Real Even Symmetric Ternary Forms

Let Sen, m denote the set of all real symmetric forms of degree m = 2d. Let PSen, m and ΣSen, m denote the cones of positive semidefinite (psd) and sum of squares (sos) elements of Sen, m, respectively. For m = 2 or 4, these cones coincide. For m = 6, they do not, and were analyzed in Even Symmetric Sextics, by M. D. Choi, T. Y. Lam, and B. Reznick (1987, Math. Z.195, pp. 559–580). We present an easily checked, necessary and sufficient condition for an even symmetric n-ary octic to be in PSen, 8 and for an even symmetric ternary decic to be in PSe3, 10; we also show that there is no corresponding condition for even symmetric ternary forms of degree greater than 10. We proceed to discuss the extremal elements of the cones PSe3, 8 and PSe3, 10. This leads to the question: how many of these extremal forms have sos representations? We prove that PSe3, 8 = ΣSe3, 8 and demonstrate that PSe3, 10\ΣSe3, 10 is nonempty, providing many new examples of psd forms which are not sos. We give a graphic representation of ternary forms which also indicates whether or not an element of Se3, 8 or Se3, 10 is psd. We interpret elements of PSe3, m as inequalities; in particular, we give all symmetric polynomial inequalities of degree ≤ 5 satisfied by the sides of a triangle.