We propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence the Hilbert series have a rational sum. Efficient variants of the methods are also developed for the truncations of infinite-dimensional algebras which provide approximations of possibly irrational Hilbert series. Moreover, we provide a characterization of the finite-dimensional algebras in terms of the nilpotency of a key matrix involved in the computations. Finally, we present a well-tested and complete implementation for the computation of graded and multigraded Hilbert series which has been developed in the kernel of the computer algebra system Singular (for the details, see preprint[1]).
{"title":"Computing noncommutative Hilbert series","authors":"R. L. Scala, S. K. Tiwari","doi":"10.1145/3338637.3338645","DOIUrl":"https://doi.org/10.1145/3338637.3338645","url":null,"abstract":"We propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence the Hilbert series have a rational sum. Efficient variants of the methods are also developed for the truncations of infinite-dimensional algebras which provide approximations of possibly irrational Hilbert series. Moreover, we provide a characterization of the finite-dimensional algebras in terms of the nilpotency of a key matrix involved in the computations. Finally, we present a well-tested and complete implementation for the computation of graded and multigraded Hilbert series which has been developed in the kernel of the computer algebra system Singular (for the details, see preprint[1]).","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"12 1","pages":"136-138"},"PeriodicalIF":0.0,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79709956","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 structures of lexicographic (LEX) Gröbner bases were studied first by Lazard [4] for bivariate ideals and then extended to general zero-dimensional multivariate (radical) ideals [3, 6, 2]. Based on the structures of LEX Gröbner bases, algorithms have been proposed to compute triangular decompositions out of LEX Gröbner bases for zero-dimensional ideals [5, 2]. The relationships between LEX Gröbner bases and Ritt characteristic sets were explored in [1] and then made clearer in [8] with the concept of W-characteristic sets.
{"title":"On W-characteristic sets of lexicographic Gröbner bases","authors":"Chenqi Mou, Dongming Wang","doi":"10.1145/3338637.3338647","DOIUrl":"https://doi.org/10.1145/3338637.3338647","url":null,"abstract":"The structures of lexicographic (LEX) Gröbner bases were studied first by Lazard [4] for bivariate ideals and then extended to general zero-dimensional multivariate (radical) ideals [3, 6, 2]. Based on the structures of LEX Gröbner bases, algorithms have been proposed to compute triangular decompositions out of LEX Gröbner bases for zero-dimensional ideals [5, 2]. The relationships between LEX Gröbner bases and Ritt characteristic sets were explored in [1] and then made clearer in [8] with the concept of W-characteristic sets.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"41 1","pages":"142-144"},"PeriodicalIF":0.0,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81600498","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}
Seung Gyu Hyun, Vincent Neiger, Hamid Rahkooy, É. Schost
Overview. Computing the Gröbner basis of an ideal with respect to a term ordering is an essential step in solving systems of polynomials; in what follows, we restrict our attention to systems with finitely many solutions. Certain term orderings, such as the degree reverse lexicographical ordering (degrevlex), make the computation of the Gröbner basis faster, while other orderings, such as the lexicographical ordering (lex), make it easier to find the coordinates of the solutions. Thus, one typically first computes a Gröbner basis for the degrevlex ordering, and then converts it to either a lex Gröbner basis or a related representation, such as Rouillier's Rational Univariate Representation [8].
