Positive trace polynomials and the universal Procesi–Schacher conjecture

Igor Klep, Špela Špenko, Jurij Volčič

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

Positivstellensätze are fundamental results in real algebraic geometry providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this article, Positivstellensätze for trace polynomials positive on semialgebraic sets of n × n 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 Schmüdgen- and Putinar-type Positivstellensätze 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 Positivstellensätze is indispensable.

Original languageEnglish
Pages (from-to)1101-1134
Number of pages34
JournalProceedings of the London Mathematical Society
Volume117
Issue number6
DOIs
StatePublished - 1 Dec 2018

Keywords

  • 13J30 (primary)
  • 14P10 (secondary)
  • 16R30
  • 16W10

ASJC Scopus subject areas

  • Mathematics (all)

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