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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Gröbner bases and gradings for partial difference ideals
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by Roberto La Scala PDF
Math. Comp. 84 (2015), 959-985 Request permission

Abstract:

In this paper we introduce a working generalization of the theory of Gröbner bases for algebras of partial difference polynomials with constant coefficients. One obtains symbolic (formal) computation for systems of linear or non-linear partial difference equations arising, for instance, as discrete models or by the discretization of systems of differential equations. From an algebraic viewpoint, the algebras of partial difference polynomials are free objects in the category of commutative algebras endowed with the action by endomorphisms of a monoid isomorphic to $\mathbb {N}^r$. Then, the investigation of Gröbner bases in this context contributes also to the current research trend consisting in studying polynomial rings under the action of suitable symmetries that are compatible with effective methods. Since the algebras of difference polynomials are not Noetherian, we propose in this paper a theory for grading them that provides a Noetherian subalgebra filtration. This implies that the variants of Buchberger’s algorithm we developed for difference ideals terminate in the finitely generated graded case when truncated up to some degree. Moreover, even in the non-graded case, we provide criterions for certifying completeness of eventually finite Gröbner bases when they are computed within sufficiently large bounded degrees. We generalize also the concepts of homogenization and saturation, and related algorithms, to the context of difference ideals. The feasibility of the proposed methods is shown by an implementation in Maple that is the first to provide computations for systems of non-linear partial difference equations. We make use of a test set based on the discretization of concrete systems of non-linear partial differential equations.
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Additional Information
  • Roberto La Scala
  • Affiliation: Dipartimento di Matematica, via Orabona 4, 70125 Bari, Italia
  • Email: roberto.lascala@uniba.it
  • Received by editor(s): December 14, 2011
  • Received by editor(s) in revised form: November 6, 2012, May 30, 2013, and July 9, 2013
  • Published electronically: July 31, 2014
  • Additional Notes: The author was partially supported by Università di Bari
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 84 (2015), 959-985
  • MSC (2010): Primary 12H10; Secondary 13P10, 16W22, 16W50
  • DOI: https://doi.org/10.1090/S0025-5718-2014-02859-7
  • MathSciNet review: 3290971