MSc Thesis Defense: Ata Marangoz, STARTER POLYNOMIALS FOR FINDING RATIONAL CYCLES, AND ZSIGMONDY SETS OF CRITICAL ORBITS OF A FAMILY OF POLYNOMIALS

STARTER POLYNOMIALS FOR FINDING RATIONAL CYCLES, AND ZSIGMONDY SETS OF CRITICAL ORBITS OF A FAMILY OF POLYNOMIALS

Ata Marangoz
Mathematics, MSc. Thesis, 2025

Thesis Jury

Assoc. Prof. Mohammad Sadek (Thesis Supervisor)

Asst. Prof. Ayesha Asloob Topaçoğlu

Asst. Prof. Eda Yıldız Yılmaz

Date & Time: December 17th, 2025 – 2:40 PM

Place: FASS G025

Keywords : Arithmetic Dynamics, K-Rational Cycles, Starter Polynomials, Rigid Divisibility Sequences, Zsigmondy Sets

Abstract

This thesis investigates the arithmetic dynamical properties of polynomials over various fields. Focusing on two main problems, the locations of preperiodic rational points and the numerical properties of critical orbits of a family of polynomials where good reducibility is dependent only on the constant term. For the first problem, to overcome the computational expense of dynatomic polynomials and their factorizations, a new method named "Starter Polynomials" is introduced. This method coincides with Lagrange interpolation in specific cases. This method is applied to investigate cycles of length 1,2,3 and preperiodic points of type 2_1 of the map z^2+c. Also as an example of higher degree polynomials, the cycles of length 2 are investigated for the map z^3+cz. This method can be generalized to rational maps and as an example the family of maps kz+b/z is inspected and its 2-cycles are found. It is established that the critical orbit of the map z^2+c is never a 3-cycle. In the second part, the family of polynomials where f(z)=u_dz^d+u_{d-1}z^{d-1}+… +u_2z^2+c is inspected. It is proven that the sequence (A_n) of the numerators of the terms of the critical orbit constitutes a rigid divisibility sequence. Using this property and an analytic investigation on the specific case d>=3, c>2, u_d=1 it is proven that the Zsigmondy set of the critical orbit of such maps is empty. Meaning that every term introduces a new prime. For other possible cases various tools are introduced and executed with examples.