Improved effective estimates of Pólya's Theorem for quadratic forms

Abstract

Following de Loera and Santos, the Pólya exponent of a $n$-ary real form (i.e. a homogeneous polynomial in $n$ variables with real coefficients) $f$ is the infimum of the upward closed set of nonnegative integers $m$ such that $(x_1+\cdots +x_n{)}^{m}f$ strictly has positive coefficients. By a theorem of Pólya, a form assumes only positive values over the standard $(n-1)$-simplex in Euclidean $n$-space if and only if its Pólya exponent is finite. In this note, we compute an upper bound of the Pólya exponent of a quadratic form $f$ that assumes only positive values over the standard simplex. Our bound improves a previous upper bound due to de Klerk, Laurent and Parrilo. For example, for the binary quadratic form $f_{\kappa }={\lambda }^2{x}_1^2-2\kappa \lambda x_1{x}_2+x_2^2$, which assumes only positive values over the standard $1$-simplex whenever $0\le \kappa <1<\lambda$, our upper bound of its Pólya's exponent is $O(1/\lambda )$ times that of de Klerk, Laurent and Parrilo's as $\lambda$ tends to infinity.

arXiv commentThe main result of this paper is not new. Essentially, this result was already gotten by Bomze and de Klerk (Journal of Global Optimization, 2002, see Theorems 3.1 and 3.2 and their proofs)