Pólya's Theorem with zeros

Mari Castle, Victoria Powers, Bruce Reznick

Research output: Contribution to journalArticlepeer-review


Let R[X] be the real polynomial ring in n variables. Pólya's Theorem says that if a homogeneous polynomial pεR[X] is positive on the standard n-simplex δn, then for sufficiently large N all the coefficients of (X1+...+Xn)Np are positive. We give a complete characterization of forms, possibly with zeros on δn, for which there exists N so that all coefficients of (X1+...+Xn)Np have only nonnegative coefficients, along with a bound on the N needed.

Original languageEnglish (US)
Pages (from-to)1039-1048
Number of pages10
JournalJournal of Symbolic Computation
Issue number9
StatePublished - Sep 2011


  • Positive polynomial
  • Pólya's Theorem
  • Sums of squares

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics


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