In this note, we present a variant of a probabilistic algorithm by Cuyt and Lee for the sparse interpolation of multivariate rational functions. We also present an analogous method for the computation of sparse gcds.
{"title":"On sparse interpolation of rational functions and gcds","authors":"J. Hoeven, Grégoire Lecerf","doi":"10.1145/3466895.3466896","DOIUrl":"https://doi.org/10.1145/3466895.3466896","url":null,"abstract":"In this note, we present a variant of a probabilistic algorithm by Cuyt and Lee for the sparse interpolation of multivariate rational functions. We also present an analogous method for the computation of sparse gcds.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"55 1","pages":"1 - 12"},"PeriodicalIF":0.1,"publicationDate":"2020-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3466895.3466896","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46329629","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}
CQF is a free, open-source Magma package for doing computations in quadratic forms theory. We present some selected ingredients of the package.
CQF是一个免费的、开源的Magma包，用于二次型理论的计算。我们介绍了包装的一些精选成分。
{"title":"CQF Magma package","authors":"P. Koprowski","doi":"10.1145/3427218.3427224","DOIUrl":"https://doi.org/10.1145/3427218.3427224","url":null,"abstract":"CQF is a free, open-source Magma package for doing computations in quadratic forms theory. We present some selected ingredients of the package.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"53 - 56"},"PeriodicalIF":0.1,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3427218.3427224","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42753804","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}
{"title":"The 2019 Richard D. Jenks memorial prize","authors":"M. Monagan","doi":"10.1145/3427218.3427226","DOIUrl":"https://doi.org/10.1145/3427218.3427226","url":null,"abstract":"","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"61 - 61"},"PeriodicalIF":0.1,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3427218.3427226","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42693273","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 is quite common that search algorithms for those solutions of difference and differential equations and systems that belong to a fixed class of functions are designed so that nonexistence of solutions of the desired type is detected only in the last stages of the algorithm. However, performing additional tests on the intermediate results makes it possible to stop the algorithm as soon as these tests imply that no solutions of the desired type exist. This gives an opportunity to save time and other computing resources. So, it makes sense to equip algorithms with checkpoints and some tests. We consider these questions in connection with the search for rational solutions of linear homogeneous difference and differential systems with polynomial coefficients, and propose a scheme equipped with such checkpoints and tests, and also report results of experiments with our implementation of the scheme in Maple.
{"title":"Checkpoints in searching for rational solutions of linear ordinary difference and differential systems","authors":"S. Abramov, D. E. Khmelnov, A. Ryabenko","doi":"10.1145/3427218.3427219","DOIUrl":"https://doi.org/10.1145/3427218.3427219","url":null,"abstract":"It is quite common that search algorithms for those solutions of difference and differential equations and systems that belong to a fixed class of functions are designed so that nonexistence of solutions of the desired type is detected only in the last stages of the algorithm. However, performing additional tests on the intermediate results makes it possible to stop the algorithm as soon as these tests imply that no solutions of the desired type exist. This gives an opportunity to save time and other computing resources. So, it makes sense to equip algorithms with checkpoints and some tests. We consider these questions in connection with the search for rational solutions of linear homogeneous difference and differential systems with polynomial coefficients, and propose a scheme equipped with such checkpoints and tests, and also report results of experiments with our implementation of the scheme in Maple.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"18 - 29"},"PeriodicalIF":0.1,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3427218.3427219","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47265975","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}
Christina Katsamaki, F. Rouillier, Elias P. Tsigaridas, Zafeirakis Zafeirakopoulos
PTOPO is a maple package computing the topology and describing the geometry of a parametric plane curve. The algorithm behind PTOPO constructs an abstract graph that is isotopic to the curve. PTOPO exploits the benefits of the parametric representation and performs all computations in the parameter space using exact computing. PTOPO computes the topology and visualizes the curve in less than a second for most examples in the literature. Comparison of maple parametric plot vs PTOPO 1 Topology of Parametric Curves The study of parametric curves is a classical topic in computational algebra and geometry (Sendra and Winkler (1999); Boissonnat and Teillaud (2006)). The interest for computing with parametric curves has been motivated, among others, by the omnipresence of parametric representations in computer modeling and computer aided geometric design (Manocha and Canny (1992); Pérez-Dı́az (2006); Sendra et al. (2008)). ⇤Supported by the Fondation Sciences Mathématiques de Paris (FSMP)
PTOPO是一个枫包计算拓扑和描述参数平面曲线的几何形状。PTOPO背后的算法构建了一个与曲线相同的抽象图。PTOPO利用参数表示的优点，并使用精确计算在参数空间中执行所有计算。对于文献中的大多数例子，PTOPO在不到一秒钟的时间内计算拓扑并可视化曲线。参数曲线的研究是计算代数和几何中的一个经典课题(Sendra and Winkler (1999);布瓦松纳和泰约(2006))。对参数曲线计算的兴趣，除其他外，是由于在计算机建模和计算机辅助几何设计中无处不在的参数表示(Manocha和Canny (1992);Perez-Dı́阿兹(2006);Sendra等人(2008))。巴黎数学科学基金会(FSMP)支持
