On Hilbert's 17th problem in low degree

TL;DR

This paper explores improvements to Pfister's theorem for representing positive semidefinite polynomials as sums of squares of rational functions, especially in low degree cases, providing tighter bounds under certain conditions.

Contribution

It establishes new bounds for expressing positive semidefinite polynomials as sums of squares of rational functions when the degree is low, refining Pfister's theorem.

Findings

For degree d ≤ 2n-2, positive semidefinite polynomials are sums of 2^n-1 squares.

The result extends to degree 2n when n is even, or n=3 or 5.

Improves understanding of sum of squares representations in low-degree cases.

Abstract

Artin solved Hilbert's 17th problem, proving that a real polynomial in $n$ variables that is positive semidefinite is a sum of squares of rational functions, and Pfister showed that only $2^n$ squares are needed. In this paper, we investigate situations where Pfister's theorem may be improved. We show that a real polynomial of degree $d$ in $n$ variables that is positive semidefinite is a sum of $2^n-1$ squares of rational functions if $d\leq 2n-2$. If $n$ is even, or equal to $3$ or $5$, this result also holds for $d=2n$.