Research Projects

Diophantine equations and polynomials are fundamental problems in mathematics, with solving them being a significant and challenging task. They are often used to solve problems involving binary recurrence sequences and combinatorial objects. One of the most significant aspects of polynomials over integers or rationals is their algebraic aspects, such as their irreducibility and Galois groups. Despite the availability of various techniques, there is still a need for new ideas and criticisms to understand these. Another important and fundamental problem is understanding the arithmetic and algebraic aspects of iterates of polynomials. While some aspects of irreducibility of iterates of some families of polynomials are understood, there is a wide openness and less knowledge about the Galois groups of iterates. This knowledge has wider applications in other areas of number theory and mathematics. This project aims to explore different aspects of Diophantine Equations and polynomials, contributing to these fundamental questions in this important field of research. The project will focus on solving exponential diophantine equations, understanding irreducibility and Galois groups of a family of polynomials, and understanding the arithmetic and algebraic properties of iterates of polynomials over integers and rationals and over finite fields. The project aims to contribute to these fundamental questions and contribute to the field of mathematics.