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Transactions of the American Mathematical Society

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Polynomials that are positive on an interval


Authors: Victoria Powers and Bruce Reznick
Journal: Trans. Amer. Math. Soc. 352 (2000), 4677-4692
MSC (1991): Primary 14Q20; Secondary 26C99, 68W30
DOI: https://doi.org/10.1090/S0002-9947-00-02595-2
Published electronically: June 14, 2000
MathSciNet review: 1707203
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Abstract:

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 $h(x), p(x) \in \mathbb{R}[x]$ such that $\{ \alpha \in \mathbb{R} \mid h(\alpha) \geq 0 \} = [-1,1]$ and $p(x) > 0$ on $[-1,1]$, then there exist sums of squares $s(x), t(x) \in \mathbb{R}[x]$ such that $p(x) = s(x) + t(x) h(x)$. Explicit degree bounds for $s$ and $t$ are given, in terms of the degrees of $p$ and $h$ and the location of the roots of $p$. This is a special case of Schmüdgen's Theorem, and extends classical results on representations of polynomials positive on a compact interval. Polynomials positive on the non-compact interval $[0,\infty)$ are also considered.


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Additional Information

Victoria Powers
Affiliation: Department of Mathematics, Emory University, Atlanta, Georgia 30322
Email: vicki@mathcs.emory.edu

Bruce Reznick
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois
Email: reznick@math.uiuc.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02595-2
Received by editor(s): January 14, 1999
Published electronically: June 14, 2000
Additional Notes: The second author was supported in part by NSF Grant DMS 95-00507
Article copyright: © Copyright 2000 American Mathematical Society