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Abstract

A real-valued function on $\mathbb {R}^n$ is $k$-regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and can be represented as a quotient of two polynomial functions on $\mathbb {R}^n$. 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 $k$-regulous functions, introduced in this paper. Here a function on a Nash manifold $X$ is called Nash $k$-regulous if it is of class $\mathcal {C}^k$ and can be represented as a quotient of two Nash functions on $X$.