Proof of a conjecture on polynomials preserving nonnegative matrices

Abstract

We consider polynomials in R[x] which map the set of nonnegative matrices of a given order into itself. Let n be a positive integer and define Pn:={p∈R[x]:p(A)≥0,forallA≥0,A∈Rn,n}. This set plays a role in the Nonnegative Inverse Eigenvalue Problem. Clark and Paparella conjectured that Pn+1 is strictly contained in Pn. We prove this conjecture. The structure of Pn is also of interest, and for this purpose define, for any positive integer m, Pn,m={p∈R[x]:degree(p)≤mandp(A)≥0,forallA≥0,A∈Rn,n}. This is a convex cone. For 0<m<2n it is known to be polyhedral, and the question is whether for m≥2n it is not polyhedral. We answer it in the affirmative when n=2.