Fonctions Régulues

TL;DR

The paper develops a robust regulous framework that blends real algebraic geometry with continuous rational functions. It defines the rings $\mathcal{R}^k(\mathbf{R}^n)$ of $k$-regulous functions and analyzes their algebraic and topological structure, proving a strong $k$-regulous Nullstellensatz and Cartan-type theorems for affine regulous varieties. It establishes that the $k$-regulous topology is Noetherian (and hence the spectrum is Noetherian), and provides a geometric characterization of regulous closed sets as algebraically constructible, with irreducible regulous sets corresponding to Zariski-constructible irreducibles. The work also clarifies the relation between regulous functions and regular functions after blowups, and situates regulous geometry within arc-symmetric and constructible frameworks, extending Cartan-style results to this real-analytic setting. Overall, it furnishes a coherent real-algebraic theory that mirrors key aspects of algebraic geometry over algebraically closed fields while accommodating real-analytic extensions and blowup techniques.

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.

Figures (3)

• Figure 1: Courbe cubique avec un point isolé.
• Figure 2: Un fermé régulument irréductible.
• Figure 6: Une chaine d'ensembles symétriques par arcs.

Theorems & Definitions (82)

• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Proposition 2.9
• proof
• Proposition 2.11
• Proposition 2.12