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Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation最新文献

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Exponent Equations in HNN-extensions hnn扩展中的指数方程
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2022-02-08 DOI: 10.1145/3476446.3535482
Michael Figelius, Markus Lohrey
{"title":"Exponent Equations in HNN-extensions","authors":"Michael Figelius, Markus Lohrey","doi":"10.1145/3476446.3535482","DOIUrl":"https://doi.org/10.1145/3476446.3535482","url":null,"abstract":"We consider exponent equations in finitely generated groups. These are equations, where the variables appear as exponents of group elements and take values from the natural numbers. Solvability of such (systems of) equations has been intensively studied for various classes of groups in recent years. In many cases, it turns out that the set of all solutions on an exponent equation is a semilinear set that can be constructed effectively. Such groups are called knapsack semilinear. The class of knapsack semilinear groups is quite rich and it is closed under many group theoretic constructions, e.g., finite extensions, graph products, wreath products, amalgamated free products with finite amalgamated subgroups, and HNN-extensions with finite associated subgroups. On the other hand, arbitrary HNN-extensions do not preserve knapsack semilinearity. In this paper, we consider the knapsack semilinearity of HNN-extensions, where the stable letter t acts trivially by conjugation on the associated subgroup A of the base group G. We show that under some additional technical conditions, knapsack semilinearity transfers from the base group G to the HNN-extension. These additional technical conditions are satisfied in many cases, e.g., when A is a centralizer in G or A is a quasiconvex subgroup of the hyperbolic group G.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129863237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Marginal Independence Models 边际独立模型
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2021-12-19 DOI: 10.1145/3476446.3536193
T. Boege, Sonja Petrovi'c, B. Sturmfels
{"title":"Marginal Independence Models","authors":"T. Boege, Sonja Petrovi'c, B. Sturmfels","doi":"10.1145/3476446.3536193","DOIUrl":"https://doi.org/10.1145/3476446.3536193","url":null,"abstract":"We impose rank one constraints on marginalizations of a tensor, given by a simplicial complex. Following work of Kirkup and Sullivant, such marginal independence models can be made toric by a linear change of coordinates. We study their toric ideals, with emphasis on random graph models and independent set polytopes of matroids. We develop the numerical algebra of parameter estimation, using both Euclidean distance and maximum likelihood, and we present a comprehensive database of small models.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115063027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Explicit Bounds for Linear Forms in the Exponentials of Algebraic Numbers 代数数指数线性形式的显界
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2021-12-09 DOI: 10.1145/3476446.3536170
Cheng-Chao Huang
{"title":"Explicit Bounds for Linear Forms in the Exponentials of Algebraic Numbers","authors":"Cheng-Chao Huang","doi":"10.1145/3476446.3536170","DOIUrl":"https://doi.org/10.1145/3476446.3536170","url":null,"abstract":"In this paper, we study linear forms λ=β1eα1+...βmeαm, where α_i and β_i are algebraic numbers. An explicit lower bound for the absolute value of λ is proved, which is derived from \"theoreme me de Lindemann--Weierstrass effectif'' via constructive methods in algebraic computation. Besides, the existence of λ with an explicit upper bound is established on the result of counting algebraic numbers.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122931712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Solving Sums of Squares in Global Fields 求解全局域的平方和
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2021-11-16 DOI: 10.1145/3476446.3535506
P. Koprowski
{"title":"Solving Sums of Squares in Global Fields","authors":"P. Koprowski","doi":"10.1145/3476446.3535506","DOIUrl":"https://doi.org/10.1145/3476446.3535506","url":null,"abstract":"The problem of writing a totally positive element as a sum of squares has a long history in mathematics, going back to Bachet and Lagrange. While for some specific rings (like integers or polynomials over the rationals), there are known methods for decomposing an element into a sum of squares, in general, for many other important rings and fields, the problem is still widely open. In this paper, we present an explicit algorithm for decomposing an element of an arbitrary global field (either a number field or a global function field) into a sum of squares of minimal length.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134100471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Average Complexity of Matrix Reduction for Clique Filtrations 团过滤矩阵约简的平均复杂度
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2021-11-03 DOI: 10.1145/3476446.3535474
