Approximation of maps from algebraic polyhedra to real algebraic varieties

TL;DR

This paper proves that continuous maps from algebraic polyhedra to certain rational varieties can be approximated by regular maps, extending the approximation theory in real algebraic geometry.

Contribution

It establishes a new approximation result for maps into uniformly retract rational varieties, generalizing previous approximation theorems.

Findings

Any $ ext{C}^l$ map can be approximated by $ ext{K}$-regular maps in the $ ext{C}^l$ topology.

The approximation holds for maps into uniformly retract rational varieties.

The result applies to maps from simplicial complexes in $ ext{R}^n$.

Abstract

Given a finite simplicial complex $\mathcal{K}$ in $\mathbb{R}^n$ and a real algebraic variety $Y,$ by a $\mathcal{K}$-regular map $|\mathcal{K}|\rightarrow Y$ 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 $0\leq l\leq k,$ then every $\mathcal{C}^l$ map $|\mathcal{K}|\rightarrow Y$ 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 $V\rightarrow W\rightarrow V,$ where $W\subset\mathbb{R}^p$ is a Zariski open set for some $p$ depending on $y.$