# A Positivstellensatz for forms on the positive orthant

## TL;DR

This paper establishes a Positivstellensatz for forms on the positive orthant, showing that certain powers of forms with positive evaluations have eventually strictly positive coefficients, with proofs based on Handelman's results and real algebra techniques.

## Contribution

It proves that powers of forms with positive values on the positive orthant eventually have strictly positive coefficients, extending Positivstellensatz results to this setting.

## Key findings

- For a form p with p(1,...,1)>0, p^m has strictly positive coefficients for large m.
- For any form q positive on the positive orthant, p^mq has strictly positive coefficients for large m.
- Provides two proofs: one using Handelman's results, another using real algebra techniques.

## 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.