We consider computational problems concerning algebras over finite fields. In particular, we propose an algorithm for finding a small generating set for the multiplicative group of GF(p)[x]/F, where p is a prime number and F in GF(p)[x] is an arbitrary polynomial. Based on this result, a new set of expander graphs can be explicitly constructed. In addition, we present algorithms for basis construction and decomposition of a given element with respect to the basis.
{"title":"Constructing Small Generating Sets for the Multiplicative Groups of Algebras over Finite Fields","authors":"Ming-Deh A. Huang, Lian Liu","doi":"10.1145/2930889.2930921","DOIUrl":"https://doi.org/10.1145/2930889.2930921","url":null,"abstract":"We consider computational problems concerning algebras over finite fields. In particular, we propose an algorithm for finding a small generating set for the multiplicative group of GF(p)[x]/F, where p is a prime number and F in GF(p)[x] is an arbitrary polynomial. Based on this result, a new set of expander graphs can be explicitly constructed. In addition, we present algorithms for basis construction and decomposition of a given element with respect to the basis.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128235953","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}
We provide an algorithm to check whether two rational space curves are related by a similarity, i.e., whether they are equal up to position, orientation and scale. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Grobner bases for the special case of helical curves.
{"title":"Detecting Similarities of Rational Space Curves","authors":"J. G. Alcázar, Carlos Hermoso, G. Muntingh","doi":"10.1145/2930889.2930892","DOIUrl":"https://doi.org/10.1145/2930889.2930892","url":null,"abstract":"We provide an algorithm to check whether two rational space curves are related by a similarity, i.e., whether they are equal up to position, orientation and scale. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Grobner bases for the special case of helical curves.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128827569","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}
Jing-Cao Li, Cheng-Chao Huang, Ming Xu, Zhi-bin Li
We consider a class of univariate real functions---poly-powers---that extend integer exponents to real algebraic exponents for polynomials. Our purpose is to isolate positive roots of such a function into disjoint intervals, which can be further easily computed up to any desired precision. To this end, we first classify poly-powers into simple and non-simple ones, depending on the number of linearly independent exponents. For the former, we present a complete isolation method based on Gelfond--Schneider theorem. For the latter, the completeness depends on Schanuel's conjecture. Finally experiential results demonstrate the effectivity of the proposed method.
{"title":"Positive Root Isolation for Poly-Powers","authors":"Jing-Cao Li, Cheng-Chao Huang, Ming Xu, Zhi-bin Li","doi":"10.1145/2930889.2930909","DOIUrl":"https://doi.org/10.1145/2930889.2930909","url":null,"abstract":"We consider a class of univariate real functions---poly-powers---that extend integer exponents to real algebraic exponents for polynomials. Our purpose is to isolate positive roots of such a function into disjoint intervals, which can be further easily computed up to any desired precision. To this end, we first classify poly-powers into simple and non-simple ones, depending on the number of linearly independent exponents. For the former, we present a complete isolation method based on Gelfond--Schneider theorem. For the latter, the completeness depends on Schanuel's conjecture. Finally experiential results demonstrate the effectivity of the proposed method.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115399985","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}
It has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact formulae, including older and recent results, in sparse elimination. We start with root bounds and juxtapose two recent formulae: a generating function of the m-Bezout bound and a closed-form expression for the mixed volume by means of a matrix permanent. For the sparse resultant, a bevy of results have established determinantal or rational formulae for a large class of systems, starting with Macaulay. The discriminant is closely related to the resultant but admits no compact formula except for very simple cases. We offer a new determinantal formula for the discriminant of a sparse multilinear system arising in computing Nash equilibria. We introduce an alternative notion of compact formula, namely the Newton polytope of the unknown polynomial. It is possible to compute it efficiently for sparse resultants, discriminants, as well as the implicit equation of a parameterized variety. This leads us to consider implicit matrix representations of geometric objects.
