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7 results on '"dynamical Gröbner basis"'

1. The syzygy theorem for Bezout rings.

Author

Gamanda, Maroua, Lombardi, Henri, Neuwirth, Stefan, and Yengui, Ihsen

Subjects
*GROBNER bases
Abstract

We provide constructive versions of Hilbert's syzygy theorem for Z and Z/NZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict Bézout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated. [ABSTRACT FROM AUTHOR]

Published
2020
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2. The syzygy theorem for Bézout rings

Author

Stefan Neuwirth, Ihsen Yengui, Henri Lombardi, Maroua Gamanda, Département de Mathematiques [Sfax], Faculté des Sciences de Sfax, Université de Sfax - University of Sfax-Université de Sfax - University of Sfax, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), John Templeton Foundation (ID 60842), ANR-15-IDEX-0003,BFC,ISITE ' BFC(2015), Fédération Bourgogne Franche-Comté Mathématiques (BFC-Math ), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), and ANR: 15-IDEX-0003,BFC,ISITE « BFC(2015)

Subjects
Pure mathematics, [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], Schreyer's syzygy algorithm, 010103 numerical & computational mathematics, 01 natural sciences, Constructive, Valuation ring, MSC 2010: 13D02, 13P10, 13C10, 13P20, 14Q20, Finitely-generated abelian group, 0101 mathematics, Syzygy theorem, Gröbner ring, Monomial order, strict Bézout ring, Mathematics, Algebra and Number Theory, Hilbert's syzygy theorem, Mathematics::Commutative Algebra, monomial order, valuation ring, Applied Mathematics, 010102 general mathematics, free resolution, Divisibility rule, Mathematics - Commutative Algebra, 16. Peace & justice, dynamical Gröbner basis, 13D02, 13P10, 13C10, 13P20, 14Q20, Schreyer's monomial order, Computational Mathematics
Abstract

International audience; We provide constructive versions of Hilbert's syzygy theorem for Z and Z/nZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict Bézout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.

Published
2019
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3. Corrigendum to “Dynamical Gröbner bases” [J. Algebra 301 (2) (2006) 447–458] and to “Dynamical Gröbner bases over Dedekind rings” [J. Algebra 324 (1) (2010) 12–24]

Author

Yengui, Ihsen

Subjects
*LINEAR algebra, *ERROR analysis in mathematics, *NOETHERIAN rings, *GROBNER bases, *MATHEMATICAL analysis
Abstract

Abstract: In this corrigendum, I correct a mild error in a technical lemma of Yengui (2006) . This error affected some results and examples in Hadj Kacem and Yengui (2010) and Yengui (2006) . [Copyright &y& Elsevier]

Published
2011
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4. Dynamical Gröbner bases over Dedekind rings

Author

Hadj Kacem, Amina and Yengui, Ihsen

Subjects
*ALGEBRAIC geometry, *IDEALS (Algebra), *DEDEKIND rings, *SET theory, *MODULES (Algebra), *MULTIVARIATE analysis, *CONSTRUCTIVE mathematics
Abstract

Abstract: In this paper, we extend the notion of “dynamical Gröbner bases” introduced by the second author to Dedekind rings (with zero divisors). As an application, we dynamically solve the ideal membership problem and compute a generating set for the syzygy module over multivariate polynomial rings with coefficients in Dedekind rings. We also give a partial positive answer to a conjecture about Gröbner rings. [Copyright &y& Elsevier]

Published
2010
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5. Dynamical Gröbner bases

Author

Yengui, Ihsen

Subjects
*POLYNOMIALS, *KRONECKER products, *GROBNER bases, *ALGORITHMS
Abstract

Abstract: In this paper, we introduce the notion of “dynamical Gröbner bases” of polynomial ideals over a principal ring. As application, we solve dynamically a fundamental algorithmic question in the theory of multivariate polynomials over the integers called “Kronecker''s problem,” that is the problem of finding a decision procedure for the ideal membership problem for . The notions of Gröbner bases over Noetherian valuation rings and dynamical Gröbner bases over principal rings have applications in error correcting codes. [Copyright &y& Elsevier]

Published
2006
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6. Dynamical Gröbner bases over Dedekind rings

Author

Amina Hadj Kacem and Ihsen Yengui

Subjects
Discrete mathematics, Pure mathematics, Conjecture, Hilbert's syzygy theorem, Algebra and Number Theory, Mathematics::Commutative Algebra, Ideal membership problem, Dedekind sum, symbols.namesake, Gröbner basis, Dedekind rings, Principal rings, symbols, Gröbner rings, Constructive mathematics, Computer Science::Symbolic Computation, Von Neumann regular ring, Dedekind cut, Dynamical Gröbner basis, Commutative algebra, Zero divisor, Mathematics
Abstract

In this paper, we extend the notion of “dynamical Grobner bases” introduced by the second author to Dedekind rings (with zero divisors). As an application, we dynamically solve the ideal membership problem and compute a generating set for the syzygy module over multivariate polynomial rings with coefficients in Dedekind rings. We also give a partial positive answer to a conjecture about Grobner rings.

Published
2010
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7. Dynamical Gröbner bases

Author

Ihsen Yengui

Subjects
Algebra and Number Theory, Mathematics::Commutative Algebra, Ideal membership problem, Euclidean division, MathematicsofComputing_NUMERICALANALYSIS, MathematicsofComputing_GENERAL, Univariate, Monomial ideal, Field (mathematics), Lexicographical order, Valuation ring, Algebra, Valuation rings, Noetherian rings, TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY, Principal rings, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, Dynamical Gröbner basis, Commutative algebra, Chinese remainder theorem, Mathematics
Abstract

In this paper, we introduce the notion of “dynamical Gröbner bases” of polynomial ideals over a principal ring. As application, we solve dynamically a fundamental algorithmic question in the theory of multivariate polynomials over the integers called “Kronecker's problem,” that is the problem of finding a decision procedure for the ideal membership problem for Z[X1,…,Xn]. The notions of Gröbner bases over Noetherian valuation rings and dynamical Gröbner bases over principal rings have applications in error correcting codes.

Published
2006
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