Dynamical irreducibility of pure polynomials over the rational field

Official URL: https://risc01.sabanciuniv.edu/record=b2767344_(Table of contents)

Let f be a polynomial in Q[x]. We say that f is dynamically irreducible or stable over Q if all its iterates fn := f ◦ f ◦{z. . . ◦ f} n are irreducible over Q. Generally, a polynomial is called eventually stable if the number of irreducible factors of any iterate fn is bounded by some c ∈ Z+, in particular, if c = 1, then f is dynamically irreducible. A polynomial defined over Q is said to be pure with respect to a prime p if its Newton polygon consists of exactly one line, e.g., pr-Eisenstein polynomials for some r ≥ 1. In 1985, Odoni showed that Eisenstein polynomials are dynamically irreducible over Q. Ali extended this result to include pr-Eisenstein polynomials for any r ≥ 1. In this thesis, we present families of pure polynomials that are dynamically irreducible in Q[x]. Under some conditions, we characterize certain families and develop some criteria of dynamically irreducible polynomials that possess a pure iterate. In addition, we describe some iterative techniques to produce irreducible polynomials in Q[x] from pure polynomials by composition. Recently, Demark et al. investigated the eventual stability of a quadratic binomial of the form x2−1c ∈ Q[x] for some c ∈ Z\{0,−1}. In this work, we prove that pure polynomials are eventually stable in Q[x]. Also, we display a family of eventually stable polynomials that possess a pure iterate.