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An algorithm to determine the Hilbert series for graded associative algebras
Stockholm University, Faculty of Science, Department of Mathematics.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we present an algorithm to compute the Hilbert series for quotients of the free algebra with homogeneous two-sided ideals. We also give a modified version of the algorithm for quotients of the polynomial ring. The algorithms are implemented in a computer program named ``aalg'' and we compare running times with other computer algebra programs.

National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-31518OAI: oai:DiVA.org:su-31518DiVA: diva2:277496
Available from: 2009-11-19 Created: 2009-11-19 Last updated: 2011-03-14Bibliographically 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

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