Approximation of maps from algebraic polyhedra to real algebraic varieties
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912285068361728 |
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| author | Bilski, Marcin Kucharz, Wojciech |
| author_facet | Bilski, Marcin Kucharz, Wojciech |
| contents | 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 $y.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation of maps from algebraic polyhedra to real algebraic varieties Bilski, Marcin Kucharz, Wojciech Algebraic Geometry 14P05, 26C15 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 $y.$ |
| title | Approximation of maps from algebraic polyhedra to real algebraic varieties |
| topic | Algebraic Geometry 14P05, 26C15 |
| url | https://arxiv.org/abs/2410.23457 |