Show simple item record

dc.contributor.authorQuadrat, A
dc.contributor.authorRobertz, Daniel
dc.date.accessioned2017-03-25T11:27:47Z
dc.date.available2017-03-25T11:27:47Z
dc.date.issued2014-10
dc.identifier.issn0167-8019
dc.identifier.issn1572-9036
dc.identifier.urihttp://hdl.handle.net/10026.1/8690
dc.description.abstract

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series. © 2014 Springer Science+Business Media.

dc.format.extent187-234
dc.languageen
dc.language.isoen
dc.publisherSpringer Science and Business Media LLC
dc.subjectWeyl algebras
dc.subjectStafford's theorems
dc.subjectLinear systems of partial differential equations
dc.subjectD-modules
dc.subjectMathematical systems theory
dc.subjectConstructive algebra
dc.subjectSymbolic computation
dc.titleA Constructive Study of the Module Structure of Rings of Partial Differential Operators
dc.typejournal-article
dc.typeJournal Article
plymouth.author-urlhttps://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000341707200009&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=11bb513d99f797142bcfeffcc58ea008
plymouth.issue1
plymouth.volume133
plymouth.publication-statusPublished
plymouth.journalActa Applicandae Mathematicae
dc.identifier.doi10.1007/s10440-013-9864-x
plymouth.organisational-group/Plymouth
plymouth.organisational-group/Plymouth/Faculty of Science and Engineering
plymouth.organisational-group/Plymouth/Faculty of Science and Engineering/School of Engineering, Computing and Mathematics
plymouth.organisational-group/Plymouth/Users by role
plymouth.organisational-group/Plymouth/Users by role/Academics
dc.identifier.eissn1572-9036
dc.rights.embargoperiodNot known
rioxxterms.versionofrecord10.1007/s10440-013-9864-x
rioxxterms.licenseref.urihttp://www.rioxx.net/licenses/all-rights-reserved
rioxxterms.typeJournal Article/Review


Files in this item

Thumbnail
Thumbnail

This item appears in the following Collection(s)

Show simple item record


All items in PEARL are protected by copyright law.
Author manuscripts deposited to comply with open access mandates are made available in accordance with publisher policies. Please cite only the published version using the details provided on the item record or document. In the absence of an open licence (e.g. Creative Commons), permissions for further reuse of content should be sought from the publisher or author.
Theme by 
Atmire NV