Representations of non-negative polynomials having finitely many zeros

Murray Marshall[1]

  • [1] Department of Computer Science, University of Saskatchewan, Saskatoon, SK Canada, S7N 5E6

Annales de la faculté des sciences de Toulouse Mathématiques (2006)

  • Volume: 15, Issue: 3, page 599-609
  • ISSN: 0240-2963

Abstract

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Consider a compact subset K of real n -space defined by polynomial inequalities g 1 ≥ 0 , ⋯ , g s ≥ 0 . For a polynomial f non-negative on K , natural sufficient conditions are given (in terms of first and second derivatives at the zeros of f in K ) for f to have a presentation of the form f = t 0 + t 1 g 1 + ⋯ + t s g s , t i a sum of squares of polynomials. The conditions are much less restrictive than the conditions given by Scheiderer in [11, Cor. 2.6]. The proof uses Scheiderer’s main theorem in [11] as well as arguments from quadratic form theory and valuation theory. We also explain how the basic lemma of Kuhlmann, Marshall and Schwartz in [3] can be used to simplify the proof of Scheiderer’s main theorem, and compare the two approaches.

How to cite

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Marshall, Murray. "Representations of non-negative polynomials having finitely many zeros." Annales de la faculté des sciences de Toulouse Mathématiques 15.3 (2006): 599-609. <http://eudml.org/doc/10014>.

@article{Marshall2006,
abstract = {Consider a compact subset $K$ of real $n$-space defined by polynomial inequalities $g_1\ge 0, \dots ,g_s\ge 0$. For a polynomial $f$ non-negative on $K$, natural sufficient conditions are given (in terms of first and second derivatives at the zeros of $f$ in $K$) for $f$ to have a presentation of the form $f=t_0+t_1g_1+\dots +t_sg_s$, $t_i$ a sum of squares of polynomials. The conditions are much less restrictive than the conditions given by Scheiderer in [11, Cor. 2.6]. The proof uses Scheiderer’s main theorem in [11] as well as arguments from quadratic form theory and valuation theory. We also explain how the basic lemma of Kuhlmann, Marshall and Schwartz in [3] can be used to simplify the proof of Scheiderer’s main theorem, and compare the two approaches.},
affiliation = {Department of Computer Science, University of Saskatchewan, Saskatoon, SK Canada, S7N 5E6},
author = {Marshall, Murray},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {real algebraic variety; semi-algebraic set; coordinate ring; quadratic module; Archimedean; positive function; ring of continuous functions},
language = {eng},
number = {3},
pages = {599-609},
publisher = {Université Paul Sabatier, Toulouse},
title = {Representations of non-negative polynomials having finitely many zeros},
url = {http://eudml.org/doc/10014},
volume = {15},
year = {2006},
}

TY - JOUR
AU - Marshall, Murray
TI - Representations of non-negative polynomials having finitely many zeros
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2006
PB - Université Paul Sabatier, Toulouse
VL - 15
IS - 3
SP - 599
EP - 609
AB - Consider a compact subset $K$ of real $n$-space defined by polynomial inequalities $g_1\ge 0, \dots ,g_s\ge 0$. For a polynomial $f$ non-negative on $K$, natural sufficient conditions are given (in terms of first and second derivatives at the zeros of $f$ in $K$) for $f$ to have a presentation of the form $f=t_0+t_1g_1+\dots +t_sg_s$, $t_i$ a sum of squares of polynomials. The conditions are much less restrictive than the conditions given by Scheiderer in [11, Cor. 2.6]. The proof uses Scheiderer’s main theorem in [11] as well as arguments from quadratic form theory and valuation theory. We also explain how the basic lemma of Kuhlmann, Marshall and Schwartz in [3] can be used to simplify the proof of Scheiderer’s main theorem, and compare the two approaches.
LA - eng
KW - real algebraic variety; semi-algebraic set; coordinate ring; quadratic module; Archimedean; positive function; ring of continuous functions
UR - http://eudml.org/doc/10014
ER -

References

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  2. T. Jacobi, A. Prestel, Distinguished presentations of strictly positive polynomials, J. reine angew. Math. 532 (2001), 223-235 Zbl1015.14029MR1817508
  3. S. Kuhlmann, M. Marshall, N. Schwartz, Positivity, sums of squares and the multi-dimensional moment problem II Zbl1095.14055MR1926876
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  5. M. Marshall, Optimization of polynomial functions, Canad. Math. Bull. 46 (2003), 575-587 Zbl1063.14071MR2011395
  6. M. Marshall, Positive polynomials and sums of squares, (2000) 
  7. A. Prestel, C. Delzell, Positive Polynomials: From Hilbert’s 17th problem to real algebra, (2001), Springer Monographs in Mathematics Zbl0987.13016MR1829790
  8. M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J. 42 (1993), 969-984 Zbl0796.12002MR1254128
  9. C. Scheiderer, Sums of squares of regular functions on real algebraic varieties, Trans. Amer. Math. Soc. 352 (1999), 1039-1069 Zbl0941.14024MR1675230
  10. C. Scheiderer, Sums of squares on real algebraic curves, Math. Zeit. 245 (2003), 725-760 Zbl1056.14078MR2020709
  11. C. Scheiderer, Distinguished representations of non-negative polynomials Zbl1082.14058MR2142385
  12. K. Schmüdgen, The K -moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), 203-206 Zbl0744.44008MR1092173

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