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arXiv · 2201.00954

Dynamics of polynomial maps over finite fields

José Alves Oliveira·Fabio Enrique Brochero Martínez
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Abstract

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $n$ be a positive integer. In this paper, we study the digraph associated to the map $x\mapsto x^n h(x^{\frac{q-1}{m}})$, where $h(x)\in\mathbb{F}_q[x].$ We completely determine the associated functional graph of maps that satisfy a certain condition of regularity. In particular, we provide the functional graphs associated to monomial maps. As a consequence of our results, the number of connected components, length of the cycles and number of fixed points of these class of maps are provided.

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José Alves Oliveira, Fabio Enrique Brochero Martínez. 2022-01-04. Dynamics of polynomial maps over finite fields. https://arxiv.org/abs/2201.00954

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