{"title":"Compact Formulae in Sparse Elimination","authors":"I. Emiris","doi":"10.1145/2930889.2930943","DOIUrl":"https://doi.org/10.1145/2930889.2930943","url":null,"abstract":"It has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact formulae, including older and recent results, in sparse elimination. We start with root bounds and juxtapose two recent formulae: a generating function of the m-Bezout bound and a closed-form expression for the mixed volume by means of a matrix permanent. For the sparse resultant, a bevy of results have established determinantal or rational formulae for a large class of systems, starting with Macaulay. The discriminant is closely related to the resultant but admits no compact formula except for very simple cases. We offer a new determinantal formula for the discriminant of a sparse multilinear system arising in computing Nash equilibria. We introduce an alternative notion of compact formula, namely the Newton polytope of the unknown polynomial. It is possible to compute it efficiently for sparse resultants, discriminants, as well as the implicit equation of a parameterized variety. This leads us to consider implicit matrix representations of geometric objects.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128718483","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}
Symbolic summation started with Abramov (1971) for rational sequences and has been pushed forward by Gosper (1978), Zeilberger (1991), Petkovsek (1992) and Paule (1995) to tackle indefinite and definite sums for hypergeometric expressions. In the last decade the class of input sums has been extended significantly and covers, for instance, hypergeometric multi-sums, holonomic sequences, unspecified sequences, radical expressions, Stirling numbers, etc. In this talk we will focus on a new difference ring approach. The foundation was led by Karr's summation algorithm (1981) which enables one to rephrase indefinite nested sums and products in the setting of difference fields. Many new ideas have been incorporated into a strong summation theory which led to new algorithms for the summation paradigms of telescoping, creative telescoping and recurrence solving. However, this elegant difference field approach has one central drawback. Alternating signs cannot be represented in such a field: zero-divisors are introduced which can be formulated only within a ring. We will present a class of difference rings in which one can represent algorithmically indefinite nested sums and products together with the alternating sign, and more generally products over primitive roots of unity. In this setting we can represent all indefinite nested sums defined over hypergeometric expressions. In particular, this construction produces expressions in terms of sums that are all algebraically independent over each other. As a consequence, the derived output of a nested product-sum expression solves the zero-recognition problem: the computed expression evaluates to the zero-sequence if and only if the expression has been simplified to zero. In combination with improved parameterized telescoping algorithms and recurrence solvers within such difference rings we obtain an efficient summation machinery that has been built into the summation package Sigma. We will illustrate the different summation techniques by large scale problems coming from the field of particle physics.
{"title":"Symbolic Summation in Difference Rings and Applications","authors":"Carsten Schneider","doi":"10.1145/2930889.2930945","DOIUrl":"https://doi.org/10.1145/2930889.2930945","url":null,"abstract":"Symbolic summation started with Abramov (1971) for rational sequences and has been pushed forward by Gosper (1978), Zeilberger (1991), Petkovsek (1992) and Paule (1995) to tackle indefinite and definite sums for hypergeometric expressions. In the last decade the class of input sums has been extended significantly and covers, for instance, hypergeometric multi-sums, holonomic sequences, unspecified sequences, radical expressions, Stirling numbers, etc. In this talk we will focus on a new difference ring approach. The foundation was led by Karr's summation algorithm (1981) which enables one to rephrase indefinite nested sums and products in the setting of difference fields. Many new ideas have been incorporated into a strong summation theory which led to new algorithms for the summation paradigms of telescoping, creative telescoping and recurrence solving. However, this elegant difference field approach has one central drawback. Alternating signs cannot be represented in such a field: zero-divisors are introduced which can be formulated only within a ring. We will present a class of difference rings in which one can represent algorithmically indefinite nested sums and products together with the alternating sign, and more generally products over primitive roots of unity. In this setting we can represent all indefinite nested sums defined over hypergeometric expressions. In particular, this construction produces expressions in terms of sums that are all algebraically independent over each other. As a consequence, the derived output of a nested product-sum expression solves the zero-recognition problem: the computed expression evaluates to the zero-sequence if and only if the expression has been simplified to zero. In combination with improved parameterized telescoping algorithms and recurrence solvers within such difference rings we obtain an efficient summation machinery that has been built into the summation package Sigma. We will illustrate the different summation techniques by large scale problems coming from the field of particle physics.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122327709","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}
The hypergeometric function 1F1 of a matrix argument Y is a symmetric entire function in the eigenvalues y1,...,ym of Y. It appears in the distribution function of the largest eigenvalue of a Wishart matrix and its numerical evaluation is important in multivariate distribution theory. Hashiguchi et al. (J. Multivariate Analysis, 2013) proposed an efficient algorithm for evaluating the matrix 1F1 by the holonomic gradient method (HGM). The algorithm is based on the system of partial differential equations (PDEs) satisfied by the matrix 1F1 given by Muirhead (Ann. Math. Statist., 1970) and it cannot be applied to the diagonal cases, i.e. the cases where several yi's are equal because the system of PDEs has singularities on the diagonal region. Hashiguchi et al. derived an ordinary differential equation (ODE) satisfied by 1F1(y,y) in the bivariate case from some relations which are obtained by applying l'opital rule to the system of PDEs for 1F1(y1,y2). In this paper we generalize this approach for computing systems of PDEs satisfied by the matrix 1F1 for various diagonalization patterns. We show that the existence of a system of PDEs for a diagonalized 1F1 is reduced to the non-singularity of the matrices systematically derived from the diagonalization pattern. By checking the non-singularity numerically, we show that there exists a system of PDEs for a diagonalized 1F1 if the size of each diagonal block ≤ 36. We have computed an ODE for 1F1(y,...,y) up to m=22. We made a test implementation of HGM for diagonal cases and we show some numerical results.
