The analogue of Hilbert's 1888 theorem for even symmetric forms

Hilbert proved in 1888 that a positive semidefinite (psd) real form is a sum of squares (sos) of real forms if and only if n=2n=2 or d=1d=1 or (n,2d)=(3,4)(n,2d)=(3,4), where n is the number of variables and 2d the degree of the form. We study the analogue for even symmetric forms. We establish that an even symmetric n-ary 2d  -ic psd form is sos if and only if n=2n=2 or d=1d=1 or (n,2d)=(n,4)n≥3(n,2d)=(n,4)n≥3 or (n,2d)=(3,8)(n,2d)=(3,8).