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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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