Proof of a conjecture on polynomials preserving nonnegative matrices

Abstract: We consider polynomials in R[x] which map the set of nonnegative (element-wise) matrices of a given order into itself. Let n be a positive integer and define P(n)= {p in R[x] : p(A) is nonnegative (element-wise), for all A, A an n-by-n nonnegative (element-wise) matrix}. This set plays a role in the Nonnegative Inverse Eigenvalue Problem. Clark and Paparella conjectured that P(n+1) is strictly contained in P(n). We prove this conjecture.