A degree-linear upper bound for the Pólya exponent of forms with fixed arity

Abstract

Let $P={\sum }_{\omega }{a}_{\omega }{x}^{\omega }$ be a degree $d$ homogeneous polynomial in $x_0,\dots ,x_n$ with real coefficients. Pólya showed that, if $P$ is strictly positive on the standard $n$-simplex, then $(x_0+\cdots +x_n{)}^{M}P$ has strictly positive coefficients for all $M\gg 0$. The least such $M$ is known as the \emph{Pólya exponent} of $P$. de Klerk-Laurent-Parrilo showed that the Pólya exponent is $O(d^2)P_{\max }^{(0)}/\lambda (P)$, where $P_{\max }^{(0)}:={\max }_{\omega }{a}_{\omega }/\left(\genfrac{}{}{0ex}{}{d}{\omega }\right)$ and $\lambda (P)$ is the infimum of $P$ on the standard simplex. Pólya's Positivstellensatz is a consequence of the classical approximation of the multivariate hypergeometric distribution by the multinomial distribution, as was already implicitly known to de Klerk-Laurent-Sun. By applying known estimates of this approximation from probability theory, we obtain a different $O_n(d)L(P)/\lambda (P)$ upper bound of the Pólya exponent, where $L(P):=\max |a_{\omega }|/(\genfrac{}{}{0ex}{}{d}{\omega })$. An example shows that $O_n(d)$ is tight.

arXiv comment12 pages