The first public beta release of GAP 4[6] was made on July 18 1997. Since then the system has been cited in over 2400 publications, and its distribution now includes over 130 contributed extension pack- ages. This tutorial will review the special features of computational abstract algebra and how they are reflected in the system design; some areas of current algorithmic development, and some recent achievements.
GAP 4的第一个公开测试版[6]发布于1997年7月18日。从那时起，该系统已在2400多种出版物中被引用，其分布现在包括130多个贡献的扩展包。本教程将回顾计算抽象代数的特殊功能以及它们如何反映在系统设计中;当前算法发展的一些领域，以及最近取得的一些成就。
{"title":"GAP 4 at Twenty-one - Algorithms, System Design and Applications","authors":"S. Linton","doi":"10.1145/3208976.3209026","DOIUrl":"https://doi.org/10.1145/3208976.3209026","url":null,"abstract":"The first public beta release of GAP 4[6] was made on July 18 1997. Since then the system has been cited in over 2400 publications, and its distribution now includes over 130 contributed extension pack- ages. This tutorial will review the special features of computational abstract algebra and how they are reflected in the system design; some areas of current algorithmic development, and some recent achievements.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"8 1 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":"116782309","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}
In 1988, Weispfenning published a seminal paper introducing a substitution technique for quantifier elimination in the linear theories of ordered and valued fields. The original focus was on complexity bounds including the important result that the decision problem for Tarski Algebra is bounded from below by a double exponential function. Soon after, Weispfenning's group began to implement substitution techniques in software in order to study their potential applicability to real world problems. Today virtual substitution has become an established computational tool, which greatly complements cylindrical algebraic decomposition. There are powerful implementations and applications with a current focus on satisfiability modulo theory solving and qualitative analysis of biological networks.
{"title":"Thirty Years of Virtual Substitution: Foundations, Techniques, Applications","authors":"T. Sturm","doi":"10.1145/3208976.3209030","DOIUrl":"https://doi.org/10.1145/3208976.3209030","url":null,"abstract":"In 1988, Weispfenning published a seminal paper introducing a substitution technique for quantifier elimination in the linear theories of ordered and valued fields. The original focus was on complexity bounds including the important result that the decision problem for Tarski Algebra is bounded from below by a double exponential function. Soon after, Weispfenning's group began to implement substitution techniques in software in order to study their potential applicability to real world problems. Today virtual substitution has become an established computational tool, which greatly complements cylindrical algebraic decomposition. There are powerful implementations and applications with a current focus on satisfiability modulo theory solving and qualitative analysis of biological networks.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"121 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":"132538092","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}
An algorithm for interpolating a polynomial f from evaluation points whose running time depends on the sparsity t of the polynomial when it is represented as a sum of t Chebyshev Polynomials of the First Kind with non-zero scalar coefficients is given by Lakshman Y. N. and Saunders [SIAM J. Comput., vol. 24, nr. 2 (1995)]; Kaltofen and Lee [JSC, vol. 36, nr. 3--4 (2003)] analyze a randomized early termination version which computes the sparsity t. Those algorithms mirror Prony's algorithm for the standard power basis to the Chebyshev Basis of the First Kind. An alternate algorithm by Arnold's and Kaltofen's [Proc. ISSAC 2015, Sec. 4] uses Prony's original algorithm for standard power terms. Here we give sparse interpolation algorithms for generalized Chebyshev polynomials, which include the Chebyshev Bases of the Second, Third and Fourth Kind. Our algorithms also reduce to Prony's algorithm. If given on input a bound B >= t for the sparsity, our new algorithms deterministically recover the sparse representation in the First, Second, Third and Fourth Kind Chebyshev representation from exactly t + B evaluations. Finally, we generalize our algorithms to bases whose Chebyshev recurrences have parametric scalars. We also show how to compute those parameter values which optimize the sparsity of the representation in the corresponding basis, similar to computing a sparsest shift.
