On Some Algebraic and Dynamical Aspects of Families of Polynomials

Abstract

This thesis investigates several arithmetic and algebraic properties of polynomials, with a particular emphasis on irreducibility, monogenity, and the behaviour of iterated polynomial sequences. We first study truncated binomial polynomials over the rationals and establish new affirmative results concerning their irreducibility. A substantial part of the work develops a detailed analysis of Newton polygons under polynomial composition. These structural insights yield broad applications, including criteria for stability and eventual stability of large families of polynomials, as well as precise information about the degrees and number of irreducible factors appearing in their iterates. These ideas further connect to questions about ramification of primes, provide new directions toward Sookdeo's conjecture, and lead to explicit constructions of towers of number fields that are not monogenic. Further investigations address the monogenity of specific classes of polynomials, providing criteria that characterize when these polynomials generate monogenic extensions. Analytic estimates are also obtained to count such polynomials within the families considered. Finally, we obtain an upper bound for the Zsigmondy set associated with a rational critical point of a polynomial, contributing to the broader understanding of primitive prime divisors in arithmetic dynamics.