Certain Diagonal Equations over Finite Fields

Author

Christopher Sze, University of South Florida

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_q^t be the finite field with q^t elements and let F*_q^t be its multiplicative group. We study the diagonal equation ax^q−1 + by^q−1 = c, where a,b and c ∈ F*_q^t. 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*_q^t × F*_q^t 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 N₃(α, β) can be expressed in terms of the number of monic irreducible cubic polynomials over F_q with prescribed trace and norm. Consequently, N₃(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 N₂(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