Lakshman Y. N. and Saunders [SIAM J. Comput]给出了一种从评价点插值多项式f的算法，当多项式被表示为t个非零标量系数的第一类Chebyshev多项式的和时，其运行时间取决于多项式的稀疏性t。，第24卷，第2期(1995)];Kaltofen和Lee [JSC, vol. 36, nr. 3—4(2003)]分析了一个随机的早期终止版本，该版本计算稀疏性t。这些算法将proony的标准功率基算法镜像到第一类切比雪夫基。Arnold和Kaltofen的替代算法[Proc. ISSAC 2015, Sec 4]使用proony的原始算法来处理标准功率项。本文给出了广义Chebyshev多项式(包括第二类、第三类和第四类Chebyshev基)的稀疏插值算法。我们的算法也简化为普罗尼算法。如果给定一个边界B >= t的稀疏性，我们的新算法确定性地从t + B次求值中恢复第一、第二、第三和第四类Chebyshev表示中的稀疏表示。最后，我们将我们的算法推广到具有参数标量的切比雪夫递归的基。我们还展示了如何计算那些参数值，这些参数值可以在相应的基中优化表示的稀疏性，类似于计算最稀疏移位。
{"title":"Sparse Polynomial Interpolation With Arbitrary Orthogonal Polynomial Bases","authors":"E. Imamoglu, E. Kaltofen, Zhengfeng Yang","doi":"10.1145/3208976.3208999","DOIUrl":"https://doi.org/10.1145/3208976.3208999","url":null,"abstract":"An algorithm for interpolating a polynomial f from evaluation points whose running time depends on the sparsity t of the polynomial when it is represented as a sum of t Chebyshev Polynomials of the First Kind with non-zero scalar coefficients is given by Lakshman Y. N. and Saunders [SIAM J. Comput., vol. 24, nr. 2 (1995)]; Kaltofen and Lee [JSC, vol. 36, nr. 3--4 (2003)] analyze a randomized early termination version which computes the sparsity t. Those algorithms mirror Prony's algorithm for the standard power basis to the Chebyshev Basis of the First Kind. An alternate algorithm by Arnold's and Kaltofen's [Proc. ISSAC 2015, Sec. 4] uses Prony's original algorithm for standard power terms. Here we give sparse interpolation algorithms for generalized Chebyshev polynomials, which include the Chebyshev Bases of the Second, Third and Fourth Kind. Our algorithms also reduce to Prony's algorithm. If given on input a bound B >= t for the sparsity, our new algorithms deterministically recover the sparse representation in the First, Second, Third and Fourth Kind Chebyshev representation from exactly t + B evaluations. Finally, we generalize our algorithms to bases whose Chebyshev recurrences have parametric scalars. We also show how to compute those parameter values which optimize the sparsity of the representation in the corresponding basis, similar to computing a sparsest shift.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"61 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":"122118068","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}
A sum of affine powers is an expression of the form [f(x1,...,xn) = ∑i=1s αi li(x1,...,xn)ei] where li is an affine form. We propose polynomial time black-box algorithms that find the decomposition with the smallest value of s for an input polynomial f . Our algorithms work in situations where s is small enough compared to the number of variables or to the exponents ei. Although quite simple, this model is a generalization of Waring decomposition. This paper extends previous work on Waring decomposition as well as our work on univariate sums of affine powers (ISSAC'17).
{"title":"Polynomial Equivalence Problems for Sum of Affine Powers","authors":"Ignacio García-Marco, P. Koiran, Timothée Pecatte","doi":"10.1145/3208976.3208993","DOIUrl":"https://doi.org/10.1145/3208976.3208993","url":null,"abstract":"A sum of affine powers is an expression of the form [f(x1,...,xn) = ∑i=1s αi li(x1,...,xn)ei] where li is an affine form. We propose polynomial time black-box algorithms that find the decomposition with the smallest value of s for an input polynomial f . Our algorithms work in situations where s is small enough compared to the number of variables or to the exponents ei. Although quite simple, this model is a generalization of Waring decomposition. This paper extends previous work on Waring decomposition as well as our work on univariate sums of affine powers (ISSAC'17).","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"102 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":"115215258","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 a new framework for a posteriori validation of vector-valued problems with componentwise tight error enclosures, and use it to design a symbolic-numeric Newton-like validation algorithm for Chebyshev approximate solutions of coupled systems of linear ordinary differential equations. More precisely, given a coupled differential system with polynomial coefficients over a compact interval (or continuous coefficients rigorously approximated by polynomials) and componentwise polynomial approximate solutions in Chebyshev basis, the algorithm outputs componentwise rigorous upper bounds for the approximation errors, with respect to the uniform norm over the interval under consideration. A complexity analysis shows that the number of arithmetic operations needed by this algorithm (in floating-point or interval arithmetics) is proportional to the approximation degree when the differential equation is considered fixed. Finally, we illustrate the efficiency of this fully automated validation method on an example of a coupled Airy-like system.