{"title":"System of Partial Differential Equations for the Hypergeometric Function 1F1 of a Matrix Argument on Diagonal Regions","authors":"M. Noro","doi":"10.1145/2930889.2930905","DOIUrl":"https://doi.org/10.1145/2930889.2930905","url":null,"abstract":"The hypergeometric function 1F1 of a matrix argument Y is a symmetric entire function in the eigenvalues y1,...,ym of Y. It appears in the distribution function of the largest eigenvalue of a Wishart matrix and its numerical evaluation is important in multivariate distribution theory. Hashiguchi et al. (J. Multivariate Analysis, 2013) proposed an efficient algorithm for evaluating the matrix 1F1 by the holonomic gradient method (HGM). The algorithm is based on the system of partial differential equations (PDEs) satisfied by the matrix 1F1 given by Muirhead (Ann. Math. Statist., 1970) and it cannot be applied to the diagonal cases, i.e. the cases where several yi's are equal because the system of PDEs has singularities on the diagonal region. Hashiguchi et al. derived an ordinary differential equation (ODE) satisfied by 1F1(y,y) in the bivariate case from some relations which are obtained by applying l'opital rule to the system of PDEs for 1F1(y1,y2). In this paper we generalize this approach for computing systems of PDEs satisfied by the matrix 1F1 for various diagonalization patterns. We show that the existence of a system of PDEs for a diagonalized 1F1 is reduced to the non-singularity of the matrices systematically derived from the diagonalization pattern. By checking the non-singularity numerically, we show that there exists a system of PDEs for a diagonalized 1F1 if the size of each diagonal block ≤ 36. We have computed an ODE for 1F1(y,...,y) up to m=22. We made a test implementation of HGM for diagonal cases and we show some numerical results.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132056345","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}
We present a parallel GCD algorithm for sparse multivariate polynomials with integer coefficients. The algorithm combines a Kronecker substitution with a Ben-Or/Tiwari sparse interpolation modulo a smooth prime to determine the support of the GCD. We have implemented our algorithm in Cilk C. We compare it with Maple and Magma's implementations of Zippel's GCD algorithm.
