Nash regulous functions

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.