Morse theory of Euclidean distance functions from algebraic hypersurfaces
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914546623447040 |
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| author | Guidolin, Andrea Lerario, Antonio Ren, Isaac Scolamiero, Martina |
| author_facet | Guidolin, Andrea Lerario, Antonio Ren, Isaac Scolamiero, Martina |
| contents | Let $Y\subseteq \mathbb{R}^n$ be a closed definable subset and $X\subseteq \mathbb{R}^n$ be a smooth manifold. We construct a version of Morse theory for the restriction to $X$ of the Euclidean distance function from $Y$. This is done using the notion of critical points of Lipschitz functions and applying the theory of continuous selections. In this theory, nondegenerate critical points have two indices: a quadratic index (as in classical Morse theory), and a piecewise linear index (that relates to the notion of bottlenecks). This framework is flexible enough to simultaneously treat and unify the study of two cases of interest for computational algebraic geometry: bottlenecks and nearest point problems. We provide a technical toolset guaranteeing the applicability of the theory to the case where $X, Y$ are generic algebraic hypersurfaces and use it to bound the number of critical points of the distance from $Y$ restricted to $X$, among other applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08639 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Morse theory of Euclidean distance functions from algebraic hypersurfaces Guidolin, Andrea Lerario, Antonio Ren, Isaac Scolamiero, Martina Algebraic Geometry Algebraic Topology Metric Geometry Primary: 14P25. Secondary: 51F99, 55N31 Let $Y\subseteq \mathbb{R}^n$ be a closed definable subset and $X\subseteq \mathbb{R}^n$ be a smooth manifold. We construct a version of Morse theory for the restriction to $X$ of the Euclidean distance function from $Y$. This is done using the notion of critical points of Lipschitz functions and applying the theory of continuous selections. In this theory, nondegenerate critical points have two indices: a quadratic index (as in classical Morse theory), and a piecewise linear index (that relates to the notion of bottlenecks). This framework is flexible enough to simultaneously treat and unify the study of two cases of interest for computational algebraic geometry: bottlenecks and nearest point problems. We provide a technical toolset guaranteeing the applicability of the theory to the case where $X, Y$ are generic algebraic hypersurfaces and use it to bound the number of critical points of the distance from $Y$ restricted to $X$, among other applications. |
| title | Morse theory of Euclidean distance functions from algebraic hypersurfaces |
| topic | Algebraic Geometry Algebraic Topology Metric Geometry Primary: 14P25. Secondary: 51F99, 55N31 |
| url | https://arxiv.org/abs/2402.08639 |