We present a new package ZpL for the mathematical software system SageMath. It implements a sharp tracking of precision on p-adic numbers, following the theory of ultrametric precision introduced in a previous paper by the same authors. The underlying algorithms are mostly based on automatic differentiation techniques. We introduce them, study their complexity and discuss our design choices. We illustrate the benefits of our package (in comparison with previous implementations) with a large sample of examples coming from linear algebra, commutative algebra and differential equations.
{"title":"ZpL","authors":"X. Caruso, David Roe, Tristan Vaccon","doi":"10.1145/3208976.3208995","DOIUrl":"https://doi.org/10.1145/3208976.3208995","url":null,"abstract":"We present a new package ZpL for the mathematical software system SageMath. It implements a sharp tracking of precision on p-adic numbers, following the theory of ultrametric precision introduced in a previous paper by the same authors. The underlying algorithms are mostly based on automatic differentiation techniques. We introduce them, study their complexity and discuss our design choices. We illustrate the benefits of our package (in comparison with previous implementations) with a large sample of examples coming from linear algebra, commutative algebra and differential equations.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121572572","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}
This tutorial provides an introduction to the development of fast matrix algorithms based on the notions of displacement and various low-rank structures.
本教程介绍了基于位移和各种低秩结构概念的快速矩阵算法的开发。
{"title":"Fast Algorithms for Displacement and Low-Rank Structured Matrices","authors":"S. Chandrasekaran, N. Govindarajan, A. Rajagopal","doi":"10.1145/3208976.3209025","DOIUrl":"https://doi.org/10.1145/3208976.3209025","url":null,"abstract":"This tutorial provides an introduction to the development of fast matrix algorithms based on the notions of displacement and various low-rank structures.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122655022","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}
Let R be a real closed field. We consider basic semi-algebraic sets defined by n -variate equations/inequalities of s symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by 2d < n. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most 2d-1 distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by s polynomials of degree d in time (sn)O(d). This improves the state-of-the-art which is exponential in n . When the variables x1, łdots, xn are quantified and the coefficients of the input system depend on parameters y1, łdots, yt, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time (sn)O(dt).
{"title":"Real Root Finding for Equivariant Semi-algebraic Systems","authors":"C. Riener, M. S. E. Din","doi":"10.1145/3208976.3209023","DOIUrl":"https://doi.org/10.1145/3208976.3209023","url":null,"abstract":"Let R be a real closed field. We consider basic semi-algebraic sets defined by n -variate equations/inequalities of s symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by 2d < n. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most 2d-1 distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by s polynomials of degree d in time (sn)O(d). This improves the state-of-the-art which is exponential in n . When the variables x1, łdots, xn are quantified and the coefficients of the input system depend on parameters y1, łdots, yt, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time (sn)O(dt).","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114879727","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 describe, study, and experiment with an algorithm for finding all solutions of systems of polynomial equations using homotopy continuation and monodromy. This algorithm follows the framework developed by Duff et al. (2018) and can operate in the presence of a large number of failures of the homotopy continuation subroutine. We give special attention to parallelization and probabilistic analysis of a model adapted to parallelization and failures. Apart from theoretical results, we developed a simulator that allows us to run a large number of experiments without recomputing the outcomes of the continuation subroutine.
我们描述、研究并实验了一种利用同伦延拓和一格求多项式方程组所有解的算法。该算法遵循Duff et al.(2018)开发的框架，可以在同伦连续子程序出现大量故障的情况下运行。我们特别关注并行化和适合并行化和失效的模型的概率分析。除了理论结果，我们还开发了一个模拟器，使我们能够运行大量的实验，而无需重新计算延续子程序的结果。
{"title":"Monodromy Solver: Sequential and Parallel","authors":"N. Bliss, Timothy Duff, A. Leykin, J. Sommars","doi":"10.1145/3208976.3209007","DOIUrl":"https://doi.org/10.1145/3208976.3209007","url":null,"abstract":"We describe, study, and experiment with an algorithm for finding all solutions of systems of polynomial equations using homotopy continuation and monodromy. This algorithm follows the framework developed by Duff et al. (2018) and can operate in the presence of a large number of failures of the homotopy continuation subroutine. We give special attention to parallelization and probabilistic analysis of a model adapted to parallelization and failures. Apart from theoretical results, we developed a simulator that allows us to run a large number of experiments without recomputing the outcomes of the continuation subroutine.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132334239","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}
Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gröbner bases taking into account the valuation of K . Because of the use of the valuation, the theory of tropical Gröbner bases has proved to provide settings for computations over polynomial rings over a p -adic field that are more stable than that of classical Gröbner bases. Beforehand, these strategies were only available for homogeneous polynomials. In this article, we extend the F5 strategy to a new definition of tropical Gröbner bases in an affine setting. We provide numerical examples to illustrate time-complexity and p -adic stability of this tropical F5 algorithm. We also illustrate its merits as a first step before an FGLM algorithm to compute (classical) lex bases over p -adics.
