Discrete & Computational Geometry最新文献_第2页

The Tropical Non-Properness Set of a Polynomial Map 多项式映射的热带非完备性集合

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-08-08 DOI: 10.1007/s00454-024-00684-4

Boulos El Hilany

{"title":"The Tropical Non-Properness Set of a Polynomial Map","authors":"Boulos El Hilany","doi":"10.1007/s00454-024-00684-4","DOIUrl":"https://doi.org/10.1007/s00454-024-00684-4","url":null,"abstract":"We study some discrete invariants of Newton non-degenerate polynomial maps (f: {mathbb {K}}^n rightarrow {mathbb {K}}^n) defined over an algebraically closed field of Puiseux series ({mathbb {K}}), equipped with a non-trivial valuation. It is known that the set ({mathcal {S}}(f)) of points at which f is not finite forms an algebraic hypersurface in ({mathbb {K}}^n). The coordinate-wise valuation of ({mathcal {S}}(f)cap ({mathbb {K}}^*)^n) is a piecewise-linear object in ({mathbb {R}}^n), which we call the tropical non-properness set of f. We show that the tropical polynomial map corresponding to f has fibers satisfying a particular combinatorial degeneracy condition exactly over points in the tropical non-properness set of f. We then use this description to outline a polyhedral method for computing this set, and to recover the fan dual to the Newton polytope of the set at which a complex polynomial map is not finite. The proofs rely on classical correspondence and structural results from tropical geometry, combined with a new description of ({mathcal {S}}(f)) in terms of multivariate resultants.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"15 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Convex Polytopes, Dihedral Angles, Mean Curvature and Scalar Curvature 凸多边形、二面角、平均曲率和标量曲率

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-07-25 DOI: 10.1007/s00454-024-00657-7

Misha Gromov

引用次数: 0

Noncrossing Partition Lattices from Planar Configurations 来自平面配置的非交叉分割网格

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-07-24 DOI: 10.1007/s00454-024-00682-6

Stella Cohen, Michael Dougherty, Andrew D. Harsh, Spencer Park Martin

引用次数: 0

Fast Algorithms for Minimum Homology Basis 最小同源性基础的快速算法

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-07-24 DOI: 10.1007/s00454-024-00680-8

Amritendu Dhar, Vijay Natarajan, Abhishek Rathod

{"title":"Fast Algorithms for Minimum Homology Basis","authors":"Amritendu Dhar, Vijay Natarajan, Abhishek Rathod","doi":"10.1007/s00454-024-00680-8","DOIUrl":"https://doi.org/10.1007/s00454-024-00680-8","url":null,"abstract":"We study the problem of finding a minimum homology basis, that is, a lightest set of cycles that generates the 1-dimensional homology classes with (mathbb {Z}_2) coefficients in a given simplicial complex K. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al. (LATIN 2018: Theoretical Informatics, Springer International Publishing, Cham, 2018), runs in (O(N m^{omega -1} + n m g)) time, where N denotes the total number of simplices in K, m denotes the number of edges in K, n denotes the number of vertices in K, g denotes the rank of the 1-homology group of K, and (omega ) denotes the exponent of matrix multiplication. In this paper, we present three conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex K. The first algorithm runs in (tilde{O}(m^omega )) time, the second algorithm runs in (O(N m^{omega -1})) time and the third algorithm runs in (tilde{O}(N^2,g + N m g{^2} + m g{^3})) time which is nearly quadratic time when (g=O(1)). We also study the problem of finding a minimum cycle basis in an undirected graph G with n vertices and m edges. The best known algorithm for this problem runs in (O(m^omega )) time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in (tilde{O}(m^omega )) time. We also provide a practical implementation of computing the minimum homology basis for general weighted complexes. The implementation is broadly based on the algorithmic ideas described in this paper, differing in its use of practical subroutines. Of these subroutines, the more costly step makes use of a parallel implementation, thus potentially addressing the issue of scale. We compare results against the currently known state of the art implementation (ShortLoop).","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141774188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Many Equiprojective Polytopes 许多等射多边形

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-07-23 DOI: 10.1007/s00454-024-00681-7

Théophile Buffière, Lionel Pournin

引用次数: 0

Elementary Fractal Geometry. 3. Complex Pisot Factors Imply Finite Type 初级分形几何3.复皮索特因子意味着有限类型

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-07-20 DOI: 10.1007/s00454-024-00678-2

Christoph Bandt

引用次数: 0

On the Connected Blocks Polytope 关于连块多面体

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-07-11 DOI: 10.1007/s00454-024-00675-5

Justus Bruckamp, Markus Chimani, Martina Juhnke

引用次数: 0

Equality Conditions for the Fractional Superadditive Volume Inequalities 分数超容积不等式的相等条件

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-07-05 DOI: 10.1007/s00454-024-00672-8

Mark Meyer

引用次数: 0

Topological Posets and Tropical Phased Matroids 拓扑 Posets 和热带相控 Matroids

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-07-02 DOI: 10.1007/s00454-024-00668-4

Ulysses Alvarez, Ross Geoghegan

引用次数: 0

Convergence of Laplacian Eigenmaps and Its Rate for Submanifolds with Singularities 拉普拉奇特征映射的收敛性及其对具有奇点的子实体的收敛率

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-07-02 DOI: 10.1007/s00454-024-00667-5

Masayuki Aino

引用次数: 0