Perdry, H and Schuster, PM (2014) Constructing Gröbner bases for Noetherian rings. Mathematical Structures in Computer Science, 24 (2). e240206. ISSN 0960-1295
Abstract
We give a constructive proof showing that every finitely generated polynomial ideal has a Gröbner basis, provided the ring of coefficients is Noetherian in the sense of Richman and Seidenberg. That is, we give a constructive termination proof for a variant of the well-known algorithm for computing the Gröbner basis. In combination with a purely order-theoretic result we have proved in a separate paper, this yields a unified constructive proof of the Hilbert basis theorem for all Noether classes: if a ring belongs to a Noether class, then so does the polynomial ring. Our proof can be seen as a constructive reworking of one of the classical proofs, in the spirit of the partial realisation of Hilbert's programme in algebra put forward by Coquand and Lombardi. The rings under consideration need not be commutative, but are assumed to be coherent and strongly discrete: that is, they admit a membership test for every finitely generated ideal. As a complement to the proof, we provide a prime decomposition for commutative rings possessing the finite-depth property.
Metadata
Item Type: | Article |
---|---|
Authors/Creators: |
|
Copyright, Publisher and Additional Information: | © 2013, Cambridge University Press. This is an author produced version of a paper accepted for publication in Mathematical Structures in Computer Science. Uploaded with permission from the publisher. |
Dates: |
|
Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 21 Oct 2013 11:04 |
Last Modified: | 23 Apr 2017 05:15 |
Published Version: | http://dx.doi.org/10.1017/S0960129513000509 |
Status: | Published |
Publisher: | Cambridge University Press |
Identification Number: | 10.1017/S0960129513000509 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:76642 |