Nash approximation of differentiable semialgebraic maps

TL;DR

This paper proves that Nash manifolds with corners can approximate any semialgebraic map with a certain smoothness, and shows that close Nash maps are Nash homotopic, advancing approximation theory in real algebraic geometry.

Contribution

It establishes Nash manifolds with corners as universal approximation targets for semialgebraic maps of any smoothness level, and links approximation closeness to Nash homotopy.

Findings

Nash manifolds with corners are $({ m N}, u)$-approximation target spaces for all $ u\, extgreater=0$.

Close Nash maps in the semialgebraic topology are Nash homotopic.

Provides a stronger approximation result connecting topology and Nash homotopy.

Abstract

Let $T\subset{\mathbb R}^n$ be a semialgebraic set and let $\mu\ge0$ be a non-negative integer. We say that $T$ is a {\em Nash $\mu$-approximation target space} (or a $({\mathcal N},\mu)$-${\tt ats}$ for short) if it has the following universal approximation property: {\em For each $m\in{\mathbb N}$ and each locally compact semialgebraic subset $S\subset{\mathbb R}^m$, the subspace of Nash maps ${\mathcal N}(S,T)$ is dense in the space ${\mathcal S}^\mu(S,T)$ of ${\mathcal C}^\mu$ semialgebraic maps between $S$ and $T$}. A necessary condition to be a $({\mathcal N},\mu)$-${\tt ats}$ is that $T$ is locally connected by analytic paths. In this paper we show: {\em Nash manifolds with corners are $({\mathcal N},\mu)$-${\tt ats}$ for each $\mu\geq0$}. As an application of a stronger version of the previous statement, we show that if two Nash maps $f,g:S\to Q$, where $S$ is a locally compact semialgebraic set of ${\mathbb R}^m$ and $Q$ is a Nash manifold with corners, are close enough in the (strong) Whitney's semialgebraic topology of ${\mathcal S}^0(S,T)$ (and consequently they are (continuous) semialgebraically homotopic), then $f,g$ are Nash homotopic.