## Graduation Year

2009

## Document Type

Thesis

## Degree

M.A.

## Degree Granting Department

Mathematics and Statistics

## Major Professor

Xiang-Dong Hou, Ph.D.

## Committee Member

Brian Curtin, Ph.D.

## Committee Member

Stephen Suen, Ph.D.

## Keywords

Irreducible polynomial, Gaussian sum, Planar function, Hasse-Weil bound, Elliptic curve

## Abstract

Let F_{qt} be the finite field with q^{t} elements and let F*_{qt} be its multiplicative group. We study the diagonal equation ax^{q−1} + by^{q−1} = c, where a,b and c ∈ F*_{qt}. This equation can be written as x^{q−1}+αy^{q−1} = β, where α, β ∈ F ∗ q t . Let Nt(α, β) denote the number of solutions (x,y) ∈ F*_{qt} × F*_{qt} of x^{q−1} + αy^{q−1} = β and I(r; a, b) be the number of monic irreducible polynomials f ∈ Fq[x] of degree r with f(0) = a and f(1) = b. We show that N_{t}(α, β) can be expressed in terms of I(r; a, b), where r | t and a, b ∈ F*_{q} are related to α and β. A recursive formula for I(r; a, b) will be given and we illustrate this by computing I(r; a, b) for 2 ≤ r ≤ 4. We also show that N_{3}(α, β) can be expressed in terms of the number of monic irreducible cubic polynomials over F_{q} with prescribed trace and norm. Consequently, N_{3}(α, β) can be expressed in terms of the number of rational points on a certain elliptic curve. We give a proof that given any a, b ∈ F*_{q} and integer r ≥ 3, there always exists a monic irreducible polynomial f ∈ F_{q}[x] of degree r such that f(0) = a and f(1) = b. We also use the result on N_{2}(α, β) to construct a new family of planar functions.

## Scholar Commons Citation

Sze, Christopher, "Certain Diagonal Equations over Finite Fields" (2009). *USF Tampa Graduate Theses and Dissertations.*

https://digitalcommons.usf.edu/etd/39