Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Vector space bases associated to vanishing ideals of points
Stockholm University, Faculty of Science, Department of Mathematics.
2010 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 214, no 4, 309-321 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give new complexity bounds. As an application, we drastically improve the computational algebra approach to the reverse engineering of gene regulatory networks.

Place, publisher, year, edition, pages
Elsevier , 2010. Vol. 214, no 4, 309-321 p.
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-31521DOI: 10.1016/j.jpaa.2009.05.013ISI: 000273248700002OAI: oai:DiVA.org:su-31521DiVA: diva2:277499
Available from: 2009-11-19 Created: 2009-11-19 Last updated: 2017-12-12Bibliographically approved
In thesis
1. Computational algorithms for algebras
Open this publication in new window or tab >>Computational algorithms for algebras
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of six papers.

In Paper I, we give an algorithm for merging sorted lists of monomials and together with a projection technique, we obtain a new complexity bound for the Buchberger-Möller algorithm and the FGLM algorithm.

In Paper II, we discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give complexity bounds. As an application we drastically improve the computational algebra approach to the reverse engineering of gene regulatory networks.

In Paper III, we introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and, in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero.

In Paper IV, we consider a subset of projective space over a finite field and give a geometric description of the minimal degree of a non-vanishing form with respect to this subset. We also give bounds on the minimal degree in terms of the cardinality of the subset.

In Paper V, we study an associative version of an algorithm constructed to compute the Hilbert series for graded Lie algebras. In the commutative case we use Gotzmann's persistence theorem to show that the algorithm terminates in finite time.

In Paper VI, we connect the commutative version of the algorithm in Paper V with the Buchberger algorithm.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2009. 9 p.
Keyword
Hilbert function, Gröbner basis, zero-dimensional ideal, affine variety, projective variety, run-time complexity
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-31552 (URN)978-91-7155-974-6 (ISBN)
Public defence
2009-12-18, Sal 14, hus 5, Kräftriket, Stockholm, 10:00 (English)
Opponent
Supervisors
Note
At the time of doctoral defence, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript. Paper 5: Manuscript. Paper 6: ManuscriptAvailable from: 2009-11-26 Created: 2009-11-19 Last updated: 2009-11-20Bibliographically approved

Open Access in DiVA

No full text

Other links

Publisher's full text

Search in DiVA

By author/editor
Lundqvist, Samuel
By organisation
Department of Mathematics
In the same journal
Journal of Pure and Applied Algebra
Mathematics

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 83 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf