Approximation on Nash sets with monomial singularities

TL;DR

This paper extends the approximation of semialgebraic functions by Nash functions to Nash sets with monomial singularities, establishing finiteness, normality, and extension properties, and applying these to Nash manifolds with corners.

Contribution

It introduces new approximation techniques for Nash functions on sets with monomial singularities and proves extension and normality results for such sets.

Findings

Nash sets with monomial singularities can be decomposed into finitely many Nash diffeomorphic parts.

Functions Nash on irreducible components extend to the ambient Nash manifold.

Affine Nash manifolds with divisorial corners are Nash diffeomorphic under certain conditions.

Abstract

This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash subsets X of M whose singularities are monomial. To that end we discuss first "finiteness" and "weak normality" for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic, for k>m^2, are also Nash diffeomorphic.