Positive polynomials on fibre products

Salma Kuhlmann; Mihai Putinar

doi:10.1016/j.crma.2007.04.009

Optimal Control/Algebraic Geometry

Positive polynomials on fibre products
[Polynômes positifs sur des produits fibrés]

Présenté par : Michel Raynaud

Salma Kuhlmann ¹ ; Mihai Putinar ²

¹ Department of Mathematics and Statistics, University of Saskatchewan, S7N 5E6, Canada
² Mathematics Department, University of California, Santa Barbara, CA 93106, USA

Comptes Rendus. Mathématique, Volume 344 (2007) no. 11, pp. 681-684.

Résumé
Abstract

Nous présentons une interprétation algébrique (dans le langage des produits fibrés de variétés algébriques) de résultats récents en théorie de l'optimisation concernant la structure de polynômes positifs (sur un sous ensemble compact et semi-algébrique $K \subset R^{n}$ ) qui satisfont certaines conditions de séparation des variables dans leurs monômes. Ceci offre la perspective d'un traitement uniforme de tels polynômes, positifs sur K non-compact, ou non-semi-algébrique, ainsi que pour des polynômes en un nombre dénombrable de variables.

Recent investigations in optimization theory concerning the structure of positive polynomials with a sparsity pattern are interpreted in the more invariant language of (iterated) fibre products of real algebraic varieties. This opens the perspective of treating on a unifying basis the cases of positivity on unbounded supports, on non-semialgebraic supports, or of polynomials depending on countably many variables.

Reçu le : 2006-12-05
Accepté le : 2007-04-03
Publié le : 2007-05-29

DOI : 10.1016/j.crma.2007.04.009

Affiliations des auteurs :

Salma Kuhlmann ¹ ; Mihai Putinar ²

¹ Department of Mathematics and Statistics, University of Saskatchewan, S7N 5E6, Canada
² Mathematics Department, University of California, Santa Barbara, CA 93106, USA

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Salma Kuhlmann; Mihai Putinar. Positive polynomials on fibre products. Comptes Rendus. Mathématique, Volume 344 (2007) no. 11, pp. 681-684. doi : 10.1016/j.crma.2007.04.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.04.009/

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Cité par Sources :

^⁎ Partially supported by an NSERC Discovery Grant, Canada and the National Science Foundation-USA.

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