Dynamics of polynomial maps over finite fields

2022/01/04 by José Alves Oliveira, Oliveira, José Alves, Fabio Enrique Brochero Martínez +1
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Coding theory and cryptography #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Graph theory and applications

paper · pdf · doi:10.48550/arxiv.2201.00954

openalex publication_date 2022/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathbbF_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↦ xⁿ h(x^(q-1)/(m)), where h(x)∈\mathbbF_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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