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Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation最新文献

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Subdivisions for macaulay formulas of sparse systems 稀疏系统macaulay公式的细分
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3403988
Friedemann Groh
{"title":"Subdivisions for macaulay formulas of sparse systems","authors":"Friedemann Groh","doi":"10.1145/3373207.3403988","DOIUrl":"https://doi.org/10.1145/3373207.3403988","url":null,"abstract":"In a seminal article [7], D'Andrea describes a method for determining Macaulay-type formulae for the resultants of sparse polynomial systems. His algorithm works recursive, reducing the dimension n of the problem at each step. In doing do, he applies a certain coherent mixed subdivision of the given Newton polytopes into cells, each representing a system with smaller dimension. To simplify this procedure, we insert an intermediate step in which these reduced systems are transferred to the n-dimensional domain of the complete cells. As a consequence, the input system of each iteration step need not contain an additional polytope and only one system per secondary cell has to be considered. The individual subdivisions determined in various steps of the algorithm are combined into a single subdivision of the whole problem. Only then, the matrix for calculating the resultant is determined. To prove our method, we generalize a theorem of [22] on the initial form of resultants with respect to coherent mixed subdivisions.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129420121","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: 2
On the parallelization of triangular decompositions 关于三角形分解的并行化
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404065
Mohammadali Asadi, Alexander Brandt, Robert H. C. Moir, M. M. Maza, Yuzhen Xie
{"title":"On the parallelization of triangular decompositions","authors":"Mohammadali Asadi, Alexander Brandt, Robert H. C. Moir, M. M. Maza, Yuzhen Xie","doi":"10.1145/3373207.3404065","DOIUrl":"https://doi.org/10.1145/3373207.3404065","url":null,"abstract":"We discuss the parallelization of algorithms for solving polynomial systems by way of triangular decomposition. The Triangularize algorithm proceeds through incremental intersections of polynomials to produce different components (points, curves, surfaces, etc.) of the solution set. Independent components imply the opportunity for concurrency. This \"component-level\" parallelization of triangular decompositions, our focus here, belongs to the class of dynamic irregular parallelism. Potential parallel speed-up depends only on geometrical properties of the solution set (number of components, their dimensions and degrees); these algorithms do not scale with the number of processors. To manage the irregularities of component-level parallelization we combine different concurrency patterns, namely, workpile, producer-consumer, and fork/join. We report on our implementation in the freely available BPAS library. Experimentation with thousands of polynomial systems yield examples with up to 9.5× speed-up on a 12-core machine.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122005246","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
WhyMP, a formally verified arbitrary-precision integer library WhyMP，经过正式验证的任意精度整数库
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404029
G. Melquiond, Raphaël Rieu-Helft
{"title":"WhyMP, a formally verified arbitrary-precision integer library","authors":"G. Melquiond, Raphaël Rieu-Helft","doi":"10.1145/3373207.3404029","DOIUrl":"https://doi.org/10.1145/3373207.3404029","url":null,"abstract":"Arbitrary-precision integer libraries such as GMP are a critical building block of computer algebra systems. GMP provides state-of-the-art algorithms that are intricate enough to justify formal verification. In this paper, we present a C library that has been formally verified using the Why3 verification platform in about four person-years. This verification deals not only with safety, but with full functional correctness. It has been performed using a mixture of mechanically checked handwritten proofs and automated theorem proving. We have implemented and verified a nontrivial subset of GMP's algorithms, including their optimizations and intricacies. Our library provides the same interface as GMP and is almost as efficient for smaller inputs. We detail our verification methodology and the algorithms we have implemented, and include some benchmarks to compare our library with GMP.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126936378","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: 2
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation 第45届符号与代数计算国际研讨会论文集
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207
{"title":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","authors":"","doi":"10.1145/3373207","DOIUrl":"https://doi.org/10.1145/3373207","url":null,"abstract":"","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131625925","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: 1
Reflections on elimination theory 关于消除理论的思考
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3403977
David A. Cox
{"title":"Reflections on elimination theory","authors":"David A. Cox","doi":"10.1145/3373207.3403977","DOIUrl":"https://doi.org/10.1145/3373207.3403977","url":null,"abstract":"My lecture will survey developments in elimination theory from Newton and Bézout up to modern times. I will discuss the dominance of elimination theory in the 19th century and the challenges it faced in the 20th century with the rise of abstract algebraic geometry. I will also mention the role of the ISSAC community and some personal history.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125146866","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
On parameterized complexity of the word search problem in the Baumslag-Gersten group Baumslag-Gersten群中词搜索问题的参数化复杂度
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404042
A. Myasnikov, Andrey Nikolaev
