On Hilbert’s 17th problem in low degree

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 $d=2n$.