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
SN - 0002-9947
VL - 352
SP - 4677
EP - 4692
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 10
ER -