{"title":"On Affine Tropical F5 Algorithms","authors":"Tristan Vaccon, Thibaut Verron, K. Yokoyama","doi":"10.1145/3208976.3209012","DOIUrl":"https://doi.org/10.1145/3208976.3209012","url":null,"abstract":"Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gröbner bases taking into account the valuation of K . Because of the use of the valuation, the theory of tropical Gröbner bases has proved to provide settings for computations over polynomial rings over a p -adic field that are more stable than that of classical Gröbner bases. Beforehand, these strategies were only available for homogeneous polynomials. In this article, we extend the F5 strategy to a new definition of tropical Gröbner bases in an affine setting. We provide numerical examples to illustrate time-complexity and p -adic stability of this tropical F5 algorithm. We also illustrate its merits as a first step before an FGLM algorithm to compute (classical) lex bases over p -adics.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124187895","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}
M. Bender, J. Faugère, Angelos Mantzaflaris, Elias P. Tsigaridas
A fundamental problem in computational algebraic geometry is the computation of the resultant. A central question is when and how to compute it as the determinant of a matrix whose elements are the coefficients of the input polynomials up-to sign. This problem is well understood for unmixed multihomogeneous systems, that is for systems consisting of multihomogeneous polynomials with the same support. However, little is known for mixed systems, that is for systems consisting of polynomials with different supports. We consider the computation of the multihomogeneous resultant of bilinear systems involving two different supports. We present a constructive approach that expresses the resultant as the exact determinant of a Koszul resultant matrix, that is a matrix constructed from maps in the Koszul complex. % We exploit the resultant matrix to propose an algorithm to solve such systems. In the process we extend the classical eigenvalues and eigenvectors criterion to a more general setting. Our extension of the eigenvalues criterion applies to a general class of matrices, including the Sylvester-type and the Koszul-type ones.
{"title":"Bilinear Systems with Two Supports: Koszul Resultant Matrices, Eigenvalues, and Eigenvectors","authors":"M. Bender, J. Faugère, Angelos Mantzaflaris, Elias P. Tsigaridas","doi":"10.1145/3208976.3209011","DOIUrl":"https://doi.org/10.1145/3208976.3209011","url":null,"abstract":"A fundamental problem in computational algebraic geometry is the computation of the resultant. A central question is when and how to compute it as the determinant of a matrix whose elements are the coefficients of the input polynomials up-to sign. This problem is well understood for unmixed multihomogeneous systems, that is for systems consisting of multihomogeneous polynomials with the same support. However, little is known for mixed systems, that is for systems consisting of polynomials with different supports. We consider the computation of the multihomogeneous resultant of bilinear systems involving two different supports. We present a constructive approach that expresses the resultant as the exact determinant of a Koszul resultant matrix, that is a matrix constructed from maps in the Koszul complex. % We exploit the resultant matrix to propose an algorithm to solve such systems. In the process we extend the classical eigenvalues and eigenvectors criterion to a more general setting. Our extension of the eigenvalues criterion applies to a general class of matrices, including the Sylvester-type and the Koszul-type ones.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126816583","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}
Curtis Bright, I. Kotsireas, A. Heinle, Vijay Ganesh
We provide a complete enumeration of all complex Golay pairs of length up to 25, verifying that complex Golay pairs do not exist in lengths 23 and 25 but do exist in length 24. This independently verifies work done by F. Fiedler in 2013 that confirms the 2002 conjecture of Craigen, Holzmann, and Kharaghani that complex Golay pairs of length 23 don't exist. Our enumeration method relies on the recently proposed SAT+CAS paradigm of combining computer algebra systems with SAT solvers to take advantage of the advances made in the fields of symbolic computation and satisfiability checking. The enumeration proceeds in two stages: First, we use a fine-tuned computer program and functionality from computer algebra systems to construct a list containing all sequences which could appear as the first sequence in a complex Golay pair (up to equivalence). Second, we use a programmatic SAT solver to construct all sequences (if any) that pair off with the sequences constructed in the first stage to form a complex Golay pair.
