Real Even Symmetric Ternary Forms

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