Differentiable approximation of continuous semialgebraic maps

TL;DR

This paper studies how to approximate continuous semialgebraic maps with differentiable ones, providing complete solutions for the case of first-order differentiability and density results under certain conditions for higher orders.

Contribution

It offers a complete affirmative solution for $ u=1$ and establishes density results for $ u extgreater= 2$ in specific semialgebraic contexts, using new approximation techniques.

Findings

Uniform approximation is always possible for $ u=1$.

Density results hold when target is a polyhedron or Nash set for $ u extgreater= 2$.

Results are sharp with explicit examples.

Abstract

In this work we approach the problem of approximating uniformly continuous semialgebraic maps $f:S\to T$ from a compact semialgebraic set $S$ to an arbitrary semialgebraic set $T$ by semialgebraic maps $g:S\to T$ that are differentiable of class~${\mathcal C}^\nu$ for a fixed integer $\nu\geq1$. As the reader can expect, the difficulty arises mainly when one tries to keep the same target space after approximation. For $\nu=1$ we give a complete affirmative solution to the problem: such a uniform approximation is always possible. For $\nu \geq 2$ we obtain density results in the two following relevant situations: either $T$ is compact and locally ${\mathcal C}^\nu$ semialgebraically equivalent to a polyhedron, for instance when $T$ is a compact polyhedron; or $T$ is an open semialgebraic subset of a Nash set, for instance when $T$ is a Nash set. Our density results are based on a recent ${\mathcal C}^1$-triangulation theorem for semialgebraic sets due to Ohmoto and Shiota, and on new approximation techniques we develop in the present paper. Our results are sharp in a sense we specify by explicit examples.