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Even symmetric sextics. (English) Zbl 0654.10024
Even symmetric sextics over the real numbers can be represented in the form \[ f(x)=\alpha \sum^{n}_{i=1}x^ 6_ i+\quad \beta \sum_{i\neq j}x^ 4_ i x^ 2_ j+\gamma \sum_{i<j<k}x^ 2_ i x^ 2_ j x^ 2_ k \] (up to a permutation of the variables). f is positive semidefinite (psd) if f(x)\(\geq 0\) for all \(x\in {\mathbb{R}}^ n\). f is a sum of squares (sos) if \(f=\sum h^ 2_ k\) with forms \(h_ k\). Depending on three parameters the even symmetric sextics correspond naturally with the quadratic polynomials. A quadratic polynomial is identified with its triple of coefficients. The sets \(P_ n\) and \(\Sigma_ n\) of quadratic polynomials corresponding to the psd and sos forms are investigated. Both are closed semi-algebraic cones in \({\mathbb{R}}^ 3\). The cones \(P_ n\) and \(\Sigma_ n\) are shown to be of quadratic polynomials nonnegative on \(\{\) 1,...,n\(\}\) and \(\{\) \(1\}\) \(\cup [2,n]\), resp. The extremal points of both cones and inequalities defining \(P_ n\) and \(\Sigma_ n\) as semi-algebraic subsets of \({\mathbb{R}}^ 3\) are determined.
Reviewer: N.Schwartz

11E76 Forms of degree higher than two
14Pxx Real algebraic and real-analytic geometry
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