Real Even Symmetric Forms

Abstract

Let F$\sb{n,m}$ denote the set of all real forms of degree m in n variables. In 1888, Hilbert proved that a form P $\in$ F$\sb{n,m}$ which is positive semidefinite (psd) must have a representation as a sum of squares (sos) of forms if and only if n = 2, m = 2, or (n,m) = (3,4). No concrete example of a psd form which is not sos was known until the late 1960's. We denote by S$\sbsp{n,m}{e}$ the set of all real symmetric forms of degree m = 2d. Let PS$\sbsp{n,m}{e}$ and $\Sigma$S$\sbsp{n,m}{e}$ denote the cones of psd and sos elements of S$\sbsp{n,m}{e},$ 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, Math. Z. 195 (1987), pp. 559-580.

We present an easily-checked, necessary and sufficient condition for an even symmetric n-ary octic to be in PS$\sbsp{n,8}{e}$ and for an even symmetric ternary decic to be in PS$\sbsp{3,10}{e},$ and 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 PS$\sbsp{3,8}{e},$ PS$\sbsp{3,10}{e}$ and PS$\sbsp{4,8}{e}.$ This leads to the question: how many of these extremal forms have sos representations? We prove that PS$\sbsp{3,8}{e}$ = $\Sigma$S$\sbsp{3,8}{e},$ a companion result to Hilbert's theorem noted above, with regard to psd ternary quartics. We also demonstrate that neither PS$\sbsp{3,10}{e}\\\Sigma$S$\sbsp{3,10}{e}$ nor PS$\sbsp{4,8}{e}\\\Sigma$S$\sbsp{4,8}{e}$ is empty, providing many new examples of psd forms which are not sos.

We give a graphic representation with examples of ternary forms which also indicates whether or not an element of S$\sbsp{3,8}{e}$ or S$\sbsp{3,10}{e}$ is psd. We interpret elements of PS$\sbsp{n,m}{e}$ as inequalities; in particular, we give all symmetric polynomial inequalities of degree $\le$5 satisfied by the sides of a triangle.