Polynomials that are positive on an interval

Victoria Powers, Bruce Reznick

Research output: Contribution to journalArticlepeer-review


This paper discusses representations of polynomials that are positive on intervals of the real line. An elementary and constructive proof of the following is given: If h(x),p(x) ∈ℝ[x] such that {α ∈ ℝ | h(α) ≥ 0} = [-1,1] and p(x) 0 on [-1,1], then there exist sums of squares s(x), t(x) ∈ ℝ[x] such that p(x) = s(x) + t(x)h(x). Explicit degree bounds for s and t arc given, in terms of the degrees of p and h and the location of the roots of p. This is a special case of Schmudgen's Theorem, and extends classical results on representations of polynomials positive on a compact interval. Polynomials positive on the non-compact interval [0, ∞) are also considered.

Original languageEnglish (US)
Pages (from-to)4677-4692
Number of pages16
JournalTransactions of the American Mathematical Society
Issue number10
StatePublished - 2000

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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