{"title":"PTOPO: a maple package for the topology of parametric curves","authors":"Christina Katsamaki, F. Rouillier, Elias P. Tsigaridas, Zafeirakis Zafeirakopoulos","doi":"10.1145/3427218.3427223","DOIUrl":"https://doi.org/10.1145/3427218.3427223","url":null,"abstract":"PTOPO is a maple package computing the topology and describing the geometry of a parametric plane curve. The algorithm behind PTOPO constructs an abstract graph that is isotopic to the curve. PTOPO exploits the benefits of the parametric representation and performs all computations in the parameter space using exact computing. PTOPO computes the topology and visualizes the curve in less than a second for most examples in the literature. Comparison of maple parametric plot vs PTOPO 1 Topology of Parametric Curves The study of parametric curves is a classical topic in computational algebra and geometry (Sendra and Winkler (1999); Boissonnat and Teillaud (2006)). The interest for computing with parametric curves has been motivated, among others, by the omnipresence of parametric representations in computer modeling and computer aided geometric design (Manocha and Canny (1992); Pérez-Dı́az (2006); Sendra et al. (2008)). ⇤Supported by the Fondation Sciences Mathématiques de Paris (FSMP)","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"49-52"},"PeriodicalIF":0.1,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3427218.3427223","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64034674","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}
Antonio Jiménez-Pastor, A. Bostan, F. Chyzak, Pierre Lairez
We present in this extended abstract a new software designed to work with generating functions that count walks in the quarter plane. With this software we offer a cohesive package that brings together all the required procedures for manipulating these generating functions, as well as a unified interface to deal with them. We also display results that this package offers on a public webpage.
{"title":"The Sage package comb_walks for walks in the quarter plane","authors":"Antonio Jiménez-Pastor, A. Bostan, F. Chyzak, Pierre Lairez","doi":"10.1145/3427218.3427220","DOIUrl":"https://doi.org/10.1145/3427218.3427220","url":null,"abstract":"We present in this extended abstract a new software designed to work with generating functions that count walks in the quarter plane. With this software we offer a cohesive package that brings together all the required procedures for manipulating these generating functions, as well as a unified interface to deal with them. We also display results that this package offers on a public webpage.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"30 - 38"},"PeriodicalIF":0.1,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3427218.3427220","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46492496","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 introduce an algorithm for computing the value of all monomials in the Chow group of zero cycles in the moduli space of stable curves.
我们介绍了一种计算稳定曲线模空间中零环Chow群中所有单项式值的算法。
{"title":"A calculus for monomials in Chow group of zero cycles in the moduli space of stable curves","authors":"Jiayue Qi","doi":"10.1145/3457341.3457344","DOIUrl":"https://doi.org/10.1145/3457341.3457344","url":null,"abstract":"We introduce an algorithm for computing the value of all monomials in the Chow group of zero cycles in the moduli space of stable curves.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"91 - 94"},"PeriodicalIF":0.1,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3457341.3457344","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46735245","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 divisor class group of a hyperelliptic curve defined over a finite field is a finite abelian group at the center of a number of important open questions in algebraic geometry, number theory and cryptography. Many of these problems lend themselves to numerical investigation, and as emphasized by Sutherland [14, 13], fast arithmetic in the divisor class group is crucial for their efficiency. Besides, implementations of these fundamental operations are at the core of the algebraic geometry packages of widely-used computer algebra systems such as Magma and Sage.
{"title":"Improved divisor arithmetic on generic hyperelliptic curves","authors":"Sebastian Lindner, L. Imbert, M. Jacobson","doi":"10.1145/3457341.3457345","DOIUrl":"https://doi.org/10.1145/3457341.3457345","url":null,"abstract":"The divisor class group of a hyperelliptic curve defined over a finite field is a finite abelian group at the center of a number of important open questions in algebraic geometry, number theory and cryptography. Many of these problems lend themselves to numerical investigation, and as emphasized by Sutherland [14, 13], fast arithmetic in the divisor class group is crucial for their efficiency. Besides, implementations of these fundamental operations are at the core of the algebraic geometry packages of widely-used computer algebra systems such as Magma and Sage.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"95 - 99"},"PeriodicalIF":0.1,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3457341.3457345","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44761796","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}
Transformation of a polynomial ODE system to a special quadratic form has been successfully used recently as a preprocessing step for model order reduction methods. However, to the best of our knowledge, there has been no practical algorithm for performing this step automatically with any optimality guarantees. We present an algorithm that, given a system of polynomial ODEs, finds a transformation into a quadratic ODE system by introducing new variables which are monomials of the original variables. The algorithm is guaranteed to produce an optimal transformation of this form. The algorithm is implemented, and we demonstrate it on examples from the literature.
