Abstract
The thesis consists of an introduction and the following four papers:
Paper I: Using resultants for SAGBI basis verification in the univariate polynomial ring.
Authors: Anna Torstensson, Victor Ufnarovski and Hans Öfverbeck.
Abstract: A resultanttype identity for univariate polynomials is proved and used to characterise SAGBI bases of subalgebras generated by two polynomials. A new equivalent condition, expressed in terms of the degree of a field extension, for a pair of univariate polynomials to form a SAGBI basis is derived.
Paper II: Automaton Presentations of Noncommutative Invariant Rings.
Author: Hans Öfverbeck.
Abstract: We introduce a new presentation of the noncommutative invariant ring of a finite permutation group. The core of the presentation is a finite state automaton which represents the leading words of the invariants. As an application we describe how the automaton presentation can be used to calculate the Hilbert series of the invariant ring. We also discuss how the automaton presentation can be used to find a free set of generators of the invariant ring.
Paper III: How to Calculate the Intersection of a Subalgebra and an Ideal.
Author: Hans Öfverbeck.
Abstract: A new characterisation of the intersection of an ideal and a subalgebra of a commutative polynomial ring is presented. This characterisation is used as the foundation for a pseudoalgorithm to calculate the intersection of a subalgebra and an ideal. The pseudoalgorithm uses SAGBIGröbner bases, and indirectly SAGBI bases. The article also contains a presentation of an implementation in Maple of the SAGBI and SAGBIGröbner basis construction algorithms, and a description of how this implementation can be used for calculating the intersection of an ideal and a subalgebra. A
comparison with a previously known method to calculate the
intersection of a subalgebra and an ideal is included.
Paper IV: A note on Computing SAGBIGröbner bases in a Polynomial Ring over a Field.
Author: Hans Öfverbeck.
Abstract: The purpose of this note is to present an observation, a sort of
SAGBIGr{"o}bner analogue of Buchberger's first criterion, which justifies substantial shrinking of the so called syzygy family of a pair of polynomials. Fewer elements in the syzygy family means that fewer syzygypolynomials need to be checked in the SAGBIGröbner basis construction/verification algorithm, thus decreasing the time needed for computation.
Paper I: Using resultants for SAGBI basis verification in the univariate polynomial ring.
Authors: Anna Torstensson, Victor Ufnarovski and Hans Öfverbeck.
Abstract: A resultanttype identity for univariate polynomials is proved and used to characterise SAGBI bases of subalgebras generated by two polynomials. A new equivalent condition, expressed in terms of the degree of a field extension, for a pair of univariate polynomials to form a SAGBI basis is derived.
Paper II: Automaton Presentations of Noncommutative Invariant Rings.
Author: Hans Öfverbeck.
Abstract: We introduce a new presentation of the noncommutative invariant ring of a finite permutation group. The core of the presentation is a finite state automaton which represents the leading words of the invariants. As an application we describe how the automaton presentation can be used to calculate the Hilbert series of the invariant ring. We also discuss how the automaton presentation can be used to find a free set of generators of the invariant ring.
Paper III: How to Calculate the Intersection of a Subalgebra and an Ideal.
Author: Hans Öfverbeck.
Abstract: A new characterisation of the intersection of an ideal and a subalgebra of a commutative polynomial ring is presented. This characterisation is used as the foundation for a pseudoalgorithm to calculate the intersection of a subalgebra and an ideal. The pseudoalgorithm uses SAGBIGröbner bases, and indirectly SAGBI bases. The article also contains a presentation of an implementation in Maple of the SAGBI and SAGBIGröbner basis construction algorithms, and a description of how this implementation can be used for calculating the intersection of an ideal and a subalgebra. A
comparison with a previously known method to calculate the
intersection of a subalgebra and an ideal is included.
Paper IV: A note on Computing SAGBIGröbner bases in a Polynomial Ring over a Field.
Author: Hans Öfverbeck.
Abstract: The purpose of this note is to present an observation, a sort of
SAGBIGr{"o}bner analogue of Buchberger's first criterion, which justifies substantial shrinking of the so called syzygy family of a pair of polynomials. Fewer elements in the syzygy family means that fewer syzygypolynomials need to be checked in the SAGBIGröbner basis construction/verification algorithm, thus decreasing the time needed for computation.
Original language  English 

Qualification  Doctor 
Awarding Institution 

Supervisors/Advisors 

Award date  2006 May 5 
Publisher  
ISBN (Print)  9162867709 
Publication status  Published  2006 
Bibliographical note
Defence detailsDate: 20060505
Time: 10:15
Place: Centre for Mathematical Sciences Sölvegatan 18, Lund room MH:C
External reviewer(s)
Name: Robbiano, Lorenzo
Title: Professor
Affiliation: Dipartimento di Matematica, Genova, Italy

Subject classification (UKÄ)
 Mathematics
Free keywords
 Matematik
 Mathematics
 elimination
 intersection
 automata
 resultants
 noncommutative invariants