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Mathematics of Computation

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The syzygy theorem for Bézout rings


Authors: Maroua Gamanda, Henri Lombardi, Stefan Neuwirth and Ihsen Yengui
Journal: Math. Comp. 89 (2020), 941-964
MSC (2010): Primary 13D02; Secondary 13P10, 13C10, 13P20, 14Q20
DOI: https://doi.org/10.1090/mcom/3466
Published electronically: September 19, 2019
MathSciNet review: 4044457
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Abstract: We provide constructive versions of Hilbert's syzygy theorem for $ \mathbb{Z}$ and $ \mathbb{Z}/N\mathbb{Z}$ 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.


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Additional Information

Maroua Gamanda
Affiliation: Département de mathématiques, Faculté des sciences de Sfax, Université de Sfax, 3000 Sfax, Tunisia
Email: marwa.gmenda@hotmail.com

Henri Lombardi
Affiliation: Laboratoire de mathématiques de Besançon, Université Bourgogne Franche-Comté, 25030 Besançon Cedex, France
Email: henri.lombardi@univ-fcomte.fr

Stefan Neuwirth
Affiliation: Laboratoire de mathématiques de Besançon, Université Bourgogne Franche-Comté, 25030 Besançon Cedex, France
Email: stefan.neuwirth@univ-fcomte.fr

Ihsen Yengui
Affiliation: Département de mathématiques, Faculté des sciences de Sfax, Université de Sfax, 3000 Sfax, Tunisia
Email: ihsen.yengui@fss.rnu.tn

DOI: https://doi.org/10.1090/mcom/3466
Keywords: Syzygy theorem, free resolution, monomial order, Schreyer's monomial order, Schreyer's syzygy algorithm, dynamical Gr\"obner basis, valuation ring, Gr\"obner ring, strict B\'ezout ring
Received by editor(s): August 17, 2017
Received by editor(s) in revised form: February 10, 2018, and May 11, 2019
Published electronically: September 19, 2019
Additional Notes: The third author was supported in part by the French “Investissements d’avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03).
The fourth author was supported in part by the John Templeton Foundation (ID 60842).
Article copyright: © Copyright 2019 the authors