{"title":"A Newton-like Validation Method for Chebyshev Approximate Solutions of Linear Ordinary Differential Systems","authors":"F. Bréhard","doi":"10.1145/3208976.3209000","DOIUrl":"https://doi.org/10.1145/3208976.3209000","url":null,"abstract":"We provide a new framework for a posteriori validation of vector-valued problems with componentwise tight error enclosures, and use it to design a symbolic-numeric Newton-like validation algorithm for Chebyshev approximate solutions of coupled systems of linear ordinary differential equations. More precisely, given a coupled differential system with polynomial coefficients over a compact interval (or continuous coefficients rigorously approximated by polynomials) and componentwise polynomial approximate solutions in Chebyshev basis, the algorithm outputs componentwise rigorous upper bounds for the approximation errors, with respect to the uniform norm over the interval under consideration. A complexity analysis shows that the number of arithmetic operations needed by this algorithm (in floating-point or interval arithmetics) is proportional to the approximation degree when the differential equation is considered fixed. Finally, we illustrate the efficiency of this fully automated validation method on an example of a coupled Airy-like system.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"51 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":"129293404","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}
In this paper, we propose algorithms for computing minimal associated primes of ideals in polynomial rings over Q and computing radicals of ideals in polynomial rings over a field. They apply Chinese Remainder Theorem (CRT) to Laplagne's algorithm which computes minimal associated primes without producing redundant components and computes radicals. CRT reconstructs an object in a ring from its modular images in the quotient rings modulo some ideals. In Laplagne's algorithm, ideals are decomposed over rational function fields by regarding some variables as parameters. In our new algorithms, we compute the minimal associated primes and the radical of < φ(G) > for a given ideal I= < G >, where φ is a substitution map for a parameter. Then we construct candidates of the minimal associated primes and the radical of I by applying CRT for those of < φ(G) >'s. In order for this method to work correctly, the shape of each modular component must coincide with that of the corresponding component of the ideal for computations of minimal associated primes, and radicals of modular images of given ideals must coincide with modular images of radicals of given ideals for radical computations. The former is realized with a high probability because a multivariate irreducible polynomial over Q remains irreducible after a substitution of integers for variables with a high probability and the latter is realized except for a finite number of moduli.
