Analytic and differentiable functions vanishing on an algebraic set

Abstract

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