Diophantine equations over finite fields and applications

Abstract

The first part of this thesis focuses on counting the number of solutions to Diophantine equations over finite fields for families that are closely related to diagonal equations, presenting significant results in this area. The second part of the thesis is centered on counting low-weight codewords in cyclic codes, presenting some novel results. The introductory chapter lays the groundwork by introducing fundamental concepts that serve as the basis for the subsequent chapters. Chapter 2 studies fully triangular polynomials, characterized by expressions of the form \begin{equation*} f(x_1, \dots, x_n) = a_1 x_1^{d_{1,1}} + a_2 x_1^{d_{1,2}} x_2^{d_{2,2}} + \dots + a_n x_1^{d_{1,n}}\cdots x_n^{d_{n,n}} - b, \end{equation*} where $a_i \in \F_q^*$ and $b \in \F_q$, with $d_{i,j} > 0$ for all $1 \leq i \leq j \leq n$. For these polynomials, explicit formulas for counting the number of solutions are derived under arithmetic conditions that relate them to diagonal polynomials of degrees 1 and 2. Chapter 3 shifts the focus to full polynomials, defined as \begin{equation*} f(x_1, \dots, x_n) = a_1 x_1^{d_{1,1}} \cdots x_n^{d_{1,n}} + \dots + a_s x_1^{d_{s,1}}\cdots x_n^{d_{s,n}} - b, \end{equation*} where $a_i \in \F_q^*$ and $b \in \F_q$, with $d_{i,j} > 0$ for $1 < i, j < n$. This chapter investigates the solution count for these polynomials when they are related to diagonal polynomials of the form $$g(x_1, \dots, x_s) = a_1 x_1^d + \cdots + a_s x_s^d - b,$$ where $n \ge s$, and the exponents and coefficients are subject to certain arithmetic conditions. Lastly, in Chapter 4 we produce many results concerning weight $2$ and $3$ codewords in cyclic codes, along with a theorem linking the solution count of systems of diagonal equations over finite fields with characteristic $2$ to the weight distribution of corresponding binary cyclic codes. Chapter 2 is based on the contents of our published paper, while chapters 3 and 4 present the contents of articles that are yet to be published. In the annex, we present SageMath commands used to create and verify the examples that are discussed throughout the thesis.