Toward involutive bases over effective rings
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Toward involutive bases over effective rings Michela Ceria1 · Teo Mora2 Received: 16 November 2019 / Revised: 1 May 2020 / Accepted: 13 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper we extend the theory of involutive divisions to the case of monomials with coefficients over effective rings. Moreover, as regards involutive bases, we study the computation of weak involutive bases and sketch a conjecture on strong involutive bases. Keywords Involutive divisions · Weak involutive bases · Local involutivity Mathematics Subject Classification 13P10
1 Introduction As remarked by Schwartz in [46], Janet [24] anticipated a preliminary version of Buchberger Theory for polynomial rings over a field, including even Buchberger Second Criterion (for a survey on this topic see [31, Vol. 4, 57.1.1]). From 1996, the Dubna group [18–22, 52, 53] studied under the name of involutive bases Janet’s approach as an alternative to Buchberger’s for efficiently producing Gröbner bases. Seiler’s [47] studies on the argument pointed to different problems and potential investigations, in particular toward solvable polynomial rings and the distinction between strong and weak involutive bases.
This research has been partially funded by GNSAGA—Istituto Nazionale di Alta Matematica “Francesco Severi”. The first author is thankful to this institution for its support. * Michela Ceria [email protected] Teo Mora [email protected] 1
Department of Computer Science, University of Milan, Via Celoria 18, 20133 Milano, Italy
2
Department of Mathematics, University of Genoa, Via Dodecaneso 35, Genova, Italy
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M. Ceria, T. Mora
Recently, Mora [31, IV.50] [30] extended Buchberger Theory on each ring effectively given through its Zacharias representation [32, 51], reformulating Buchberger Algorithm on the basis of Möller Lifting Theorem [34]. The reformulation by Ceria [10] of Weispfenning completion in terms of Weispfenning Multiplication [50] gave an algorithm for bilateral Gröbner bases [13], which is more efficient than Mora’s proposal. A recent paper [35] which was reformulating Weispfenning completion within Mora’s theory toward a wider class of Noetherian rings than solvable polynomial rings, suggested us to reinvestigate weak/strong involutive bases over effective rings. What is clear is that a. involutive bases, requiring a Noetherian associated graded ring with d d T ∶= {X1 1 ⋯ Xn n | (d1 , … , dn ) ∈ ℕn } as R-basis, could be used only in a class of multivariate Ore extensions which could, at most, slightly enlarge the setting proposed in [35]; b. Janet and Riquier gave an introduction to what will be called Buchberger reduction and a criterion that it is at least equivalente to Buchberger’s. Moreover, they give Janet bases, which can be seen (for the DegLex ordering) a pre-version of Gröbner bases, together with Buchberger algorithm. The restriction to the DegLex ordering has been removed by Buchberger, as well as the redundancy on leading terms that is proper of Jan
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