{"title":"A Fast Parallel Sparse Polynomial GCD Algorithm","authors":"Jiaxiong Hu, M. Monagan","doi":"10.1145/2930889.2930903","DOIUrl":"https://doi.org/10.1145/2930889.2930903","url":null,"abstract":"We present a parallel GCD algorithm for sparse multivariate polynomials with integer coefficients. The algorithm combines a Kronecker substitution with a Ben-Or/Tiwari sparse interpolation modulo a smooth prime to determine the support of the GCD. We have implemented our algorithm in Cilk C. We compare it with Maple and Magma's implementations of Zippel's GCD algorithm.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"529 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123356900","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}
The 2016 International Symposium on Symbolic and Algebraic Computation (ISSAC 2016) is the premier conference for research in symbolic computation and computer algebra. ISSAC 2016, held at Wilfrid Laurier University, Canada, is the 41st meeting in the series. ISSAC was directly preceded by a series of conferences with the names SYMSAC, EUROCAL, EUROSAM and EUROCAM, initiated by the seminal ACM Symposium on Symbolic and Algebraic Manipulation (SYMSAM- 1966) held 50 years ago. The ISSAC conference is a showcase for original research contributions on all aspects of computer algebra and symbolic mathematical computation, including: Algorithmic aspects: Exact and symbolic linear, polynomial and differential algebra Symbolic-numeric, homotopy, perturbation and series methods Computational algebraic geometry, group theory and number theory Computer arithmetic Summation, recurrence equations, integration, solution of ODEs and PDEs Symbolic methods in other areas of pure and applied mathematics Complexity of algebraic algorithms Software aspects: Design of symbolic computation packages and systems Language design and type systems for symbolic computation Data representation Consideration for modern hardware Algorithm implementation and performance tuning Mathematical user interfaces Application aspects: Applications stretching the current limits of computer algebra algorithms or systems, use computer algebra in new areas or new ways, or apply it in situations with broad impact. The ISSAC Program Committee adhered to the highest standards and practices in the evaluation of submitted papers, and we are very pleased with the quality of the papers appearing at the conference. All papers submitted to ISSAC 2016 were judged, and accepted or rejected, solely according to their scientific novelty and excellence. Each submitted paper was assigned to three members of the Program Committee, and a minimum of three referee reports were obtained for each submission. Ultimately 50 papers were accepted for presentation and publication in the proceedings. The authors of accepted papers are from 17 countries. In addition to the contributed research presentations, the program of ISSAC 2016 features 3 invited talks given by Ioannis Z. Emiris, J. Ian Munro, and Carsten Schneider, 3 tutorials given by Clemens G. Raab, Daniel Robertz, and Georg Regensburger, 2 satellite workshops: "Milestones in Computer Algebra" and "The Waterloo Workshop on Computer Algebra," as well as a poster session and a software exhibits session.
2016年符号与代数计算国际研讨会(ISSAC 2016)是符号计算和计算机代数研究的首要会议。ISSAC 2016在加拿大威尔弗里德劳里埃大学举行，是该系列的第41次会议。ISSAC的前身是一系列名为SYMSAC, EUROCAL, EUROSAM和EUROCAM的会议，这些会议是由50年前举行的具有开创性的ACM符号和代数操作研讨会(SYMSAM- 1966)发起的。ISSAC会议展示了计算机代数和符号数学计算各个方面的原创研究贡献，包括:精确和符号线性、多项式和微分代数符号数值、同伦、摄动和级数方法计算代数几何、群论和数论计算机算术求和、递归方程、积分、ode和PDEs的解纯数学和应用数学其他领域的符号方法代数算法的复杂性软件方面:符号计算包和系统的设计符号计算的语言设计和类型系统数据表示对现代硬件的考虑算法实现和性能调优数学用户界面应用方面:扩展计算机代数算法或系统的当前限制的应用，在新的领域或新的方法中使用计算机代数，或将其应用于具有广泛影响的情况。ISSAC项目委员会在评审提交的论文时坚持了最高的标准和做法，我们对会议上出现的论文的质量感到非常满意。所有提交给ISSAC 2016的论文都将根据其科学新颖性和卓越性进行评判，并被接受或拒绝。每篇提交的论文被分配给三名计划委员会成员，并且每次提交至少获得三份裁判报告。最终有50篇论文被接受在会议上发表。接受论文的作者来自17个国家。除了贡献的研究报告外，ISSAC 2016计划还包括Ioannis Z. Emiris, J. Ian Munro和Carsten Schneider的3次邀请演讲，Clemens G. Raab, Daniel Robertz和Georg Regensburger的3次教程，2个卫星研讨会:“计算机代数的里程碑”和“计算机代数的滑铁卢研讨会”，以及海报会议和软件展览会议。
{"title":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","authors":"S. Abramov, E. Zima, X. Gao","doi":"10.1145/2930889","DOIUrl":"https://doi.org/10.1145/2930889","url":null,"abstract":"The 2016 International Symposium on Symbolic and Algebraic Computation (ISSAC 2016) is the premier conference for research in symbolic computation and computer algebra. ISSAC 2016, held at Wilfrid Laurier University, Canada, is the 41st meeting in the series. ISSAC was directly preceded by a series of conferences with the names SYMSAC, EUROCAL, EUROSAM and EUROCAM, initiated by the seminal ACM Symposium on Symbolic and Algebraic Manipulation (SYMSAM- 1966) held 50 years ago. \u0000 \u0000The ISSAC conference is a showcase for original research contributions on all aspects of computer algebra and symbolic mathematical computation, including: Algorithmic aspects: \u0000Exact and symbolic linear, polynomial and differential algebra \u0000Symbolic-numeric, homotopy, perturbation and