{"title":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","authors":"","doi":"10.1145/3208976","DOIUrl":"https://doi.org/10.1145/3208976","url":null,"abstract":"","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":"116589264","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 deterministic algorithm for deciding if a polynomial ideal, with coefficients in an algebraically closed field K of characteristic zero, of which we know just some very limited data, namely: the number n of variables, and some upper bound for the geometric degree of its zero set in Kn, is or not the zero ideal. The algorithm performs just a finite number of decisions to check wheather a point is or not in the zero set of the ideal. Moreover, we extend this technique to test, in the same fashion, if the elimination of some variables in the given ideal yields or not the zero ideal. Finally, the role of this technique in the context of automated theorem proving of elementary geometry statements, is presented, with references to recent documents describing the excellent performance of the already existing prototype version, implemented in GeoGebra.
{"title":"The Importance of Being Zero","authors":"T. Recio, J. Sendra, Carlos Villarino","doi":"10.1145/3208976.3208981","DOIUrl":"https://doi.org/10.1145/3208976.3208981","url":null,"abstract":"We present a deterministic algorithm for deciding if a polynomial ideal, with coefficients in an algebraically closed field K of characteristic zero, of which we know just some very limited data, namely: the number n of variables, and some upper bound for the geometric degree of its zero set in Kn, is or not the zero ideal. The algorithm performs just a finite number of decisions to check wheather a point is or not in the zero set of the ideal. Moreover, we extend this technique to test, in the same fashion, if the elimination of some variables in the given ideal yields or not the zero ideal. Finally, the role of this technique in the context of automated theorem proving of elementary geometry statements, is presented, with references to recent documents describing the excellent performance of the already existing prototype version, implemented in GeoGebra.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"21 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":"115560446","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 five methods for computation of limits of real multivariate rational functions. The methods do not require any assumptions about the rational function and compute the lower limit and the upper limit. All methods are based on the cylindrical algebraic decomposition (CAD) algorithm, but use different formulations of the problem. We give an empirical comparison of the methods on a large set of examples.
{"title":"Comparison of CAD-based Methods for Computation of Rational Function Limits","authors":"A. Strzebonski","doi":"10.1145/3208976.3208982","DOIUrl":"https://doi.org/10.1145/3208976.3208982","url":null,"abstract":"We present five methods for computation of limits of real multivariate rational functions. The methods do not require any assumptions about the rational function and compute the lower limit and the upper limit. All methods are based on the cylindrical algebraic decomposition (CAD) algorithm, but use different formulations of the problem. We give an empirical comparison of the methods on a large set of examples.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"42 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":"114552793","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 give a new algorithm for the formal reduction of linear differential systems with Laurent series coefficients. We show how to obtain a decomposition of Balser, Jurkat and Lutz using eigenring techniques. We establish structural information on the obtained indecomposable subsystems and retrieve information on their invariants such as ramification. We show why classical algorithms then perform well on these subsystems. We also give precise estimates of the precision on the power series which is required in each step of our algorithm. The algorithm is implemented in Maple. We give examples in [14].
{"title":"A New Approach for Formal Reduction of Singular Linear Differential Systems Using Eigenrings","authors":"M. Barkatou, Joelle Saadé, Jacques-Arthur Weil","doi":"10.1145/3208976.3209016","DOIUrl":"https://doi.org/10.1145/3208976.3209016","url":null,"abstract":"We give a new algorithm for the formal reduction of linear differential systems with Laurent series coefficients. We show how to obtain a decomposition of Balser, Jurkat and Lutz using eigenring techniques. We establish structural information on the obtained indecomposable subsystems and retrieve information on their invariants such as ramification. We show why classical algorithms then perform well on these subsystems. We also give precise estimates of the precision on the power series which is required in each step of our algorithm. The algorithm is implemented in Maple. We give examples in [14].","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"27 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":"125997004","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 consider the problem of computing the nearest matrix polynomial with a non-trivial Smith Normal Form. We show that computing the Smith form of a matrix polynomial is amenable to numeric computation as an optimization problem. Furthermore, we describe an effective optimization technique to find a nearby matrix polynomial with a non-trivial Smith form. The results are later generalized to include the computation of a matrix polynomial having a maximum specified number of ones in the Smith Form (i.e., with a maximum specified McCoy rank). We discuss the geometry and existence of solutions and how our results can used for a backwards error analysis. We develop an optimization-based approach and demonstrate an iterative numerical method for computing a nearby matrix polynomial with the desired spectral properties. We also describe the implementation of our algorithms and demonstrate the robustness with examples in Maple.