{"title":"Optimal monomial quadratization for ODE systems","authors":"A. Bychkov, G. Pogudin","doi":"10.1145/3457341.3457350","DOIUrl":"https://doi.org/10.1145/3457341.3457350","url":null,"abstract":"Transformation of a polynomial ODE system to a special quadratic form has been successfully used recently as a preprocessing step for model order reduction methods. However, to the best of our knowledge, there has been no practical algorithm for performing this step automatically with any optimality guarantees. We present an algorithm that, given a system of polynomial ODEs, finds a transformation into a quadratic ODE system by introducing new variables which are monomials of the original variables. The algorithm is guaranteed to produce an optimal transformation of this form. The algorithm is implemented, and we demonstrate it on examples from the literature.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"119 - 123"},"PeriodicalIF":0.1,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3457341.3457350","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46707568","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 p be a prime of the form p = σ2k + 1 with σ small and let Fp denote the finite field with p elements. Let P(z) be a polynomial of degree d in Fp[z] with d distinct roots in Fp. For p =5 · 255 + 1 we can compute the roots of such polynomials of degree 109. We believe we are the first to factor such polynomials of size one billion. We used a multi-core computer with two 10 core Intel Xeon E5 2680 v2 CPUs and 128 gigabytes of RAM. The factorization takes just under 4,000 seconds on 10 cores and uses 121 gigabytes of RAM. We used the tangent Graeffe root finding algorithm from [27, 19] which is a factor of O(log d) faster than the Cantor-Zassenhaus algorithm. We implemented the tangent Graeffe algorithm in C using our own library of 64 bit integer FFT based in-place polynomial algorithms then parallelized the FFT and main steps using Cilk C. In this article we discuss the steps of the tangent Graeffe algorithm, the sub-algorithms that we used, how we parallelized them, and how we organized the memory so we could factor a polynomial of degree 109. We give both a theoretical and practical comparison of the tangent Graeffe algorithm with the Cantor-Zassenhaus algorithm for root finding. We improve the complexity of the tangent Graeffe algorithm by a factor of 2. We present a new in-place product tree multiplication algorithm that is fully parallelizable. We present some timings comparing our software with Magma's polynomial factorization command. Polynomial root finding over smooth finite fields is a key ingredient for algorithms for sparse polynomial interpolation that are based on geometric sequences. This application was also one of our main motivations for the present work.
{"title":"Computing one billion roots using the tangent Graeffe method","authors":"J. Hoeven, M. Monagan","doi":"10.1145/3457341.3457342","DOIUrl":"https://doi.org/10.1145/3457341.3457342","url":null,"abstract":"Let p be a prime of the form p = σ2k + 1 with σ small and let Fp denote the finite field with p elements. Let P(z) be a polynomial of degree d in Fp[z] with d distinct roots in Fp. For p =5 · 255 + 1 we can compute the roots of such polynomials of degree 109. We believe we are the first to factor such polynomials of size one billion. We used a multi-core computer with two 10 core Intel Xeon E5 2680 v2 CPUs and 128 gigabytes of RAM. The factorization takes just under 4,000 seconds on 10 cores and uses 121 gigabytes of RAM. We used the tangent Graeffe root finding algorithm from [27, 19] which is a factor of O(log d) faster than the Cantor-Zassenhaus algorithm. We implemented the tangent Graeffe algorithm in C using our own library of 64 bit integer FFT based in-place polynomial algorithms then parallelized the FFT and main steps using Cilk C. In this article we discuss the steps of the tangent Graeffe algorithm, the sub-algorithms that we used, how we parallelized them, and how we organized the memory so we could factor a polynomial of degree 109. We give both a theoretical and practical comparison of the tangent Graeffe algorithm with the Cantor-Zassenhaus algorithm for root finding. We improve the complexity of the tangent Graeffe algorithm by a factor of 2. We present a new in-place product tree multiplication algorithm that is fully parallelizable. We present some timings comparing our software with Magma's polynomial factorization command. Polynomial root finding over smooth finite fields is a key ingredient for algorithms for sparse polynomial interpolation that are based on geometric sequences. This application was also one of our main motivations for the present work.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"65 - 85"},"PeriodicalIF":0.1,"publicationDate":"2020-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3457341.3457342","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43840894","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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