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Fonctions régulues

Goulwen Fichou, Johannes Huisman, Frédéric Mangolte and Jean-Philippe Monnier

Abstract

Nous étudions l’anneau des fonctions rationnelles qui se prolongent par continuité sur n. Nous établissons plusieurs propriétés algébriques de cet anneau dont un Nullstellensatz fort. Nous étudions les propriétés schématiques associées et montrons une version régulue des théorèmes A et B de Cartan. Nous caractérisons géométriquement les idéaux premiers de cet anneau à travers leurs lieux d’annulation et montrons que les fermés régulus coïncident avec les fermés algébriquement constructibles.

We study the ring of rational functions admitting a continuous extension to the real affine space. We establish several properties of this ring. In particular, we prove a strong Nullstellensatz. We study the scheme theoretic properties and prove regulous versions of Theorems A and B of Cartan. We also give a geometrical characterization of prime ideals of this ring in terms of their zero-locus and relate them to euclidean closed Zariski-constructible sets.

Funding statement: Ce travail a bénéficié d’un support partiel provenant du contrat ANR BirPol ANR-11-JS01-004-01.

Acknowledgements

Nous remercions J. Kollár pour nous avoir transmis une version préliminaire de son article, ainsi que S. Cantat, M. Coste, L. Evain, W. Kucharz, K. Kurdyka, D. Naie et A. Parusiński pour l’intérêt précoce qu’ils ont porté à nos travaux et pour leurs suggestions qui ont contribué à améliorer ce texte. Merci aussi à F. Broglia et F. Acquistapace pour nous avoir signalé les références [2, 1]. La version finale de cet article doit beaucoup au referee dont la lecture attentive et les remarques ont été très constructives.

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Received: 2013-05-09
Revised: 2014-03-16
Published Online: 2015-01-10
Published in Print: 2016-09-01

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