year 2016
DOI 10.1016/j.laa.2016.02.020
Journal Linear Algebra and its Applications
Volume 497
Month 05
Abstract Let $f=(f_1,..,f_n)$ be a system of $n$ complex homogeneous polynomials in $n$ variables of degree $d$. We call λ ∈ $\mathbb C$ an eigenvalue of $f$ if there exists a nonzero $v$ ∈ $\mathbb C$ with $f(v)=$ λ $v$, generalizing the case of eigenvalues of matrices ($d=1$). We derive the distribution of λ when the $f_i$ are independently chosen at random according to the unitary invariant Weyl distribution and determine the limit distribution for $n\to\infty$.
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