Toward involutive bases over effective rings
type: Item
instance of: scholarly article
title: Toward involutive bases over effective rings
zbMATH Open document ID: 1474.13054
author: Michela Ceria
author: Ferdinando Mora
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 2021-05-14T00:00:00Z
review text: The origins of the Janet-Riquier theory of involutive divisions come from a paper of \textit{M. Janet} [C. R. Acad. Sci., Paris 170, 1101--1103 (1920; JFM 47.0440.03)] and gave rise to the notion of involutive basis. In that paper the author developed an algebraic theory that turned out to be useful for the study of linear systems of partial differential equations, also from a computational symbolic point of view. Then, the idea of involutive division have been further improved by several authors also in other frameworks. For example, recently, involutive divisions have been exploited in order to develop computational methods for the study of Hilbert schemes in a free-term order setting (see [\textit{M. Ceria} et al., J. Symb. Comput. 68, Part 2, 87--108 (2015; Zbl 1311.13036); \textit{M. Ceria} et al., J. Symb. Comput. 95, 100--133 (2019; Zbl 1430.13043)] and the reference therein). Today it is established that Janet's studies anticipated the ideas of Gröbner bases theory, in particular with the notions of weak and strong involutive bases (see [\textit{W. M. Seiler}, Appl. Algebra Eng. Commun. Comput. 20, No. 3--4, 207--259 (2009; Zbl 1175.13012); Appl. Algebra Eng. Commun. Comput. 20, No. 3--4, 261--338 (2009; Zbl 1175.13011)]). Indeed, the notion of involutive divisor gives rice to a decomposition of the set of terms of a monomial ideal, which can be detected by involutive bases and is a partition in the strong case. In this context, the authors of the paper under review focus their attention on the very interesting problem that consists in making all the theory to be applicable to monomials with coefficients over an effective ring. As it is clearly pointed out in the Introduction, with this aim in mind, four important goals can be considered: (1) to adapt the detection of involutive divisors over a field to monomials over an effective ring, if possible; (2) to investigate the case of strong involutive bases; (3) to reformulate the algorithm of W. Seiler for comp…
reviewed by: Francesca Cioffi
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 7348008
zbMATH Keywords: involutive divisions
zbMATH Keywords: weak involutive bases
zbMATH Keywords: local involutivity
describes a project that uses: Plural
MaRDI profile type: Publication
full work available at URL: s00200-020-00448-6
OpenAlex ID: W3049626666
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DOI: 10.1007/S00200-020-00448-6
description: scientific article; zbMATH DE number 7348008