{"title":"Sparse FGLM using the block Wiedemann algorithm","authors":"Seung Gyu Hyun, Vincent Neiger, Hamid Rahkooy, É. Schost","doi":"10.1145/3338637.3338641","DOIUrl":"https://doi.org/10.1145/3338637.3338641","url":null,"abstract":"Overview. Computing the Gröbner basis of an ideal with respect to a term ordering is an essential step in solving systems of polynomials; in what follows, we restrict our attention to systems with finitely many solutions. Certain term orderings, such as the degree reverse lexicographical ordering (degrevlex), make the computation of the Gröbner basis faster, while other orderings, such as the lexicographical ordering (lex), make it easier to find the coordinates of the solutions. Thus, one typically first computes a Gröbner basis for the degrevlex ordering, and then converts it to either a lex Gröbner basis or a related representation, such as Rouillier's Rational Univariate Representation [8].","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"48 1","pages":"123-125"},"PeriodicalIF":0.0,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90509221","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 symbolic-numeric sparse interpolation of multivariate polynomials. The problem is to find the coefficients and the exponents of a given black-box polynomial [EQUATION] by evaluating the value of f(x1,..., xn) at any point in Cn in floating-point arithmetic and by using the conditions of the input.
{"title":"Robust algorithms for sparse interpolation of multivariate polynomials","authors":"Dai Numahata, Hiroshi Sekigawa","doi":"10.1145/3338637.3338648","DOIUrl":"https://doi.org/10.1145/3338637.3338648","url":null,"abstract":"We consider the problem of symbolic-numeric sparse interpolation of multivariate polynomials. The problem is to find the coefficients and the exponents of a given black-box polynomial [EQUATION] by evaluating the value of <i>f</i>(<i>x</i><sub>1</sub>,..., <i>x<sub>n</sub></i>) at any point in C<sup><i>n</i></sup> in floating-point arithmetic and by using the conditions of the input.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"169 1","pages":"145-147"},"PeriodicalIF":0.0,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86897064","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 method of computing residue integrals with integration over certain cycles for systems of transcendental equations is presented. Such integrals are connected to the power sums of roots for a certain system of equations. The described approach can be used for developing methods for the elimination of unknowns from transcendental systems.
{"title":"On computing residue integrals for a class of nonlinear transcendental systems","authors":"A. Kytmanov, A. Kytmanov, E. K. Myshkina","doi":"10.1145/3338637.3338644","DOIUrl":"https://doi.org/10.1145/3338637.3338644","url":null,"abstract":"A method of computing residue integrals with integration over certain cycles for systems of transcendental equations is presented. Such integrals are connected to the power sums of roots for a certain system of equations. The described approach can be used for developing methods for the elimination of unknowns from transcendental systems.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"5 1","pages":"133-135"},"PeriodicalIF":0.0,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90922843","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 integer points of polyhedral sets are of interest in many areas of mathematical sciences, see for instance the landmark textbooks of A. Schrijver [18] and A. Barvinok [3], as well as the compilation of articles [4]. One of these areas is the analysis and transformation of computer programs. For instance, integer programming [6] is used by P. Feautrier in the scheduling of for-loop nests [7], Barvinok's algorithm [2] for counting integer points in polyhedra is adapted by M. Köppe and S. Verdoolaege in [15] to answer questions like how many memory locations are touched by a for-loop nest. In [16], W. Pugh proposes an algorithm, called the Omega Test, for testing whether a polyhedron has integer points. In the same paper, W. Pugh shows how to use the Omega Test for performing dependence analysis [16] in for-loop nests. In [17], W. Pugh also suggests, without stating a formal algorithm, that the Omega Test could be used for quantifier elimination on Presburger formulas. This observation is a first motivation for the work presented here.
