Graduation Year
2023
Document Type
Dissertation
Degree
Ph.D.
Degree Name
Doctor of Philosophy (Ph.D.)
Degree Granting Department
Mathematics and Statistics
Major Professor
Xiang-dong Hou, Ph.D.
Committee Member
Brian Curtin, Ph.D.
Committee Member
Giacomo Micheli, Ph.D.
Committee Member
Dmytro Savchuk, Ph.D.
Committee Member
Qiang Wang, Ph.D.
Keywords
Carlitz’s formula, absolutely irreducible, Hasse-Weil bound, permutation polynomial, power sum
Abstract
Rational functions over a finite field Fq induce mappings from the projective line P1(Fq) to itself. Rational functions that permute the projective line are called permutation rational functions (PRs). The notion of permutation rational functions is a natural extension of the permutation polynomials which have been studied for over a century. Recently, PRs of degrees up to four have been determined. This dissertation is a project aimed at determining PRs of degree five.
Rational functions of degree five (excluding those that are equivalent to polynomials) are divided into five cases according to the factorization of their denominators. Our main results can be summarized as follows:1. We showed that in Cases I and II, there are no PRs, whenever q is sufficiently large. 2. In Case III, we completely determined all PRs. When q is odd, there is an infinite family; when q is even, there are two infinite families. 3. In Case IV, we determined all PRs under an additional condition. There is an infinite family each for odd q and for even q.
Our approach is based on a combination of two methods. One uses the Carlitz power sum formula, which is a new technique that is particularly effective for PRs of low degrees. The other method relies on the Hasse-Weil bound on the number of zeros of an absolutely irreducible polynomial in two variables. This is a well-known technique that allows people to relate permutation properties of a rational function with the factorization of an associated polynomial.
Scholar Commons Citation
Sze, Christopher, "Rational Functions of Degree Five That Permute the Projective Line Over a Finite Field" (2023). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/9934