{"title":"Computing Nearby Non-trivial Smith Forms","authors":"M. Giesbrecht, Joseph Haraldson, G. Labahn","doi":"10.1145/3208976.3209024","DOIUrl":"https://doi.org/10.1145/3208976.3209024","url":null,"abstract":"We consider the problem of computing the nearest matrix polynomial with a non-trivial Smith Normal Form. We show that computing the Smith form of a matrix polynomial is amenable to numeric computation as an optimization problem. Furthermore, we describe an effective optimization technique to find a nearby matrix polynomial with a non-trivial Smith form. The results are later generalized to include the computation of a matrix polynomial having a maximum specified number of ones in the Smith Form (i.e., with a maximum specified McCoy rank). We discuss the geometry and existence of solutions and how our results can used for a backwards error analysis. We develop an optimization-based approach and demonstrate an iterative numerical method for computing a nearby matrix polynomial with the desired spectral properties. We also describe the implementation of our algorithms and demonstrate the robustness with examples in Maple.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"12 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":"125370763","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 algorithms and heuristics that allow us to express arbitrary elements of SLn(Z) and Sp2n (Z) as products of generators in particular "standard" generating sets. For elements obtained experimentally as random products, it produces product expressions whose lengths are competitive with the input lengths.
{"title":"Constructive Membership Tests in Some Infinite Matrix Groups","authors":"A. Hulpke","doi":"10.1145/3208976.3208983","DOIUrl":"https://doi.org/10.1145/3208976.3208983","url":null,"abstract":"We describe algorithms and heuristics that allow us to express arbitrary elements of SLn(Z) and Sp2n (Z) as products of generators in particular \"standard\" generating sets. For elements obtained experimentally as random products, it produces product expressions whose lengths are competitive with the input lengths.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"66 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":"116618596","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}
Straightening is the most fundamental symbolic manipulation in bracket algebra. Young's classical algorithm and White's more recent algorithm have poor performance in straightening bracket polynomials of degree >4. Rota's straightening algorithm based on Capelli operator is generally superior to the former two in speed, but still performs badly when the degree reaches 5. In this paper, a new operator is defined in bracket algebra based on tableau manipulations, and is simpler than Capelli operator. A new straightening algorithm is then proposed, and is superior to the above three algorithms by a speedup of two order of magnitude on average by testing over 500 examples in the past two years.
{"title":"Fast Straightening Algorithm for Bracket Polynomials Based on Tableau Manipulations","authors":"Changpeng Shao, Hongbo Li","doi":"10.1145/3208976.3208978","DOIUrl":"https://doi.org/10.1145/3208976.3208978","url":null,"abstract":"Straightening is the most fundamental symbolic manipulation in bracket algebra. Young's classical algorithm and White's more recent algorithm have poor performance in straightening bracket polynomials of degree >4. Rota's straightening algorithm based on Capelli operator is generally superior to the former two in speed, but still performs badly when the degree reaches 5. In this paper, a new operator is defined in bracket algebra based on tableau manipulations, and is simpler than Capelli operator. A new straightening algorithm is then proposed, and is superior to the above three algorithms by a speedup of two order of magnitude on average by testing over 500 examples in the past two years.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"40 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":"115486544","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}
Homotopy continuation is a well-known method in numerical root-finding. Recently, certified algorithms for homotopy continuation based on Smale's alpha-theory have been developed. This approach enforces very strong requirements at each step, leading to small step sizes. In this paper, we propose an approach that is independent of alpha-theory. It is based on the weaker notion of well-isolated approximations to the roots. We apply it to univariate polynomials and provide experimental evidence of its feasibility.
