The bottleneck degree of algebraic varieties
| Autor: | Sandra Di Rocco, Madeleine Weinstein, David Eklund |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Discrete mathematics
Algebra and Number Theory Degree (graph theory) Applied Mathematics 010102 general mathematics Nonlinear dimensionality reduction Algebraic variety 010103 numerical & computational mathematics Reach 01 natural sciences Bottleneck Manifold learning Mathematics - Algebraic Geometry Euclidean geometry Line (geometry) FOS: Mathematics Geometry and Topology 0101 mathematics Algebraic Geometry (math.AG) Polar classes Mathematics |
| Zdroj: | Di Rocco, S, Eklund, D & Weinstein, M 2020, ' The bottleneck degree of algebraic varieties ', SIAM Journal on Applied Algebra and Geometry, vol. 4, no. 1, pp. 227-253 . https://doi.org/10.1137/19M1265776 |
| DOI: | 10.1137/19M1265776 |
| Popis: | A bottleneck of a smooth algebraic variety $X \subset \mathbb{C}^n$ is a pair of distinct points $(x,y) \in X$ such that the Euclidean normal spaces at $x$ and $y$ contain the line spanned by $x$ and $y$. The narrowness of bottlenecks is a fundamental complexity measure in the algebraic geometry of data. In this paper we study the number of bottlenecks of affine and projective varieties, which we call the bottleneck degree. The bottleneck degree is a measure of the complexity of computing all bottlenecks of an algebraic variety, using for example numerical homotopy methods. We show that the bottleneck degree is a function of classical invariants such as Chern classes and polar classes. We give the formula explicitly in low dimension and provide an algorithm to compute it in the general case. Major revision. New introduction. Added some new illustrative lemmas and figures. Added pseudocode for the algorithm to compute bottleneck degree. Fixed some typos |
| Databáze: | OpenAIRE |
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