Approximation of maps between real algebraic varieties

TL;DR

This paper characterizes real algebraic varieties that can be approximated by regular maps, focusing on the approximation property and its variants, with implications for both smooth and continuous maps.

Contribution

It provides a complete characterization of varieties with the approximation property and explores conditions for regular approximation with interpolation.

Findings

Characterization of varieties with the approximation property

Conditions for approximation combined with interpolation

Extensions to singular real algebraic varieties

Abstract

A nonsingular real algebraic variety Y is said to have the approximation property if for every real algebraic variety X the following holds: if f:X-->Y is a C^inf map that is homotopic to a regular map, then f can be approximated in the C^inf topology by regular maps. In this paper, we characterize the varieties Y with the approximation property. We also characterize the varieties Y with the approximation property combined with a suitable interpolation condition. Some of our results have variants concerning the regular approximation of continuous maps defined on (possibly singular) real algebraic varieties.