# Approximation by piecewise-regular maps

## TL;DR

This paper proves that smooth maps from compact subsets of real algebraic varieties into uniformly rational varieties can be approximated by piecewise-regular maps, with implications for algebraizing topological vector bundles.

## Contribution

It establishes a general approximation theorem for C^l maps into uniformly rational varieties using piecewise-regular maps, extending previous approximation results.

## Key findings

- C^l maps can be approximated by piecewise-regular maps of class C^k
- Results apply to algebraization of topological vector bundles
- Provides a new method for approximation in real algebraic geometry

## Abstract

A real algebraic variety W of dimension m is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of R^m. Let l be any nonnegative integer. We prove that every map of class C^l from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the C^l topology by piecewise-regular maps of class C^k, where k is an arbitrary integer greater than or equal to l. Next we derive consequences regarding algebraization of topological vector bundles.