Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation最新文献_第2页

Constructing All Composition Series of a Finite Group 构造有限群的所有复合级数

Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2015-06-24 DOI: 10.1145/2755996.2756642

A. Hulpke

引用次数: 0

Real Quantifier Elimination by Computation of Comprehensive Gröbner Systems 综合Gröbner系统的实量词消去计算

Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2015-06-24 DOI: 10.1145/2755996.2756646

Ryoya Fukasaku, Hidenao Iwane, Yosuke Sato

引用次数: 23

On þ-adic Expansions of Algebraic Integers 代数整数的正交展开式

Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2015-06-24 DOI: 10.1145/2755996.2756675

Hsing-Hau Chen, Ming-Deh A. Huang

{"title":"On þ-adic Expansions of Algebraic Integers","authors":"Hsing-Hau Chen, Ming-Deh A. Huang","doi":"10.1145/2755996.2756675","DOIUrl":"https://doi.org/10.1145/2755996.2756675","url":null,"abstract":"It is well known that every rational integer has a finite or periodic p-adic expansion. In this paper a more general notion of Þ-adic expansion is introduced for algebraic integers, where given a number field K and a principal prime ideal Þ in K, a different choice of generator for Þ is allowed in each stage of the expansion. With the notion of Þ-adic expansion, we prove that there is always a finite or periodic Þ-adic expansion for every algebraic integer. Moreover, we prove a bound on the periodicity of the Þ-adic expansion that depends only on the number field K and the prime ideal Þ. The proof yields an algorithm for constructing such a Þ-adic expansion for elements in the ring O of algebraic integers of K, through finding an approximation to the closest vector on the lattice spanned by the unit group of O. As a special case we prove that, similar to rational integers, Gaussian integers are finite or periodic not only in Þ-adic expansion but also in π-adic expansion, where a fixed generator π for Þ is used in each stage of the expansion. Moreover, the time complexity of finding a π-adic expansion for a Gaussian integer is polynomial in the length of input, the period, and p, where p is the rational prime contained in Þ. We implement the algorithm for some quadratic number fields and provide examples which illustrate that the Þ-adic expansion of the elements in O is either finite or periodic.","PeriodicalId":182805,"journal":{"name":"Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114971860","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

Open Non-uniform Cylindrical Algebraic Decompositions 开放非均匀圆柱代数分解

Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2015-06-24 DOI: 10.1145/2755996.2756654

Christopher W. Brown

引用次数: 28

Optimizing a Parametric Linear Function over a Non-compact Real Algebraic Variety 非紧实代数变量上参数线性函数的优化

Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2015-06-24 DOI: 10.1145/2755996.2756666

Feng Guo, M. S. E. Din, Chu Wang, L. Zhi

{"title":"Optimizing a Parametric Linear Function over a Non-compact Real Algebraic Variety","authors":"Feng Guo, M. S. E. Din, Chu Wang, L. Zhi","doi":"10.1145/2755996.2756666","DOIUrl":"https://doi.org/10.1145/2755996.2756666","url":null,"abstract":"We consider the problem of optimizing a parametric linear function over a non-compact real trace of an algebraic set. Our goal is to compute a representing polynomial which defines a hypersurface containing the graph of the optimal value function. Rostalski and Sturmfels showed that when the algebraic set is irreducible and smooth with a compact real trace, then the least degree representing polynomial is given by the defining polynomial of the irreducible hypersurface dual to the projective closure of the algebraic set. First, we generalize this approach to non-compact situations. We prove that the graph of the opposite of the optimal value function is still contained in the affine cone over a dual variety similar to the one considered in compact case. In consequence, we present an algorithm for solving the considered parametric optimization problem for generic parameters' values. For some special parameters' values, the representing polynomials of the dual variety can be identically zero, which give no information on the optimal value. We design a dedicated algorithm that identifies those regions of the parameters' space and computes for each of these regions a new polynomial defining the optimal value over the considered region.","PeriodicalId":182805,"journal":{"name":"Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"149 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132299261","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

Graph-Coloring Ideals: Nullstellensatz Certificates, Gröbner Bases for Chordal Graphs, and Hardness of Gröbner Bases 图着色理想:Nullstellensatz证书，Gröbner弦图的基础，和Gröbner基础的硬度

Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2015-06-24 DOI: 10.1145/2755996.2756639

J. D. Loera, S. Margulies, Michael Oesterle, Eric Riedl, D. Rolnick, Gwen Spencer, Despina Stasi, Jon Swenson

引用次数: 16

An Introduction to Finite Element Methods 有限元方法导论

Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2015-06-24 DOI: 10.1145/2755996.2756685

V. Pillwein

引用次数: 5

Building Bridges between Symbolic Computation and Satisfiability Checking 在符号计算和可满足性检验之间架起桥梁

Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2015-06-24 DOI: 10.1145/2755996.2756636

E. Ábrahám

{"title":"Building Bridges between Symbolic Computation and Satisfiability Checking","authors":"E. Ábrahám","doi":"10.1145/2755996.2756636","DOIUrl":"https://doi.org/10.1145/2755996.2756636","url":null,"abstract":"The satisfiability problem is the problem of deciding whether a logical formula is satisfiable. For first-order arithmetic theories, in the early 20th century some novel solutions in form of decision procedures were developed in the area of mathematical logic. With the advent of powerful computer architectures, a new research line started to develop practically feasible implementations of such decision procedures. Since then, symbolic computation has grown to an extremely successful scientific area, supporting all kinds of scientific computing by efficient computer algebra systems. Independently, around 1960 a new technology called SAT solving started its career. Restricted to propositional logic, SAT solvers showed to be very efficient when employed by formal methods for verification. It did not take long till the power of SAT solving for Boolean problems had been extended to cover also different theories. Nowadays, fast SAT-modulo-theories (SMT) solvers are available also for arithmetic problems. Due to their different roots, symbolic computation and SMT solving tackle the satisfiability problem differently. We discuss differences and similarities in their approaches, highlight potentials of combining their strengths, and discuss the challenges that come with this task.","PeriodicalId":182805,"journal":{"name":"Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122672146","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: 40

Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences 计算线性代数Gröbner线性递归多维序列的基

Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2015-06-24 DOI: 10.1145/2755996.2756673

Jérémy Berthomieu, Brice Boyer, J. Faugère

{"title":"Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences","authors":"Jérémy Berthomieu, Brice Boyer, J. Faugère","doi":"10.1145/2755996.2756673","DOIUrl":"https://doi.org/10.1145/2755996.2756673","url":null,"abstract":"Sakata generalized the Berlekamp--Massey algorithm to n dimensions in~1988. The Berlekamp--Massey--Sakata (BMS) algorithm can be used for finding a Grbner basis of a 0-dimensional ideal of relations verified by a table. We investigate this problem usingö linear algebra techniques, with motivations such as accelerating change of basis algorithms (FGLM) or improving their complexity. We first define and characterize multidimensional linear recursive sequences for 0-dimensional ideals. Under genericity assumptions, we propose a randomized preprocessing of the table that corresponds to performing a linear change of coordinates on the polynomials associated with the linear recurrences. This technique then essentially reduces our problem to using the efficient 1-dimensional Berlekamp--Massey (BM) algorithm. However, the number of probes to the table in this scheme may be elevated. We thus consider the table in the black-box model: we assume probing the table is expensive and we minimize the number of probes to the table in our complexity model. We produce an FGLM-like algorithm for finding the relations in the table, which lets us use linear algebra techniques. Under some additional assumptions, we make this algorithm adaptive and reduce further the number of table probes. This number can be estimated by counting the number of distinct elements in a multi-Hankel matrix (a multivariate generalization of Hankel matrices); we can relate this quantity with the geometry of the final staircase. Hence, in favorable cases such as convex ones, the complexity is essentially linear in the size of the output. Finally, when using the LEX ordering, we can make use of fast structured linear algebra similarly to the Hankel interpretation of Berlekamp--Massey.","PeriodicalId":182805,"journal":{"name":"Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"213 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117315320","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: 21

Implementation of the DKSS Algorithm for Multiplication of Large Numbers 大数乘法的DKSS算法的实现

Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2015-06-24 DOI: 10.1145/2755996.2756643

Christoph Lüders

引用次数: 6