Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T05:38:03.824Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximating Positive Polynomials Using Sums of Squares

Published online by Cambridge University Press:  20 November 2018

M. Marshall
M. Marshall*
Affiliation:
Department of Computer Science University of Saskatchewan Saskatoon, Saskatchewan S7N 5E6, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The paper considers the relationship between positive polynomials, sums of squares and the multi-dimensional moment problem in the general context of basic closed semi-algebraic sets in real $n$-space. The emphasis is on the non-compact case and on quadratic module representations as opposed to quadratic preordering presentations. The paper clarifies the relationship between known results on the algebraic side and on the functional-analytic side and extends these results in a variety of ways.

Keywords

14P1044A60
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 3 , 01 September 2003 , pp. 400 - 418
Copyright
Copyright © Canadian Mathematical Society 2003

References

[1] Becker, E. and Powers, V., Sums of powers in rings and the real holomorphy ring. J. Reine Angew. Math 480 (1996), 71–103.Google Scholar
[2] Bochnak, J., Coste, M. and Roy, M.-F., Géométrie algébrique réelle. Springer-Verlag, 1987.Google Scholar
[3] Choquet, G., Lectures on analysis, Volume II. Benjamin Math. Lecture Note Series, 1969.Google Scholar
[4] Haviland, E. K., On the momentum problem for distribution functions in more than one dimension. Amer. J. Math. 57 (1935), 562–572.Google Scholar
[5] Haviland, E. K., On the momentum problem for distribution functions in more than one dimension II. Amer. J. Math. 58 (1936), 164–168.Google Scholar
[6] Jacobi, T., A representation theorem for certain partially ordered commutative rings. Math. Z. 237(2001).Google Scholar
[7] Jacobi, T. and Prestel, A., Distinguished representations of strictly positive polynomials. J. Reine Angew.Math. 532 (2001), 223–235.Google Scholar
[8] Kelley, J. L. and Srinivasan, T. P., Measure and Integral, Volume 1. Graduate Texts in Math. 116, Springer-Verlag, 1988.Google Scholar
[9] Kuhlmann, S. and Marshall, M., Positivity, sums of squares and the multi-dimensional moment problem. Trans. Amer.Math. Soc. 354(2002), 4285 4301.Google Scholar
[10] Marshall, M., Extending the archimedean Positivstellensatz to the non-compact case. Canad. Math. Bull. 44(2001), 223 230.Google Scholar
[11] Marshall, M., Positive polynomials and sums of squares. Dottorato de Ricerca in Matematica, Dept. di Mat., Univ. Pisa, 2000.Google Scholar
[12] Marshall, M., A general representation theorem for partially ordered commutative rings. Math. Z. 242(2002), 217 225.Google Scholar
[13] Parrilo, P. and Sturmfels, B., Minimizing polynomial functions. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, to appear.Google Scholar
[14] Powers, V. and Scheiderer, C., The moment problem for non-compact semialgebraic sets. Adv. Geom. 1(2001), 71 88.Google Scholar
[15] Prestel, A., Lectures on formally real Fields. Springer Lecture Notes in Math. 1093, 1984.Google Scholar
[16] Putinar, M., Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. (3) 42(1993), 969 984.Google Scholar
[17] Putinar, M. and F.-H. Vasilescu, Solving the moment problem by dimension extension. Ann. of Math. 149(1999), 1087 1107.Google Scholar
[18] Scheiderer, C., Sums of squares of regular functions on real algebraic varieties. Trans. Amer. Math. Soc. 352(1999), 1030 1069.Google Scholar
[19] Schmdgen, K., The K-moment problem for compact semi-algebraic sets. Math. Ann. 289(1991), 203 206.Google Scholar
[20] Schweighofer, M., Iterated rings of bounded elements and generalizations of Schmdgen's Positivstellensatz. J. Reine Angew.Math. 554(2003), 19 45.Google Scholar
[21] Wörmann, T., Short algebraic proofs of theorems of Schmdgen and Plya. preprint.Google Scholar