{"title":"On parameterized complexity of the word search problem in the Baumslag-Gersten group","authors":"A. Myasnikov, Andrey Nikolaev","doi":"10.1145/3373207.3404042","DOIUrl":"https://doi.org/10.1145/3373207.3404042","url":null,"abstract":"We consider the word search problem in the Baumslag-Gersten group GB. We show that the parameterized complexity of this problem, where the area of van Kampen diagram serves as a parameter, is polynomial in the length of the input and the parameter. This contrasts the well-known result that the Dehn function and the time complexity of the word search problem in GB are non-elementary.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122398540","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: 1
Computing the real isolated points of an algebraic hypersurface 计算代数超曲面的实孤立点
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404049
H. P. Le, M. S. E. Din, T. Wolff
{"title":"Computing the real isolated points of an algebraic hypersurface","authors":"H. P. Le, M. S. E. Din, T. Wolff","doi":"10.1145/3373207.3404049","DOIUrl":"https://doi.org/10.1145/3373207.3404049","url":null,"abstract":"Let R be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in Rn given as the vanishing set of a polynomial system. This problem plays an important role for studying rigidity properties of mechanism in material designs. In this paper, we design an algorithm which solves this problem. It is based on the computations of critical points as well as roadmaps for answering connectivity queries in real algebraic sets. This leads to a probabilistic algorithm of complexity (nd)O (n log(n)) for computing the real isolated points of real algebraic hypersurfaces of degree d. It allows us to solve in practice instances which are out of reach of the state-of-the-art.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122110358","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
Computation of free non-commutative gröbner bases over Z with Singular:Letterplace Z上具有奇异位的自由非交换gröbner基的计算
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404052
V. Levandovskyy, Tobias Metzlaff, Karim Abou Zeid
{"title":"Computation of free non-commutative gröbner bases over Z with Singular:Letterplace","authors":"V. Levandovskyy, Tobias Metzlaff, Karim Abou Zeid","doi":"10.1145/3373207.3404052","DOIUrl":"https://doi.org/10.1145/3373207.3404052","url":null,"abstract":"The extension of Gröbner bases concept from polynomial algebras over fields to polynomial rings over rings allows to tackle numerous applications, both of theoretical and of practical importance. Gröbner and Gröbner-Shirshov bases can be defined for various non-commutative and even non-associative algebraic structures. We study the case of associative rings and aim at free algebras over principal ideal rings. We concentrate ourselves on the case of commutative coefficient rings without zero divisors (i.e. a domain). Even working over Z allows one to do computations, which can be treated as universal for fields of arbitrary characteristic. By using the systematic approach, we revisit the theory and present the algorithms in the implementable form. We show drastic differences in the behavior of Gröbner bases between free algebras and algebras, close to commutative. Even the formation of critical pairs has to be reengineered, together with the criteria for their quick discarding. We present an implementation of algorithms in the Singular subsystem called Letterplace, which internally uses Letterplace techniques (and Letterplace Gröbner bases), due to La Scala and Levandovskyy. Interesting examples accompany our presentation.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116336475","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: 6
Ubiquity of the exponent of matrix multiplication 矩阵乘法指数的普遍性
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3403979
Lek-Heng Lim, Ke Ye
{"title":"Ubiquity of the exponent of matrix multiplication","authors":"Lek-Heng Lim, Ke Ye","doi":"10.1145/3373207.3403979","DOIUrl":"https://doi.org/10.1145/3373207.3403979","url":null,"abstract":"The asymptotic exponent of matrix multiplication is the smallest ω such that one may multiply two n × n matrices or invert an n × n matrix in O(nω+ε)-complexity for ε > 0 arbitrarily small. One of the biggest open problem in complexity theory and numerical linear algebra is its conjectured value ω = 2. This article is about the universality of ω. We will show that ω is not only the asymptotic exponent for the product operation in matrix algebras but also that for various infinite families of Lie algebras, Jordan algebras, and Clifford algebras. In addition, we will show that ω is not just the asymptotic exponent for matrix product and inversion but also that for the evaluation of any matrix-valued polynomial and rational functions of matrix variables.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124027565","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
Letterplace Letterplace
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404056
V. Levandovskyy, H. Schönemann, Karim Abou Zeid
{"title":"Letterplace","authors":"V. Levandovskyy, H. Schönemann, Karim Abou Zeid","doi":"10.1145/3373207.3404056","DOIUrl":"https://doi.org/10.1145/3373207.3404056","url":null,"abstract":"We present the newest release of the subsystem of Singular called Letterplace which exists since 2009. It is devoted to computations with finitely presented associative algebras over fields and offers Gröbner(-Shirshov) bases over free algebras via the Letterplace correspondence of La Scala and Levandovskyy. This allows to use highly tuned commutative data structures internally and to reuse parts of existing algorithms in the non-commutative situation. The present version has been deeply reengineered, based on the experience with earlier and experimental versions. We offer an unprecedented functionality, some of which for the first time in the history of computer algebra. In particular, we present tools for elimination theory (via truncated Gröbner bases and via supporting several kinds of elimination orderings), dimension theory (Gel'fand-Kirillov and global dimension), and for homological algebra (such as syzygy bimodules and lifts for ideals and bimodules) to name a few. Another article in this issue is devoted to the extension of Gröbner bases to the coefficients in principal ideal rings including Z, which is also a part of this release. We report on comparison with other systems and on some advances in the theory. Quite nontrivial examples illustrate the abilities of the system.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121282668","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
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