Making Non-Negative Polynomials into Sums of Squares

TL;DR

This paper explores linear operators that transform non-negative polynomials into sums of squares, providing explicit constructions and conditions, and investigates the structure of such operators within the framework of Lie groups and algebras.

Contribution

It introduces explicit operators converting non-negative polynomials into sums of squares and analyzes their properties within Lie group frameworks.

Findings

Explicit operators A such that e^A maps positive polynomials to sums of squares.

Conditions under which e^A maps Pos(K) into sums of squares.

No bijective linear operator can map Pos(K) into sums of squares for compact K.

Abstract

We investigate linear operators $A:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$. We give explicit operators $A$ such that, for fixed $d\in\mathbb{N}_0$ and closed $K\subseteq\mathbb{R}^n$, $e^A\mathrm{Pos}(K)_{\leq 2d}\subseteq\sum\mathbb{R}[x_1,\dots,x_n]_{\leq d}^2$. We give an explicit operator $A$ such that $e^A\mathrm{Pos}(\mathbb{R}^n)\subseteq\sum\mathbb{R}[x_1,\dots,x_n]^2$. For $K\subseteq\mathbb{R}^n$, we give a condition such that $A$ exists with $e^A\mathrm{Pos}(K)\subseteq\sum\mathbb{R}[x_1,\dots,x_n]^2$. We show that, for compact $K\subseteq\mathbb{R}^n$, there is no bijective linear operator $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ with $T\mathrm{Pos}(K)\subseteq\sum\mathbb{R}[x_1,\dots,x_n]^2$. In the framework of regular Fr\'echet Lie groups and Lie algebras we investigate the linear operators $A$ such that $e^{tA}:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ is well-defined for all $t\in\mathbb{R}$. We give a three-line-proof of Stochel's Theorem.