Representations of non-negative polynomials having finitely many zeros
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 3, p. 599-609

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.

Soit K une partie compacte de R n définie par les inégalités polynomiales g 1 0,...,g s 0. Pour un polynôme positif f sur K, des conditions suffisantes naturelles sont dégagées (en termes des dérivées premières et secondes en les zéros de f dans K) pour que f puisse se représenter sous la forme f=t 0 +t 1 g 1 ++t s g s , où les t i sont des sommes de carrés de polynômes. Les conditions sont bien plus générales que celles mises en évidence par Scheiderer dans [11, Cor. 2.6]. La démonstration utilise le théorème principal de Scheiderer [11] ainsi que des arguments de la théorie des formes quadratiques et de celle de la valuation. L’article explique également comment le lemme fondamental de Kuhlmann, Marshall et Schwartz [3] peut être mis à profit pour simplifier le théorème principal de Scheiderer, et compare les deux approches.

@article{AFST_2006_6_15_3_599_0,
     author = {Marshall, Murray},
     title = {Representations of non-negative polynomials having finitely many zeros},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 15},
     number = {3},
     year = {2006},
     pages = {599-609},
     doi = {10.5802/afst.1131},
     mrnumber = {2246416},
     zbl = {1130.13015},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2006_6_15_3_599_0}
}
Marshall, Murray. Representations of non-negative polynomials having finitely many zeros. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 3, pp. 599-609. doi : 10.5802/afst.1131. http://www.numdam.org/item/AFST_2006_6_15_3_599_0/

[1] Jacobi, T. A representation theorem for certain partially ordered commutative rings, Math. Zeit., Tome 237 (2001), pp. 223-235 | MR 1838311 | Zbl 0988.54039

[2] Jacobi, T.; Prestel, A. Distinguished presentations of strictly positive polynomials, J. reine angew. Math., Tome 532 (2001), pp. 223-235 | MR 1817508 | Zbl 1015.14029

[3] Kuhlmann, S.; Marshall, M.; Schwartz, N. Positivity, sums of squares and the multi-dimensional moment problem II (Advances in Geometry, to appear) | MR 1926876 | Zbl 02242759

[4] Lasserre, J. B. Optimization globale et théorie des moments, C. R. Acad. Sci. Paris, Série I, Tome 331 (2000), pp. 929-934 | MR 1806434 | Zbl 1016.90031

[5] Marshall, M. Optimization of polynomial functions, Canad. Math. Bull., Tome 46 (2003), pp. 575-587 | MR 2011395 | Zbl 1063.14071

[6] Marshall, M. Positive polynomials and sums of squares, Univ. Pisa (2000) (Ph. D. Thesis)

[7] Prestel, A.; Delzell, C. Positive Polynomials: From Hilbert’s 17th problem to real algebra, Springer Monographs in Mathematics (2001) | MR 1829790 | Zbl 0987.13016

[8] Putinar, M. Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J., Tome 42 (1993), pp. 969-984 | MR 1254128 | Zbl 0796.12002

[9] Scheiderer, C. Sums of squares of regular functions on real algebraic varieties, Trans. Amer. Math. Soc., Tome 352 (1999), pp. 1039-1069 | MR 1675230 | Zbl 0941.14024

[10] Scheiderer, C. Sums of squares on real algebraic curves, Math. Zeit., Tome 245 (2003), pp. 725-760 | MR 2020709 | Zbl 1056.14078

[11] Scheiderer, C. Distinguished representations of non-negative polynomials (to appear) | MR 2142385 | Zbl 02196738

[12] Schmüdgen, K. The K-moment problem for compact semi-algebraic sets, Math. Ann., Tome 289 (1991), pp. 203-206 | MR 1092173 | Zbl 0744.44008