{"title":"Computing the integer points of a polyhedron","authors":"Rui-Juan Jing, M. M. Maza","doi":"10.1145/3338637.3338642","DOIUrl":"https://doi.org/10.1145/3338637.3338642","url":null,"abstract":"The integer points of polyhedral sets are of interest in many areas of mathematical sciences, see for instance the landmark textbooks of A. Schrijver [18] and A. Barvinok [3], as well as the compilation of articles [4]. One of these areas is the analysis and transformation of computer programs. For instance, integer programming [6] is used by P. Feautrier in the scheduling of for-loop nests [7], Barvinok's algorithm [2] for counting integer points in polyhedra is adapted by M. Köppe and S. Verdoolaege in [15] to answer questions like how many memory locations are touched by a for-loop nest. In [16], W. Pugh proposes an algorithm, called the Omega Test, for testing whether a polyhedron has integer points. In the same paper, W. Pugh shows how to use the Omega Test for performing dependence analysis [16] in for-loop nests. In [17], W. Pugh also suggests, without stating a formal algorithm, that the Omega Test could be used for quantifier elimination on Presburger formulas. This observation is a first motivation for the work presented here.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"80 1","pages":"126-129"},"PeriodicalIF":0.0,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75944523","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 [2], we proposed the EZ-GCD algorithm based on extended Hensel construction, in order to compute the GCD of multivariate polynomials. The extended Hensel construction is a specify factorization into algebraic function, and it is efficient for sparse multivariate polynomials. However, it is slower than Maple's GCD routine. In this paper, we improve our EZ-GCD algorithm efficiency.
{"title":"Improvement of EZ-GCD algorithm based on extended hensel construction","authors":"Masaru Sanuki","doi":"10.1145/3338637.3338649","DOIUrl":"https://doi.org/10.1145/3338637.3338649","url":null,"abstract":"In [2], we proposed the EZ-GCD algorithm based on extended Hensel construction, in order to compute the GCD of multivariate polynomials. The extended Hensel construction is a specify factorization into algebraic function, and it is efficient for sparse multivariate polynomials. However, it is slower than Maple's GCD routine. In this paper, we improve our EZ-GCD algorithm efficiency.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"35 1","pages":"148-150"},"PeriodicalIF":0.0,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72797471","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}
Each quarter we are pleased to present abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates who have defended in the past two years (and their supervisors), to submit their abstracts for publication in CCA. Please send abstracts to the CCA editors for consideration.
{"title":"Abstracts of recent doctoral dissertations in computer algebra","authors":"Jiaxiong Hu, A. Panferov","doi":"10.1145/3338637.3338651","DOIUrl":"https://doi.org/10.1145/3338637.3338651","url":null,"abstract":"Each quarter we are pleased to present abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates who have defended in the past two years (and their supervisors), to submit their abstracts for publication in CCA.\u0000 Please send abstracts to the CCA editors for consideration.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"1 1","pages":"154-155"},"PeriodicalIF":0.0,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77939793","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 being developed to compute geometric Galois groups of polynomials over Q(t) is discussed. This algorithm also computes a field extension such that the factorization of a polynomial over this field extension is the absolute factorization of the polynomial.
{"title":"Computation of geometric Galois groups and absolute factorizations","authors":"Nicole Sutherland","doi":"10.1145/3338637.3338650","DOIUrl":"https://doi.org/10.1145/3338637.3338650","url":null,"abstract":"An algorithm being developed to compute geometric Galois groups of polynomials over Q(t) is discussed. This algorithm also computes a field extension such that the factorization of a polynomial over this field extension is the absolute factorization of the polynomial.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"91 1","pages":"151-153"},"PeriodicalIF":0.0,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86707989","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 Couveignes-Rostovtsev-Stolbunov key-exchange protocol based on isogenies of elliptic curves is of interest because it may resist quantum attacks, but its efficient implementation remains a challenge. We briefly present the computations involved, and efficient algorithms to achieve the critical steps, with timing results for our implementations in Sage and Julia/Nemo.
{"title":"Isogeny-based cryptography in Julia/Nemo: a case study","authors":"J. Kieffer, L. Feo","doi":"10.1145/3338637.3338643","DOIUrl":"https://doi.org/10.1145/3338637.3338643","url":null,"abstract":"The Couveignes-Rostovtsev-Stolbunov key-exchange protocol based on isogenies of elliptic curves is of interest because it may resist quantum attacks, but its efficient implementation remains a challenge. We briefly present the computations involved, and efficient algorithms to achieve the critical steps, with timing results for our implementations in Sage and Julia/Nemo.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"30 1","pages":"130-132"},"PeriodicalIF":0.0,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80765723","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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