Distribution of the eigenvalues of a random system of homogeneous polynomials

Let f=(f1,…,fn)f=(f1,…,fn) be a system of n complex homogeneous polynomials in n variables of degree d  . We call λ∈Cλ∈C an eigenvalue of f   if there exists v∈Cn\{0}v∈Cn\{0} with f(v)=λvf(v)=λv, generalizing the case of eigenvalues of matrices (d=1d=1). We derive the distribution of λ   when the fifi are independently chosen at random according to the unitary invariant Weyl distribution and determine the limit distribution for n→∞n→∞.