A Positivstellensatz for forms on the positive orthant

Abstract

Let $p$ be a nonconstant form in $\mathbb{R}[x_1,\dots,x_n]$ with $p(1,\dots,1)>0$. If $p^m$ has strictly positive coefficients for some integer $m\ge1$, we show that $p^m$ has strictly positive coefficients for all sufficiently large $m$. More generally, for any such $p$, and any form $q$ that is strictly positive on $(\mathbb{R}_+)^n\setminus\{0\}$, we show that the form $p^mq$ has strictly positive coefficients for all sufficiently large $m$. This result can be considered as a strict Positivstellensatz for forms relative to $(\mathbb{R}_+)^n\setminus\{0\}$. We give two proofs, one based on results of Handelman, the other on techniques from real algebra.