Approximation by piecewise-regular maps

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 satisfying k ≥ l. Next we derive consequences regarding algebraization of topological vector bundles.