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Real even symmetric ternary forms. (English) Zbl 0970.11013
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.

##### MSC:
 11E76 Forms of degree higher than two 11E10 Forms over real fields 13J30 Real algebra
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##### References:
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