Nash regulous functions

TL;DR

This paper introduces Nash k-regulous functions, extending the concept of k-regulous functions to Nash manifolds, and explores their properties including analogues of classical theorems like Nullstellensatz and Cartan's theorems A and B.

Contribution

The paper defines Nash k-regulous functions on Nash manifolds and establishes foundational results, extending classical algebraic geometry theorems to this new setting.

Findings

Nash k-regulous functions are of class C^k and can be expressed as quotients of Nash functions.

Nullstellensatz and Cartan's theorems A and B are valid for Nash k-regulous functions.

The framework generalizes previous results on regulous functions to Nash manifolds.

Abstract

A real-valued function on R^n is k-regulous, where k is a nonnegative integer, if it is of class C^k and can be represented as a quotient of two polynomial functions on R^n. Several interesting results involving such functions have been obtained recently. Some of them (Nullstellensatz, Cartan's theorems A and B, etc.) can be carried over to a new setting of Nash regulous functions, introduced in this paper. Here a function on a Nash manifold X is called Nash k-regulous if it is of class C^k and can be represented as a quotient of two Nash functions on X.