Barbara Giunti, Guillaume Houry, Michael Kerber
{"title":"Average Complexity of Matrix Reduction for Clique Filtrations","authors":"Barbara Giunti, Guillaume Houry, Michael Kerber","doi":"10.1145/3476446.3535474","DOIUrl":"https://doi.org/10.1145/3476446.3535474","url":null,"abstract":"We study the algorithmic complexity of computing persistent homology of a randomly chosen filtration. Specifically, we prove upper bounds for the average fill-up (number of non-zero entries) of the boundary matrix on Erdös-Rényi and Vietoris-Rips filtrations after matrix reduction. Our bounds show that, in both cases, the reduced matrix is expected to be significantly sparser than what the general worst-case predicts. Our method is based on a link between the fillup of the boundary matrix and expected Betti numbers of random filtrations. Our bound for Vietoris-Rips complexes is asymptotically tight up to logarithmic factors. We also provide an Erdös-Rényi filtration realising the worst-case.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129724538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices 有限生成矩阵群的Zariski闭包的计算
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2021-06-03 DOI: 10.1145/3476446.3536172
Klara Nosan, Amaury Pouly, S. Schmitz, M. Shirmohammadi, J. Worrell
{"title":"On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices","authors":"Klara Nosan, Amaury Pouly, S. Schmitz, M. Shirmohammadi, J. Worrell","doi":"10.1145/3476446.3536172","DOIUrl":"https://doi.org/10.1145/3476446.3536172","url":null,"abstract":"We investigate the complexity of computing the Zariski closure of a finitely generated group of matrices. The Zariski closure was previously shown to be computable by Derksen, Jeandel, and Koiran, but the termination argument for their algorithm appears not to yield any complexity bound. In this paper we follow a different approach and obtain a bound on the degree of the polynomials that define the closure. Our bound shows that the closure can be computed in elementary time. We also obtain upper bounds on the length of chains of linear algebraic groups.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133476876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Border Basis Computation with Gradient-Weighted Normalization 梯度加权归一化边界基计算
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2021-01-02 DOI: 10.1145/3476446.3535476
Hiroshi Kera
{"title":"Border Basis Computation with Gradient-Weighted Normalization","authors":"Hiroshi Kera","doi":"10.1145/3476446.3535476","DOIUrl":"https://doi.org/10.1145/3476446.3535476","url":null,"abstract":"Normalization of polynomials plays a vital role in the approximate basis computation of vanishing ideals. Coefficient normalization, which normalizes a polynomial with its coefficient norm, is the most common method in computer algebra. This study proposes the gradient-weighted normalization method for the approximate border basis computation of vanishing ideals, inspired by recent developments in machine learning. The data-dependent nature of gradient-weighted normalization leads to better stability against perturbation and consistency in the scaling of input points, which cannot be attained by coefficient normalization. Only a subtle change is needed to introduce gradient normalization in the existing algorithms with coefficient normalization. The analysis of algorithms still works with a small modification, and the order of magnitude of time complexity of algorithms remains unchanged. We also prove that, with coefficient normalization, which does not provide the scaling consistency property, scaling of points (e.g., as a preprocessing) can cause an approximate basis computation to fail. This study is the first to theoretically highlight the crucial effect of scaling in approximate basis computation and presents the utility of data-dependent normalization.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123305350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Existence of Quantum Symmetries for Graphs on Up to Seven Vertices: A Computer based Approach 七顶点图的量子对称性的存在性:一个基于计算机的方法
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2019-06-28 DOI: 10.1145/3476446.3535481
V. Levandovskyy, C. Eder, Andreas Steenpaß, Simon Schmidt, J. Schanz, Moritz Weber
{"title":"Existence of Quantum Symmetries for Graphs on Up to Seven Vertices: A Computer based Approach","authors":"V. Levandovskyy, C. Eder, Andreas Steenpaß, Simon Schmidt, J. Schanz, Moritz Weber","doi":"10.1145/3476446.3535481","DOIUrl":"https://doi.org/10.1145/3476446.3535481","url":null,"abstract":"The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph: The question is whether or not the associative algebra associated to the quantum automorphism group is commutative. We use noncommutative Gröbner bases in order to tackle this problem; the implementation uses Gap and Singular:Letterplace. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116197430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
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