Positive semidefinite diagonal minus tail forms are sums of squares

Abstract

By a diagonal minus tail form (of even degree) we understand a real homogeneous polynomial F(x1, ..., xn) = F(x) = D(x) − T(x), where the diagonal part D(x) is a sum of terms of the form bix2d i with all bi ≥ 0 and the tail T(x) a sum of terms ai1i2...inxi1 1 ...xin n with ai1i2...in > 0 and at least two i ≥ 1. We show that an arbitrary change of the signs of the tail terms of a positive semidefinite diagonal minus tail form will result in a sum of squares of polynomials.