Fonctions R\'egulues

TL;DR

This paper investigates the properties of the ring of rational functions that extend continuously over real affine space, establishing key algebraic and geometric theorems and characterizations.

Contribution

It introduces the regulous functions ring, proves a strong Nullstellensatz, and provides geometric characterizations of prime ideals, extending classical theorems to this context.

Findings

Proved a strong Nullstellensatz for regulous functions

Characterized prime ideals via zero-loci and Zariski-constructible sets

Established regulous versions of Cartan's Theorems A and B

Abstract

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 Nullstelensatz. 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.