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Nash approximation of differentiable semialgebraic maps

Antonio Carbone, José F. Fernando

TL;DR

This work establishes that Nash manifolds with corners are $({\mathcal N},\mu)$-${\tt ats}$ for all $\mu\ge0$, meaning every ${\mathcal S}^{\mu}$-map from a locally compact semialgebraic set to such a target can be approximated by Nash maps in the strong Whitney topology. The authors develop a robust toolkit—including pushing corners inside interiors with controlled derivatives, relative Nash approximation, and a trimmed Whitney topology—to prove universal approximation and to derive Nash homotopies from semialgebraic homotopies. These results bridge semialgebraic and Nash categories for manifold-with-corners targets, with direct applications to Nash homotopy theory and potential extensions to broader tame settings. The approach leverages pushing/diffeomorphism constructions, Nash envelopes, and compatibility with ${\mathcal S}^{\mu}$-topologies, providing a flexible framework for approximating differentiable semialgebraic maps by Nash maps in intricate geometric contexts.

Abstract

Let $T\subset{\mathbb R}^n$ be a semialgebraic set and let $μ\ge0$ be a non-negative integer. We say that $T$ is a {\em Nash $μ$-approximation target space} (or a $({\mathcal N},μ)$-${\tt ats}$ 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 ${\mathcal S}^μ(S,T)$ of ${\mathcal C}^μ$ semialgebraic maps between $S$ and $T$}. A necessary condition to be a $({\mathcal N},μ)$-${\tt ats}$ is that $T$ is locally connected by analytic paths. In this paper we show: {\em Nash manifolds with corners are $({\mathcal N},μ)$-${\tt ats}$ for each $μ\geq0$}. 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.

Nash approximation of differentiable semialgebraic maps

TL;DR

This work establishes that Nash manifolds with corners are - for all , meaning every -map from a locally compact semialgebraic set to such a target can be approximated by Nash maps in the strong Whitney topology. The authors develop a robust toolkit—including pushing corners inside interiors with controlled derivatives, relative Nash approximation, and a trimmed Whitney topology—to prove universal approximation and to derive Nash homotopies from semialgebraic homotopies. These results bridge semialgebraic and Nash categories for manifold-with-corners targets, with direct applications to Nash homotopy theory and potential extensions to broader tame settings. The approach leverages pushing/diffeomorphism constructions, Nash envelopes, and compatibility with -topologies, providing a flexible framework for approximating differentiable semialgebraic maps by Nash maps in intricate geometric contexts.

Abstract

Let be a semialgebraic set and let be a non-negative integer. We say that is a {\em Nash -approximation target space} (or a - for short) if it has the following universal approximation property: {\em For each and each locally compact semialgebraic subset , the subspace of Nash maps is dense in the space of semialgebraic maps between and }. A necessary condition to be a - is that is locally connected by analytic paths. In this paper we show: {\em Nash manifolds with corners are - for each }. As an application of a stronger version of the previous statement, we show that if two Nash maps , where is a locally compact semialgebraic set of and is a Nash manifold with corners, are close enough in the (strong) Whitney's semialgebraic topology of (and consequently they are (continuous) semialgebraically homotopic), then are Nash homotopic.
Paper Structure (27 sections, 28 theorems, 102 equations, 2 figures)

This paper contains 27 sections, 28 theorems, 102 equations, 2 figures.

Table of Contents

  1. Introduction
  2. State of the art
  3. Main results
  4. Applications of the main results and further developments
  5. Nash homotopies
  6. Further developments
  7. Whitney's semialgebraic topology
  8. Mostowski's trick
  9. Whitney's semialgebraic topology
  10. Extension of ${\mathcal{S}}^{\mu}$ maps and invariance of the topology
  11. Continuity of composition homomorphisms
  12. ${\mathcal{S}}^\mu$ and Nash embeddings of Nash manifolds
  13. Trimmed Whitney's semialgebraic topology
  14. Construction of the trimmed Whitney's semialgebraic topology
  15. Nash manifolds with corners
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let ${\EuScript S}\subset{\mathbb R}^m$ be a locally compact semialgebraic set, let ${\EuScript Q}$ be a Nash manifold with corners and let $f:{\EuScript S}\to{\EuScript Q}$ be an ${\mathcal{S}}^{\mu}$ map. Then there exist Nash maps $g:{\EuScript S}\to\operatorname{Int}({\EuScript Q})$ arbitrarily

Figures (2)

  • Figure 3.1: The Nash manifold with corners ${\EuScript Q}:=\{{\tt x}\geq 0,{\tt y}^2\leq{\tt x}^2-{\tt x}^4\}$.
  • Figure 7.1: Semialgebraic set ${\EuScript T}\subset{\mathbb R}^2$

Theorems & Definitions (61)

  • Theorem 1.1: Nash manifold with corners
  • Theorem 1.2: Relative Nash approximation
  • Theorem 1.3: Pushing a Nash manifold with corners inside its interior
  • Theorem 1.4: Nash approximation of homotopies
  • Corollary 1.5
  • Corollary 1.6
  • Lemma 1.7
  • Lemma 1.8
  • Conjecture 1.9
  • Remark 1.10
  • ...and 51 more