Canonical Bases for Algebraic Computations
Author
Summary, in English
In the first paper we investigate, for quotients of the non-commutative polynomial ring, a property that implies finiteness of Gröbner bases computation, and examine its connection with Noetherianity. We propose a Gröbner bases theory for factor algebras, of particular interest for one-sided ideals, and show a few applications, e.g. how to compute (one-sided) syzygy modules. The material of the third paper is in some sense related to the contents of this first paper; in the third paper, the theory of SAGBI bases is extended to factor algebras.
The second and fourth paper concerns composition of polynomials. In the first of those two papers, we give sufficient and necessary conditions on a set of polynomials to guarantee that the property of being a non-commutative Gröbner basis is preserved after composition by this set. The latter paper treats the same problem for SAGBI bases.
In the fifth paper we introduce the concept of bi-automaton algebras, generalizing the automaton algebras previously defined by Ufnarovski. A bi-automaton algebra is a quotient of the free algebra, defined by a binomial ideal admitting a Gröbner basis which can be encoded as a regular set; we call such a Gröbner basis regular. We give several examples of bi-automaton algebras, and show how automata connected to regular Gröbner bases can be used to perform reduction.
In the last paper we investigate various important properties of regular languages associated with quotients of the free associative algebra. We suggest a generalization of a graph for normal words introduced by Ufnarovski, applicable to testing Noetherian properties of automaton algebras. Finally we show an alternative way to compute the generators for the Jacobson radical of any automaton monomial algebra.
Department/s
Publishing year
2001
Language
English
Publication/Series
Doctoral Theses in Mathematical Sciences
Document type
Dissertation
Publisher
Centre for Mathematical Sciences, Lund University
Topic
- Mathematics
Keywords
- gruppteori
- algebra
- algebraisk geometri
- fältteori
- Talteori
- group theory
- algebraic geometry
- field theory
- Matematik
- Number Theory
- regular languages
- Mathematics
- composition of polynomials
- factor algebras
- Gröbner bases
- SAGBI bases
Status
Published
Research group
- Algebra
Supervisor
- [unknown] [unknown]
ISBN/ISSN/Other
- ISSN: 1404-0034
- ISBN: 91-628-4969-7
- LUTFMA-1012-2001
Defence date
5 October 2001
Defence time
10:15
Defence place
Matematikcentrum, sal C
Opponent
- Joachim Apel (Prof)