本文给出了计算Q上多项式环上理想的最小关联素数和计算域上多项式环上理想的根的算法。他们将中国剩余定理(CRT)应用于计算最小关联素数而不产生冗余分量和计算根号的拉普拉斯算法。CRT从其在商环中的模像对某些理想模重建环中的对象。在Laplagne算法中，将一些变量作为参数，在有理函数域上进行理想分解。在我们的新算法中，我们计算了给定理想I= < G >的最小关联素数和< φ(G) >的根，其中φ是参数的替换映射。然后对< φ(G) > s的最小伴生素数和I的根数应用CRT构造了最小伴生素数的候选项。为了使这种方法正确地工作，每个模分量的形状必须与计算最小关联素数的理想的相应分量的形状相一致，并且给定理想的模象的根必须与给定理想的根的模象相一致。前者实现的概率很大，因为用整数替换变量后，Q上的多元不可约多项式仍然不可约，而后者实现的概率只有有限个模。
{"title":"Modular Algorithms for Computing Minimal Associated Primes and Radicals of Polynomial Ideals","authors":"T. Aoyama, M. Noro","doi":"10.1145/3208976.3209014","DOIUrl":"https://doi.org/10.1145/3208976.3209014","url":null,"abstract":"In this paper, we propose algorithms for computing minimal associated primes of ideals in polynomial rings over Q and computing radicals of ideals in polynomial rings over a field. They apply Chinese Remainder Theorem (CRT) to Laplagne's algorithm which computes minimal associated primes without producing redundant components and computes radicals. CRT reconstructs an object in a ring from its modular images in the quotient rings modulo some ideals. In Laplagne's algorithm, ideals are decomposed over rational function fields by regarding some variables as parameters. In our new algorithms, we compute the minimal associated primes and the radical of < φ(G) > for a given ideal I= < G >, where φ is a substitution map for a parameter. Then we construct candidates of the minimal associated primes and the radical of I by applying CRT for those of < φ(G) >'s. In order for this method to work correctly, the shape of each modular component must coincide with that of the corresponding component of the ideal for computations of minimal associated primes, and radicals of modular images of given ideals must coincide with modular images of radicals of given ideals for radical computations. The former is realized with a high probability because a multivariate irreducible polynomial over Q remains irreducible after a substitution of integers for variables with a high probability and the latter is realized except for a finite number of moduli.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"41 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":"129195606","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}
Simply put, a sparse polynomial is one whose zero coefficients are not explicitly stored. Such objects are ubiquitous in exact computing, and so naturally we would like to have efficient algorithms to handle them. However, with this compact storage comes new algorithmic challenges, as fast algorithms for dense polynomials may no longer be efficient. In this tutorial we examine the state of the art for sparse polynomial algorithms in three areas: arithmetic, interpolation, and factorization. The aim is to highlight recent progress both in theory and in practice, as well as opportunities for future work.
{"title":"What Can (and Can't) we Do with Sparse Polynomials?","authors":"Daniel S. Roche","doi":"10.1145/3208976.3209027","DOIUrl":"https://doi.org/10.1145/3208976.3209027","url":null,"abstract":"Simply put, a sparse polynomial is one whose zero coefficients are not explicitly stored. Such objects are ubiquitous in exact computing, and so naturally we would like to have efficient algorithms to handle them. However, with this compact storage comes new algorithmic challenges, as fast algorithms for dense polynomials may no longer be efficient. In this tutorial we examine the state of the art for sparse polynomial algorithms in three areas: arithmetic, interpolation, and factorization. The aim is to highlight recent progress both in theory and in practice, as well as opportunities for future work.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"4 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":"122704411","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 year we are celebrating the 10th anniversary of a dramatic revolution in combinatorial geometry, fueled by the infusion of techniques from algebraic geometry and algebra that have proven effective in solving a variety of hard problems that were thought to be unreachable with more traditional techniques. The new era has begun with two groundbreaking papers of Guth and Katz, the second of which has (almost completely) solved the celebrated distinct distances problem of Paul Erdös, open since 1946. In this talk I will survey, as time permits, some of the progress that has been made since then, including a variety of problems on distinct and repeated distances and other configurations, on incidences between points and lines, curves, and surfaces in two, three, and higher dimensions, on polynomials vanishing on Cartesian products with applications, and on cycle elimination for lines and triangles in three dimensions.
