Nash approximation of differentiable semialgebraic maps
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918296481169408 |
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| author | Carbone, Antonio Fernando, José F. |
| author_facet | Carbone, Antonio Fernando, José F. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_13164 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nash approximation of differentiable semialgebraic maps Carbone, Antonio Fernando, José F. Algebraic Geometry Primary: 14P10, 14P20, 41A99, Secondary: 41A10, 41A20, 58A07 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. |
| title | Nash approximation of differentiable semialgebraic maps |
| topic | Algebraic Geometry Primary: 14P10, 14P20, 41A99, Secondary: 41A10, 41A20, 58A07 |
| url | https://arxiv.org/abs/2601.13164 |