{"title":"Enumeration of Complex Golay Pairs via Programmatic SAT","authors":"Curtis Bright, I. Kotsireas, A. Heinle, Vijay Ganesh","doi":"10.1145/3208976.3209006","DOIUrl":"https://doi.org/10.1145/3208976.3209006","url":null,"abstract":"We provide a complete enumeration of all complex Golay pairs of length up to 25, verifying that complex Golay pairs do not exist in lengths 23 and 25 but do exist in length 24. This independently verifies work done by F. Fiedler in 2013 that confirms the 2002 conjecture of Craigen, Holzmann, and Kharaghani that complex Golay pairs of length 23 don't exist. Our enumeration method relies on the recently proposed SAT+CAS paradigm of combining computer algebra systems with SAT solvers to take advantage of the advances made in the fields of symbolic computation and satisfiability checking. The enumeration proceeds in two stages: First, we use a fine-tuned computer program and functionality from computer algebra systems to construct a list containing all sequences which could appear as the first sequence in a complex Golay pair (up to equivalence). Second, we use a programmatic SAT solver to construct all sequences (if any) that pair off with the sequences constructed in the first stage to form a complex Golay pair.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124392017","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}
Hermite reduction is a classical algorithmic tool in symbolic integration. It is used to decompose a given rational function as a sum of a function with simple poles and the derivative of another rational function. We extend Hermite reduction to arbitrary linear differential operators instead of the pure derivative, and develop efficient algorithms for this reduction. We then apply the generalized Hermite reduction to the computation of linear operators satisfied by single definite integrals of D-finite functions of several continuous or discrete parameters. The resulting algorithm is a generalization of reduction-based methods for creative telescoping.
{"title":"Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions","authors":"A. Bostan, F. Chyzak, Pierre Lairez, B. Salvy","doi":"10.1145/3208976.3208992","DOIUrl":"https://doi.org/10.1145/3208976.3208992","url":null,"abstract":"Hermite reduction is a classical algorithmic tool in symbolic integration. It is used to decompose a given rational function as a sum of a function with simple poles and the derivative of another rational function. We extend Hermite reduction to arbitrary linear differential operators instead of the pure derivative, and develop efficient algorithms for this reduction. We then apply the generalized Hermite reduction to the computation of linear operators satisfied by single definite integrals of D-finite functions of several continuous or discrete parameters. The resulting algorithm is a generalization of reduction-based methods for creative telescoping.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132652383","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}
One of the biggest open problems in computational algebra is the design of efficient algorithms for Gröbner basis computations that take into account the sparsity of the input polynomials. We can perform such computations in the case of unmixed polynomial systems, that is systems with polynomials having the same support, using the approach of Faugère, Spaenlehauer, and Svartz [ISSAC'14]. We present two algorithms for sparse Gröbner bases computations for mixed systems. The first one computes with mixed sparse systems and exploits the supports of the polynomials. Under regularity assumptions, it performs no reductions to zero. For mixed, square, and 0-dimensional multihomogeneous polynomial systems, we present a dedicated, and potentially more efficient, algorithm that exploits different algebraic properties that performs no reduction to zero. We give an explicit bound for the maximal degree appearing in the computations.
{"title":"Towards Mixed Gröbner Basis Algorithms: the Multihomogeneous and Sparse Case","authors":"M. Bender, J. Faugère, Elias P. Tsigaridas","doi":"10.1145/3208976.3209018","DOIUrl":"https://doi.org/10.1145/3208976.3209018","url":null,"abstract":"One of the biggest open problems in computational algebra is the design of efficient algorithms for Gröbner basis computations that take into account the sparsity of the input polynomials. We can perform such computations in the case of unmixed polynomial systems, that is systems with polynomials having the same support, using the approach of Faugère, Spaenlehauer, and Svartz [ISSAC'14]. We present two algorithms for sparse Gröbner bases computations for mixed systems. The first one computes with mixed sparse systems and exploits the supports of the polynomials. Under regularity assumptions, it performs no reductions to zero. For mixed, square, and 0-dimensional multihomogeneous polynomial systems, we present a dedicated, and potentially more efficient, algorithm that exploits different algebraic properties that performs no reduction to zero. We give an explicit bound for the maximal degree appearing in the computations.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134436776","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}
As a typical application, the Lenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL) is used to compute a reduced basis of the orthogonal lattice for a given integer matrix, via reducing a special kind of lattice bases. With such bases in input, we propose a new technique for bounding from above the number of iterations required by the LLL algorithm. The main technical ingredient is a variant of the classical LLL potential, which could prove useful to understand the behavior of LLL for other families of input bases.
{"title":"Computing an LLL-reduced Basis of the Orthogonal Latice","authors":"Jingwei Chen, D. Stehlé, G. Villard","doi":"10.1145/3208976.3209013","DOIUrl":"https://doi.org/10.1145/3208976.3209013","url":null,"abstract":"As a typical application, the Lenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL) is used to compute a reduced basis of the orthogonal lattice for a given integer matrix, via reducing a special kind of lattice bases. With such bases in input, we propose a new technique for bounding from above the number of iterations required by the LLL algorithm. The main technical ingredient is a variant of the classical LLL potential, which could prove useful to understand the behavior of LLL for other families of input bases.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134117379","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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