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Seminars

Number Theory Seminar

Periodicity and Dynamical Systems of Dickson Polynomials in Finite Fields

2025/09/12 (Fri.) 10:30~11:30
  • Date : 2025/09/12 (Fri.) 10:30~11:30
  • Location : Seminar Room 617, Institute of Mathematics (NTU Campus)
  • Speaker : Wayne Peng (National Central University)
  • Abstract : In this talk, I will present our recent investigation into the dynamical properties of Dickson polynomials over finite fields. We'll explore the periodicity and structural behavior of iterated sequences defined by \( [D_n(x, \alpha) \mod (x^q - x)]_n \), where \( D_n(x, \alpha) \) is a Dickson polynomial of the first kind. I'll demonstrate our explicit formulas for determining the period of these sequences. We'll discuss a symmetric property of polynomial coefficients that proves fundamental to our analysis. Through combinatorial and number-theoretic approaches, I'll showcase the dynamic structures, with particular emphasis on cases where \( n \) is coprime to \( q^2 - 1 \). The talk will conclude with open problems and implications for understanding algebraic structures in finite field dynamics. If the time is enough, we will also outline our algorithm for determining Dickson polynomials.
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