Making Non-Negative Polynomials into Sums of Squares

Abstract

We study linear operators $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$, especially for the purpose to move sets $S\subseteq\mathbb{R}[x_1,\dots,x_n]$ into cones $C\subseteq\mathbb{R}[x_1,\dots,x_n]$: $TS\subseteq C$. We develop the theory of (semi-)groups of operators $(e^{tA})_{t\in\mathbb{R}}$ on $\mathbb{R}[x_1,\dots,x_n]$, which requires techniques from regular Fr\'echet Lie groups. We study the special case of making non-negative polynomials $\mathrm{Pos}(K)_{\leq 2d}$ with $K\subseteq\mathbb{R}^n$ and $\mathrm{int}\, K\neq \emptyset$ into sums of squares: $\tilde{T}\mathrm{Pos}(K)_{\leq 2d}\subseteq \sum\mathbb{R}[x_1,\dots,x_n]_{\leq d}^2$. With $N:=\dim\mathbb{R}[x_1,\dots,x_n]_{\leq 2d} = \binom{n+2d}{n}$, for $\tilde{T}$, a memory of at most $2N+1$ is required. Matrix multiplications $\tilde{T}M$, $M\tilde{T}$, $\tilde{T}^{-1}M$, and $M\tilde{T}^{-1}$ of $\tilde{T}$ with any $M\in\mathbb{R}^{N\times N}$ require at most $4N^2+1$ operations. Transformations $\tilde{T}v$ and $\tilde{T}^{-1}v$ of vectors $v\in\mathbb{R}^N$ require at most $4N+1$ operations. Calculating $\tilde{T}^{-1}$ of $\tilde{T}$ requires only one (!) operation.