TY - JOUR

T1 - Polynomials that are positive on an interval

AU - Powers, Victoria

AU - Reznick, Bruce

PY - 2000

Y1 - 2000

N2 - 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.

AB - 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.

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U2 - 10.1090/s0002-9947-00-02595-2

DO - 10.1090/s0002-9947-00-02595-2

M3 - Article

AN - SCOPUS:23044523312

VL - 352

SP - 4677

EP - 4692

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 10

ER -