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

Abstract

Let $P = \sum_ω a_ωx^ω$ 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 $O(d^2) P^{(0)}_{\max}/λ(P)$, where $P^{(0)}_{\max} := \max_ωa_ω/\binom{d}ω$ and $λ(P)$ 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 $O_n(d) L(P)/λ(P)$ upper bound of the Pólya exponent, where $L(P) := \max |a_ω|/\binom{d}ω$. An example shows that $O_n(d)$ is tight.