Date of Award

Spring 2020

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

School

Mathematics and Natural Sciences

Committee Chair

John Perry

Committee Chair School

Mathematics and Natural Sciences

Committee Member 2

Bernd Schroeder

Committee Member 2 School

Mathematics and Natural Sciences

Committee Member 3

Karen Kohl

Committee Member 3 School

Mathematics and Natural Sciences

Committee Member 4

Jiu Ding

Committee Member 4 School

Mathematics and Natural Sciences

Committee Member 5

Rajeev Agrawal

Abstract

Gröbner bases are a “nice” representation for nonlinear systems of polynomials, where by “nice” we mean they have good computation properties. They have many useful applications, including decidability (whether the system has a solution or not), ideal membership (whether a given polynomial is in the system or not), and cryptography. Traditional Gröbner basis algorithms require as input an ideal and an admissible term ordering. They then determine a Gröbner basis with respect to the given ordering. Some term orderings lead to a smaller basis, but finding them traditionally requires testing many orderings and hoping for better results. A dynamic algorithm requires as input only the ideal and allows the term ordering to vary at each step of the algorithm. Previous work has shown that this often produces a smaller basis and/or finds a basis in a shorter time frame. Since some Gröbner bases under certain term orderings are extremely large, it is advantageous to find ways to compute smaller bases. The F5 algorithm is a traditional algorithm that computes a Gröbner basis in a way that attempts to avoid computing S-polynomials that reduce to zero. Since S-polynomials that reduce to zero do not add any useful information, avoiding these computations can drastically reduce the amount of work done. This work describes a dynamic F5 algorithm to compute Gröbner bases. The algorithm combines the advantages of a traditional F5 algorithm by avoiding the majority of S-polynomials that reduce to zero as well as the decrease in size that can be gained using a dynamic algorithm.

ORCID ID

https://orcid.org/0000-0001-8716-9383

Available for download on Thursday, May 14, 2020

Included in

Algebra Commons

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