Approximation of maps from algebraic polyhedra to real algebraic varieties

Abstract

Given a finite simplicial complex $\mathcal{K}$ in ${\mathbb{R}}^n$ and a real algebraic variety $Y,$ by a $\mathcal{K}$-regular map $|\mathcal{K}|\to Y$ we mean a continuous map whose restriction to every simplex in $\mathcal{K}$ is a regular map. A simplified version of our main result says that if $Y$ is a uniformly retract rational variety and if $k,l$ are integers satisfying $0\le l\le k,$ then every ${\mathcal{C}}^l$ map $|\mathcal{K}|\to Y$ can be approximated in the ${\mathcal{C}}^l$ topology by $\mathcal{K}$-regular maps of class ${\mathcal{C}}^k.$ By definition, $Y$ is uniformly retract rational if for every point $y\in Y$ there is a Zariski open neighborhood $V\subset Y$ of $y$ such that the identity map of $V$ is the composite of regular maps $V\to W\to V,$ where $W\subset {\mathbb{R}}^p$ is a Zariski open set for some $p$ depending on $y.$

20 pages, paper slightly reorganized, minor corrections made