Positive univariate trace polynomials

Igor Klep, James Eldred Pascoe, Jurij Volčič

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(xj). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.

Original languageEnglish
Pages (from-to)303-317
Number of pages15
JournalJournal of Algebra
StatePublished - 1 Aug 2021
Externally publishedYes


  • Hankel matrix
  • Positivstellensatz
  • Power mean inequality
  • Real algebraic geometry
  • Trace polynomial

ASJC Scopus subject areas

  • Algebra and Number Theory


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