series methods \u0000Computational algebraic geometry, group theory and number theory \u0000Computer arithmetic \u0000Summation, recurrence equations, integration, solution of ODEs and PDEs \u0000Symbolic methods in other areas of pure and applied mathematics \u0000Complexity of algebraic algorithms \u0000 \u0000 \u0000 \u0000Software aspects: \u0000Design of symbolic computation packages and systems \u0000Language design and type systems for symbolic computation \u0000Data representation \u0000Consideration for modern hardware \u0000Algorithm implementation and performance tuning \u0000Mathematical user interfaces \u0000 \u0000 \u0000 \u0000Application aspects: \u0000Applications stretching the current limits of computer algebra algorithms or systems, use computer algebra in new areas or new ways, or apply it in situations with broad impact. \u0000 \u0000 \u0000 \u0000The ISSAC Program Committee adhered to the highest standards and practices in the evaluation of submitted papers, and we are very pleased with the quality of the papers appearing at the conference. All papers submitted to ISSAC 2016 were judged, and accepted or rejected, solely according to their scientific novelty and excellence. Each submitted paper was assigned to three members of the Program Committee, and a minimum of three referee reports were obtained for each submission. Ultimately 50 papers were accepted for presentation and publication in the proceedings. The authors of accepted papers are from 17 countries. \u0000 \u0000In addition to the contributed research presentations, the program of ISSAC 2016 features 3 invited talks given by Ioannis Z. Emiris, J. Ian Munro, and Carsten Schneider, 3 tutorials given by Clemens G. Raab, Daniel Robertz, and Georg Regensburger, 2 satellite workshops: \"Milestones in Computer Algebra\" and \"The Waterloo Workshop on Computer Algebra,\" as well as a poster session and a software exhibits session.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"82 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123655146","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}
We present an algorithm for determining the existence of the limit of a real multivariate rational function q at a given point which is an isolated zero of the denominator of q. When the limit exists, the algorithm computes it, without making any assumption on the number of variables. A process, which extends the work of Cadavid, Molina and Velez, reduces the multivariate setting to computing limits of bivariate rational functions. By using regular chain theory and triangular decomposition of semi-algebraic systems, we avoid the computation of singular loci and the decomposition of algebraic sets into irreducible components.
{"title":"Computing Limits of Real Multivariate Rational Functions","authors":"P. Alvandi, Mahsa Kazemi, M. M. Maza","doi":"10.1145/2930889.2930938","DOIUrl":"https://doi.org/10.1145/2930889.2930938","url":null,"abstract":"We present an algorithm for determining the existence of the limit of a real multivariate rational function q at a given point which is an isolated zero of the denominator of q. When the limit exists, the algorithm computes it, without making any assumption on the number of variables. A process, which extends the work of Cadavid, Molina and Velez, reduces the multivariate setting to computing limits of bivariate rational functions. By using regular chain theory and triangular decomposition of semi-algebraic systems, we avoid the computation of singular loci and the decomposition of algebraic sets into irreducible components.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114853395","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}
Integrals are important to many applications ranging from physics over engineering to statistics. While systematic methods for symbolic computation of integrals have a long history, computer algebra tools for computation of parameter integrals are a more recent topic. For computing (definite) parameter integrals, one does not necessarily need to know an explicit antiderivative of the integrand, such computations often rely on techniques like differentiation under the integral sign instead.
{"title":"Symbolic Computation of Parameter Integrals","authors":"C. Raab","doi":"10.1145/2930889.2930940","DOIUrl":"https://doi.org/10.1145/2930889.2930940","url":null,"abstract":"Integrals are important to many applications ranging from physics over engineering to statistics. While systematic methods for symbolic computation of integrals have a long history, computer algebra tools for computation of parameter integrals are a more recent topic. For computing (definite) parameter integrals, one does not necessarily need to know an explicit antiderivative of the integrand, such computations often rely on techniques like differentiation under the integral sign instead.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121858438","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}
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