# 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

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topMarshall, 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 -

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