Darwish Mohamed, Mohamed Osama Hafez
(2022)
*Dynamical irreducibility of pure polynomials over the rational field.*
[Thesis]

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## Abstract

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.

Item Type: | Thesis |
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Uncontrolled Keywords: | Arithmetic Dynamical systems. -- Dynamically Irreducible Polynomials. -- Eventually Stable Polynomials. -- Eisenstein Criterion. -- Dumas Criterion. -- Aritmetik Dinamik Sistemler. -- Dinamik İndirgenemez Polinomlar. -- Zamanla Stabil Polinomlar. -- Eisenstein Kriteri. -- Dumas Kriteri. |

Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences |

Depositing User: | Dila Günay |

Date Deposited: | 26 Apr 2023 13:55 |

Last Modified: | 26 Apr 2023 13:55 |

URI: | https://research.sabanciuniv.edu/id/eprint/47174 |