{"title":"Algebraic Techniques in Geometry: The 10th Anniversary","authors":"M. Sharir","doi":"10.1145/3208976.3209028","DOIUrl":"https://doi.org/10.1145/3208976.3209028","url":null,"abstract":"This year we are celebrating the 10th anniversary of a dramatic revolution in combinatorial geometry, fueled by the infusion of techniques from algebraic geometry and algebra that have proven effective in solving a variety of hard problems that were thought to be unreachable with more traditional techniques. The new era has begun with two groundbreaking papers of Guth and Katz, the second of which has (almost completely) solved the celebrated distinct distances problem of Paul Erdös, open since 1946. In this talk I will survey, as time permits, some of the progress that has been made since then, including a variety of problems on distinct and repeated distances and other configurations, on incidences between points and lines, curves, and surfaces in two, three, and higher dimensions, on polynomials vanishing on Cartesian products with applications, and on cycle elimination for lines and triangles in three dimensions.","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":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125679642","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}
A new algorithm, which combines the GVW algorithm with the Mora normal form algorithm, is presented to compute the standard bases of ideals in a local ring. Since term orders in local ring are not well-orderings, there may not be a minimal signature in an infinite set, and we can not extend the GVW algorithm from a polynomial ring to a local ring directly. Nevertheless, when given an anti-graded order in R and a term-over-position order in Rm that are compatible, we can construct a special set such that it has a minimal signature, where R , Rm are a local ring and a R -module, respectively. That is, for any given polynomial v0 ın R, the set consisting of signatures of pairs (u,v)ın Rm x R has a minimal element, where the leading power products of v and v0 are equal. In this case, we prove a cover theorem in R , and use three criteria (syzygy criterion, signature criterion and rewrite criterion) to discard useless J-pairs without any reductions. Mora normal form algorithm is also extended to do regular top-reductions in Rm x R, and the correctness and termination of the algorithm are proved. The proposed algorithm has been implemented in the computer algebra system Maple, and experiment results show that most of J-pairs can be discarded by three criteria in the examples.
将GVW算法与Mora范式算法相结合，提出了一种计算局部环理想标准基的新算法。由于局部环中的项序不是良序的，因此在无限集中可能不存在最小签名，因此我们不能将GVW算法从多项式环直接推广到局部环。然而，当给定R中的反梯度阶和Rm中的项过位阶相容时，我们可以构造一个具有最小签名的特殊集合，其中R、Rm分别为局部环和R -模。也就是说，对于任何给定的多项式v0 ın R，由(u,v)ın Rm x R对的签名组成的集合有一个最小元素，其中v和v0的前导幂积相等。在这种情况下，我们在R中证明了一个覆盖定理，并使用三个准则(syzygy准则、signature准则和重写准则)在没有任何约简的情况下丢弃无用的j对。将Mora范式算法推广到Rm x R的正则上约，证明了该算法的正确性和终止性。该算法已在计算机代数系统Maple中实现，实验结果表明，在实例中，根据三个准则可以丢弃大多数j对。
{"title":"Extending the GVW Algorithm to Local Ring","authors":"Dong Lu, Dingkang Wang, Fanghui Xiao, Jie Zhou","doi":"10.1145/3208976.3208979","DOIUrl":"https://doi.org/10.1145/3208976.3208979","url":null,"abstract":"A new algorithm, which combines the GVW algorithm with the Mora normal form algorithm, is presented to compute the standard bases of ideals in a local ring. Since term orders in local ring are not well-orderings, there may not be a minimal signature in an infinite set, and we can not extend the GVW algorithm from a polynomial ring to a local ring directly. Nevertheless, when given an anti-graded order in R and a term-over-position order in Rm that are compatible, we can construct a special set such that it has a minimal signature, where R , Rm are a local ring and a R -module, respectively. That is, for any given polynomial v0 ın R, the set consisting of signatures of pairs (u,v)ın Rm x R has a minimal element, where the leading power products of v and v0 are equal. In this case, we prove a cover theorem in R , and use three criteria (syzygy criterion, signature criterion and rewrite criterion) to discard useless J-pairs without any reductions. Mora normal form algorithm is also extended to do regular top-reductions in Rm x R, and the correctness and termination of the algorithm are proved. The proposed algorithm has been implemented in the computer algebra system Maple, and experiment results show that most of J-pairs can be discarded by three criteria in the examples.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"44 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":"114227150","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 article is an extended abstract of the ISSAC 2018 talk "Polynomial systems arising from discretizing systems of nonlinear differential equations" by Andrew Sommese.
{"title":"Polynomial Systems Arising From Discretizing Systems of Nonlinear Differential Equations","authors":"A. Sommese","doi":"10.1145/3208976.3209029","DOIUrl":"https://doi.org/10.1145/3208976.3209029","url":null,"abstract":"This article is an extended abstract of the ISSAC 2018 talk \"Polynomial systems arising from discretizing systems of nonlinear differential equations\" by Andrew Sommese.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"38 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":"133062407","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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