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Copositive symmetric cubic forms. (English) Zbl 1138.11320
From the text: A homogeneous polynomial \(f\) in \(n\) variables over \(\mathbb R\) is called a form. Such a form \(f\) is said to be copositive if \(f(x_1,\dots,x_n)\geq 0\) whenever \(x_i\geq 0\) for \(i=1,\dots,n\).
The set of symmetric cubic forms in any number \(n\) of variables is a vector space over \(\mathbb R\) that is spanned by the three forms \(M_3^{(n)}\), \(M_2^{(n)}\), \(M_1^{(n)}\), and \((M_1^{(n)})^3\), where
\[ M_r^{(n)}(x_1,\dots,x_n)= x_1^r+\cdots+ x_n^r. \]
Here the author proves:
Theorem 1. Let \(f^{(n)}\) be a symmetric cubic form in \(n\) variables, say
\[ f^{(n)}= aM_3^{(n)}+ bM_2^{(n)} M_1^{(n)}+ c(M_1^{(n)})^3. \]
Then \(f^{(n)}\) is copositive if and only if \(f(n)({\mathbf v}_m^{(n)})\geq 0\) for \(m=1,\dots,n\). In other words, \(f^{(n)}\) is copositive if and only if \(a+bm+cm^2\geq 0\) for \(m=1,\dots,n\).

MSC:
11E76 Forms of degree higher than two
11E10 Forms over real fields
26C99 Polynomials, rational functions in real analysis
26D07 Inequalities involving other types of functions
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