2013/07/02 by Elías Baro, Baro, Elías, José F. Fernando +4
Computer Science · Mathematics · #14P20 #32C05 #58A07 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:14P20 #msc:32C05 #msc:58A07
paper · pdf · doi:10.48550/arxiv.1307.0724
39 pages
arxiv created 2013/07/02 · openalex publication_date 2013/07/02 · arxiv updated 2013/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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>m2, are also Nash diffeomorphic.