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2017 On Hilbert's 17th problem in low degree
Olivier Benoist
Algebra Number Theory 11(4): 929-959 (2017). DOI: 10.2140/ant.2017.11.929

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 2n 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 2n 1 squares of rational functions if d 2n 2. If n is even or equal to 3 or 5, this result also holds for d = 2n.

Citation

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Olivier Benoist. "On Hilbert's 17th problem in low degree." Algebra Number Theory 11 (4) 929 - 959, 2017. https://doi.org/10.2140/ant.2017.11.929

Information

Received: 11 July 2016; Revised: 5 January 2017; Accepted: 3 February 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 06735375
MathSciNet: MR3665641
Digital Object Identifier: 10.2140/ant.2017.11.929

Subjects:
Primary: 11E25
Secondary: 14F20, 14P99

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.11 • No. 4 • 2017
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