Extension of Hilbert's 1888 Theorem to Even Symmetric Forms

Abstract

We compare the cone of positive semidefinite (real) forms to its subcone of sum of squares of (real) forms under the additional assumption of symmetry on the given forms. The aim was to generalize a classical theorem of Hilbert from 1888, namely, a positive semidefinite form (psd) in n variables and of degree 2d is a sum of squares (sos) if and only if n=2 or d=1 or (n,2d)=(3,4); for symmetric and even symmetric forms respectively. As main results we construct explicitly psd not sos symmetric quartic forms in more than 4 variables, thereby completing the analogue of Hilbert's 1888 theorem for symmetric forms, which was asserted by Choi and Lam in 1976. Moreover, we construct psd not sos even symmetric octic forms in more than 4 variables and introduce a degree jumping principle to increase the degree of a psd not sos even symmetric form while simultaneously preserving the psd not sos even symmetric property. Finally using these constructions and techniques we present a version of Hilbert's 1888 theorem for even symmetric forms.