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On Grobner bases and Krull dimension of residue class rings of polynomial rings over integral domains

Francis, Maria and Dukkipati, Ambedkar (2018) On Grobner bases and Krull dimension of residue class rings of polynomial rings over integral domains. In: JOURNAL OF SYMBOLIC COMPUTATION, 86 . pp. 1-19.

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Official URL: http://dx.doi.org/10.1016/j.jsc.2017.03.003

Abstract

Given an ideal a in Ax(1),.... x(n)] where A is a Noetherian integral domain, we propose an approach to compute the Krull dimension of Axi,,.x(n)]/a, when the residue class ring is a free A-module. When A is a field, the Krull dimension of Ax(1),.... x(n)]/a has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetherian rings. For a Noetherian integral domain A we introduce the notion of combinatorial dimension of Ax(1),.... x(n)]/a and give a Grobner basis method to compute it for residue class rings that have a free A-module representation w.r.t. a lexicographic ordering. For such A-algebras, we derive a relation between Krull dimension and combinatorial dimension of Ax(1),.... x(n)]/a. An immediate application of this relation is that it gives a uniform method, the first of its kind, to compute the dimension of Ax(1),.... x(n)]/a without having to consider individual properties of the ideal. For A-algebras that have a free A-module representation w.r.t degree compatible monomial orderings, we introduce the concepts of Hilbert function, Hilbert series and Hilbert polynomials and show that Grobner basis methods can be used to compute these quantities. We then proceed to show that the combinatorial dimension of such A-algebras is equal to the degree of the Hilbert polynomial. This enables us to extend the relation between Krull dimension and combinatorial dimension to A-algebras with a free A-module representation w.r.t. a degree compatible ordering as well. (C) 2017 Elsevier Ltd. All rights reserved.

Item Type: Journal Article
Additional Information: Copy right for this article belongs to the ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD, 24-28 OVAL RD, LONDON NW1 7DX, ENGLAND
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Depositing User: Id for Latest eprints
Date Deposited: 20 Jan 2018 07:04
Last Modified: 20 Jan 2018 07:04
URI: http://eprints.iisc.ac.in/id/eprint/58795

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