#### Title

#### 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