An observation on positive definite forms

TL;DR

This paper proves that for two positive definite forms, the product of one with a large power of the other eventually becomes strictly inside the sums of squares cone, indicating a form of asymptotic positivity enhancement.

Contribution

It establishes a new asymptotic result about the interior of sums of squares cones for products of positive definite forms.

Findings

For large r, fg^r is in the interior of the sums of squares cone.

The result applies to positive definite forms in real polynomial rings.

Provides insight into the structure of sums of squares cones and positivity.

Abstract

Given two positive definite forms $f,\,g\in\mathbb{R}[x_1,\dots,x_n]$, we prove that $fg^r$ lies in the interior of the sums of squares cone for large $r$.