#### Vol. 11, No. 4, 2017

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On Hilbert's 17th problem in low degree

### Olivier Benoist

Vol. 11 (2017), No. 4, 929–959
##### 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 $d\le 2n-2$. If $n$ is even or equal to $3$ or $5$, this result also holds for $d=2n$.

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