MSc Thesis Defense: Montaha Naouar, DIOPHANTINE F-SETS OVER FINITE FIELDS, Date & Time: 23 June, 2026 – 1:00 PM, Place: FENS L065

DIOPHANTINE F-SETS OVER FINITE FIELDS

Montaha Naouar
Mathematics, MSc Thesis, 2026

Thesis Jury

     Assoc. Prof. Mohammad Sadek (Thesis Advisor)

  Assoc. Prof. Kağan Kurşungöz

  Asst. Prof. Kübra Benli

Date & Time: 23rd June, 2026 – 1:00 PM

Place: FENS L065

Keywords :  Finite fields, character sums,  power residues, Gaussian sums, Jacobi Sums, Hasse-Weil 

Abstract

A well-studied question in number theory is to determine how often a polynomial equation attains perfect powers in a finite field. In this thesis, we investigate certain families of Diophantine equations and estimate the number of their solutions over finite fields.

In the first part of this thesis, we give due attention to the Diophantine equation y^p=F(x_1,...,x_m), where F(x_1,...,x_m)=x_1^s+...+x_m^s is a diagonal polynomial. For certain choices of p,s> 2, we survey how to establish an upper bound on the number of solutions (x_1,...,x_m;y) of this equation over a finite field. For this purpose, we use different character sums including Gaussian sums and Jacobi sums.

The second part of the thesis focuses on Diophantine tuples in finite fields. A Diophantine tuple is a tuple of elements for which the product of any pair of distinct elements is a fixed shift of a square. If, moreover, the square of any element in the tuple is also that fixed shift of a square, then the Diophantine tuple is said to be strong. After surveying the known estimate for the number of Diophantine triples in a finite field, we obtain upper bounds for the numbers of strong Diophantine pairs and triples over finite fields by studying certain families of character sums.