Analytic and differentiable functions vanishing on an algebraic set

Abstract:

Let $U$ be an open semi-algebraic subset of ${{\mathbf {R}}^n}$ and let $X$ be a closed analytic subset of $U$ which is also a semi-algebraic set (e.g., $U = {{\mathbf {R}}^n}$ and $X$ is an algebraic subset of ${{\mathbf {R}}^n}$). It is proved that the ideal of analytic functions on $U$ vanishing on $X$ is finitely generated provided that the set $X$ is coherent. The ideal of infinitely differentiable functions on $U$ vanishing on $X$ is finitely generated if and only if the set $X$ is coherent.