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Journal article
Positive trace polynomials and the universal Procesi-Schacher conjecture
Proceedings of the London Mathematical Society, v 117(6), pp 1101-1134
01 Dec 2018
Abstract
Positivstellensatze are fundamental results in real algebraic geometry providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this article, Positivstellensatze for trace polynomials positive on semialgebraic sets of nxn matrices are provided. A Krivine-Stengle-type Positivstellensatz is proved characterizing trace polynomials nonnegative on a general semialgebraic set K using weighted sums of Hermitian squares with denominators. The weights in these certificates are obtained from generators of K and traces of Hermitian squares. For compact semialgebraic sets K Schmudgen- and Putinar-type Positivstellensatze are obtained: every trace polynomial positive on K has a sum of Hermitian squares decomposition with weights and without denominators. The methods employed are inspired by invariant theory, classical real algebraic geometry and functional analysis. Procesi and Schacher in 1976 developed a theory of orderings and positivity on central simple algebras with involution and posed a Hilbert's 17th problem for a universal central simple algebra of degree n: is every totally positive element a sum of Hermitian squares? They gave an affirmative answer for n=2. In this paper, a negative answer for n=3 is presented. Consequently, including traces of Hermitian squares as weights in the Positivstellensatze is indispensable.
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Details
- Title
- Positive trace polynomials and the universal Procesi-Schacher conjecture
- Creators
- Igor Klep - University of AucklandSpela Spenko - Vrije Universiteit BrusselJurij Volcic - Ben-Gurion University of the Negev
- Publication Details
- Proceedings of the London Mathematical Society, v 117(6), pp 1101-1134
- Publisher
- Wiley
- Number of pages
- 34
- Grant note
- University of Auckland Doctoral Scholarship 665501 / European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant Research Foundation Flanders (FWO); FWO SCHW 1723/1-1 / Deutsche Forschungsgemeinschaft (DFG); German Research Foundation (DFG) Marsden Fund Council of the Royal Society of New Zealand; Royal Society of New Zealand; Marsden Fund (NZ) J1-8132; N1-0057; P1-0222 / Slovenian Research Agency; Slovenian Research Agency - Slovenia
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000451840300001
- Scopus ID
- 2-s2.0-85057804565
- Other Identifier
- 991021861883504721
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- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics