Real even symmetric ternary forms (Q1969481)

In real algebra it is an important issue to decide which positive semi-definite forms over the real numbers can be represented as sums of squares of forms. The author studies this question for the space \(S^e_{n,m}\) of \(n\)-ary forms that have degree \(m\), are even (all variables occur only with even degrees) and symmetric (invariance under permutations of the variables). The cones of positive semi-definite (or sum of squares) forms in \(S^e_{n,m}\) are denoted by \(PS^e_{n,m}\) (or \(\Sigma S^e_{n,m})\). As a first step, tests are devised for \(S^e_{n,8}\) and \(S^e_{3,10}\) that decide whether a given form is positive semi-definite or not. The tests are used to determine subsets \(U\subseteq PS^e_{3,8}\) and \(V\subseteq PS^e_{3,10}\) that contain all extremal elements in the two cones. The set \(U\) consists entirely of sums of squares, and it follows that \(PS^e_{3,8}= \Sigma S^e_{n,8}\). On the other hand, the inclusion \(\Sigma S^e_{3,10}\subseteq PS^e_{3,10}\) is shown to be proper. For all elements of \(V\) it is decided whether they are sums of squares or not, which results in various new families of positive semi-definite forms that are not sums of squares.