{"title":"An Approach for Certifying Homotopy Continuation Paths: Univariate Case","authors":"Juan Xu, M. Burr, C. Yap","doi":"10.1145/3208976.3209010","DOIUrl":"https://doi.org/10.1145/3208976.3209010","url":null,"abstract":"Homotopy continuation is a well-known method in numerical root-finding. Recently, certified algorithms for homotopy continuation based on Smale's alpha-theory have been developed. This approach enforces very strong requirements at each step, leading to small step sizes. In this paper, we propose an approach that is independent of alpha-theory. It is based on the weaker notion of well-isolated approximations to the roots. We apply it to univariate polynomials and provide experimental evidence of its feasibility.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"66 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":"126205026","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}
D. Kapur, Dong Lu, M. Monagan, Yao Sun, Dingkang Wang
A new efficient algorithm for computing a parametric greatest common divisor (GCD) of parametric multivariate polynomials over k[u][x] is presented. The algorithm is based on a well-known simple insight that the GCD of two multivariate polynomials (non-parametric as well as parametric) can be extracted using the generator of the quotient ideal of a polynomial with respect to the second polynomial. And, further, this generator can be obtained by computing a minimal Gröbner basis of the quotient ideal. The main attraction of this idea is that it generalizes to the parametric case for which a comprehensive Gröbner basis is constructed for the parametric quotient ideal. It is proved that in a minimal comprehensive Gröbner system of a parametric quotient ideal, each branch of specializations corresponds to a principal parametric ideal with a single generator. Using this generator, the parametric GCD of that branch is obtained by division. This algorithm does not need to consider whether parametric polynomials are primitive w.r.t. the main variable. This is in sharp contrast to two algorithms recently proposed by Nagasaka (ISSAC, 2017). The resulting algorithm is not only conceptually simple to understand but is considerably efficient. The proposed algorithm and both of Nagasaka's algorithms have been implemented in Singular (available at http://www.mmrc.iss.ac.cn/~dwang/software.html), and their performance is compared on a number of examples. For more than two polynomials, this process can be repeated by considering pairs of polynomials; the efficiency in that case becomes even more evident.
{"title":"An Efficient Algorithm for Computing Parametric Multivariate Polynomial GCD","authors":"D. Kapur, Dong Lu, M. Monagan, Yao Sun, Dingkang Wang","doi":"10.1145/3208976.3208980","DOIUrl":"https://doi.org/10.1145/3208976.3208980","url":null,"abstract":"A new efficient algorithm for computing a parametric greatest common divisor (GCD) of parametric multivariate polynomials over k[u][x] is presented. The algorithm is based on a well-known simple insight that the GCD of two multivariate polynomials (non-parametric as well as parametric) can be extracted using the generator of the quotient ideal of a polynomial with respect to the second polynomial. And, further, this generator can be obtained by computing a minimal Gröbner basis of the quotient ideal. The main attraction of this idea is that it generalizes to the parametric case for which a comprehensive Gröbner basis is constructed for the parametric quotient ideal. It is proved that in a minimal comprehensive Gröbner system of a parametric quotient ideal, each branch of specializations corresponds to a principal parametric ideal with a single generator. Using this generator, the parametric GCD of that branch is obtained by division. This algorithm does not need to consider whether parametric polynomials are primitive w.r.t. the main variable. This is in sharp contrast to two algorithms recently proposed by Nagasaka (ISSAC, 2017). The resulting algorithm is not only conceptually simple to understand but is considerably efficient. The proposed algorithm and both of Nagasaka's algorithms have been implemented in Singular (available at http://www.mmrc.iss.ac.cn/~dwang/software.html), and their performance is compared on a number of examples. For more than two polynomials, this process can be repeated by considering pairs of polynomials; the efficiency in that case becomes even more evident.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"39 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":"129879801","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 present some algorithms and programs for computing generalized spectral sequences, a useful tool in Computational Algebraic Topology which provides topological information on spaces with generalized filtrations over a poset. Our programs have been implemented as a new module for the Kenzo system and solve the classical problems of spectral sequences which are differential maps and extensions. Moreover, combined with the use of effective homology and discrete vector fields, the programs make it possible to compute generalized spectral sequences of big spaces, sometimes of infinite type.
{"title":"Effective Computation of Generalized Spectral Sequences","authors":"Andrea Guidolin, A. Romero","doi":"10.1145/3208976.3208984","DOIUrl":"https://doi.org/10.1145/3208976.3208984","url":null,"abstract":"In this paper, we present some algorithms and programs for computing generalized spectral sequences, a useful tool in Computational Algebraic Topology which provides topological information on spaces with generalized filtrations over a poset. Our programs have been implemented as a new module for the Kenzo system and solve the classical problems of spectral sequences which are differential maps and extensions. Moreover, combined with the use of effective homology and discrete vector fields, the programs make it possible to compute generalized spectral sequences of big spaces, sometimes of infinite type.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"PP 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":"126679083","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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