Differentiable approximation of continuous semialgebraic maps

Abstract

In this work we approach the problem of approximating uniformly continuous semialgebraic maps f : S Ñ T from a compact semialgebraic set S to an arbitrary semialgebraic set T by semialgebraic maps g : S Ñ T that are differentiable of class C ν for a fixed integer ν ě 1. As the reader can expect, the difficulty arises mainly when one tries to keep the same target space after approximation. For ν " 1 we give a complete affirmative solution to the problem: such a uniform approximation is always possible. For ν ě 2 we obtain density results in the two following relevant situations: either T is compact and locally C ν 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 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.

Approximation by piecewise-regular maps

“…The problem of algebraic approximation of continuous maps between real algebraic varieties has been considered by several mathematicians (see [1], [4], [10], [21] and the references therein). It is well known that such maps can be approximated by continuous semialgebraic maps in the compact-open topology.…”