Combinatorial Criteria for Grobner Bases

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Title: Combinatorial Criteria for Grobner Bases
Author: Perry, John Edward
Advisors: Erich Kaltofen, Committee Member
Agnes Szanto, Committee Member
Michael Singer, Committee Member
Hoon Hong, Committee Chair
Abstract: Both the computation and the detection of Gröbner bases require a criterion that decides whether a set of polynomials is a Gröbner basis. The most fundamental decision criterion is the reduction of all S-polynomials to zero. However, S-polynomial reduction is expensive in terms of time and storage, so a number of researchers have investigated the question of when we can avoid S-polynomial reduction. Certain results can be considered "combinatorial", because they are criteria on the leading terms, which are determined by integers. Our research builds on these results; this thesis presents combinatorial criteria for Gröbner bases. The first part of this thesis reviews the relevant literature on Gröbner bases and skipping S-polynomial reduction. The second part considers criteria for skipping a fixed number of S-polynomial reductions. The first two theorems of part two show howto apply Buchberger's criteria to obtain necessary and sufficient conditions for skipping all S-polynomial reductions, and for skipping all but one S-polynomial reductions. The third theorem considers the question of skipping all but two Spolynomial reductions; we have found that this problem requires new criteria on leading terms. We provide one new criterion that solves this problem for a set of three polynomials; for larger sets, the problem remains open. The final part of this thesis considers Gröbner basis detection. After a brief review of a previous result that requires S-polynomial reduction, we provide a new result which takes a completely different approach, avoiding S-polynomial reduction completely. Throughout the latter two parts, we provide some statistical analysis and experimental results.
Date: 2005-07-21
Degree: PhD
Discipline: Mathematics
URI: http://www.lib.ncsu.edu/resolver/1840.16/5581


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