Improved effective estimates of P\'olya's Theorem for quadratic forms

TL;DR

This paper improves the upper bounds on Pólya's exponent for quadratic forms that are positive over the standard simplex, providing tighter estimates than previous results, especially as certain parameters grow large.

Contribution

The authors derive a sharper upper bound for the Pólya exponent of positive quadratic forms, enhancing previous bounds by de Klerk, Laurent, and Parrilo.

Findings

New upper bound for Pólya's exponent of quadratic forms

Bound is asymptotically tighter as parameters grow large

Application to specific binary quadratic forms

Abstract

Following de Loera and Santos, the P\'olya 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 P\'olya, a form assumes only positive values over the standard $(n - 1)$-simplex in Euclidean $n$-space if and only if its P\'olya exponent is finite. In this note, we compute an upper bound of the P\'olya 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 P\'olya's exponent is $O(1/\lambda)$ times that of de Klerk, Laurent and Parrilo's as $\lambda$ tends to infinity.