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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
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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  • S. Bernstein, Sur la représentation des polynômes positifs, Soobshch. Kharkov matem. ob-va, ser. 2, 14 (1915), 227–228.
  • Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960
  • Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635
  • Tamás Erdélyi, Estimates for the Lorentz degree of polynomials, J. Approx. Theory 67 (1991), no. 2, 187–198. MR 1133059, DOI
  • T. Erdélyi and J. Szabados, On polynomials with positive coefficients, J. Approx. Theory 54 (1988), no. 1, 107–122. MR 951032, DOI
  • J. Franel, solution, Intermèd. des math. 1 (1894), 253–254.
  • E. Goursat, solution, Intermèd. des math. 1 (1894), 251.
  • David Handelman, Representing polynomials by positive linear functions on compact convex polyhedra, Pacific J. Math. 132 (1988), no. 1, 35–62. MR 929582
  • G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed., Cambridge Univ. Press, 1952.
  • F. Hausdorff, Summationsmethoden und Momentfolgen I, Math. Zeit. 9 (1921), 74–109.
  • C. Hermite, problem, Intermèd. des math. 1 (1894), 65–66.
  • S. Karlin and L. S. Shapley, Geometry of Moment Spaces, Memoirs of the Amer. Math. Soc, 12, 1953.
  • Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0204922
  • Jesús A. de Loera and Francisco Santos, An effective version of Pólya’s theorem on positive definite forms, J. Pure Appl. Algebra 108 (1996), no. 3, 231–240. MR 1384003, DOI
  • J. A. de Loera and F. Santos, Correction to An effective version of Pòlya theorem on positive definite forms, J. Pure Appl. Alg., to appear.
  • Charles A. Micchelli and Allan Pinkus, Some remarks on nonnegative polynomials on polyhedra, Probability, statistics, and mathematics, Academic Press, Boston, MA, 1989, pp. 163–186. MR 1031284
  • G. Pólya, Über positive Darstellung von Polynomen Vierteljschr, Naturforsch. Ges. Zürich 73 (1928 141–145, in Collected Papers 2 (1974), MIT Press, 309-313.
  • G. Pólya and G. Szegö, Problems and Theorems in Analysis II, Springer-Verlag, New York, 1976.
  • V. Powers and B. Reznick, A new bound for Pólya’s Theorem with applications to polynomials positive on polyhedra, to appear in Proceedings of the MEGA 2000 conference.
  • B. Reznick, Some Concrete Aspects of Hilbert’s 17th Problem, to appear in RAGOS Proceedings, Contemp. Math. 253 (2000), 251–272.
  • J. Sadier, solution, Intermèd. des math. 1 (1894), 251–253.
  • C. Scheiderer, Sums of squares of regular functions on real algebraic varieties, to appear in Trans. Amer. Math. Soc.
  • Konrad Schmüdgen, The $K$-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), no. 2, 203–206. MR 1092173, DOI
  • Gilbert Stengle, Complexity estimates for the Schmüdgen Positivstellensatz, J. Complexity 12 (1996), no. 2, 167–174. MR 1398323, DOI

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Additional Information

Victoria Powers
Affiliation: Department of Mathematics, Emory University, Atlanta, Georgia 30322

Bruce Reznick
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois
MR Author ID: 147525

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