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
Copyright
Candice Mitchell, 2020
Recommended Citation
Mitchell, Candice, "A Dynamic F5 Algorithm" (2020). Dissertations. 1748.
https://aquila.usm.edu/dissertations/1748