Dynamics of polynomial maps over finite fields

被引:0
作者
José Alves Oliveira
F. E. Brochero Martínez
机构
[1] Universidade Federal de Lavras,Departamento de Matemática
[2] UFLA,Departamento de Matemática
[3] Universidade Federal de Minas Gerais,undefined
[4] UFMG,undefined
来源
Designs, Codes and Cryptography | 2024年 / 92卷
关键词
Functional graph; Polynomial maps; Finite fields; Finite abelian group; Primary 37P25; Secondary 05C20;
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中图分类号
学科分类号
摘要
Let Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q$$\end{document} 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↦xnh(xq-1m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\mapsto x^n h(x^{\frac{q-1}{m}})$$\end{document} over Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q$$\end{document}, where h(x)∈Fq[x].\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(x)\in \mathbb {F}_q[x].$$\end{document} 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, one have the number of connected components, length of the cycles and number of fixed points of these class of maps.
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页码:1113 / 1125
页数:12
相关论文
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