Constructive Methods for SAGBI and SAGBI-Gröbner Bases

University dissertation from Centre for Mathematical Sciences Faculty of Science Mathematics Box 118 SE-221 00 Lund Sweden

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 resultant-type 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 pseudo-algorithm to calculate the intersection of a subalgebra and an ideal. The pseudo-algorithm uses SAGBI-Gröbner bases, and indirectly SAGBI bases. The article also contains a presentation of an implementation in Maple of the SAGBI and SAGBI-Gröbner 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 SAGBI-Gröbner 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 SAGBI-Gr{"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 syzygy-polynomials need to be checked in the SAGBI-Gröbner basis construction/verification algorithm, thus decreasing the time needed for computation.

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