On the roots of certain Dickson polynomials

Aart Blokhuis; Xiwang Cao; Wun Seng Chou; Xiang Dong Hou

doi:10.1016/j.jnt.2018.01.003

On the roots of certain Dickson polynomials

Aart Blokhuis
, Xiwang Cao
, Wun Seng Chou
, Xiang Dong Hou

Research output: Contribution to journal › Article › Academic › peer-review

5 Citations (Scopus)

Abstract

Let n be a positive integer, q=2ⁿ, and let F_q be the finite field with q elements. For each positive integer m, let D_m(X) be the Dickson polynomial of the first kind of degree m with parameter 1. Assume that m>1 is a divisor of q+1. We study the existence of α∈F_q ^⁎ such that D_m(α)=D_m(α⁻¹)=0. We also explore the connections of this question to an open question by Wiedemann and a game called “Button Madness”.

Original language	English
Pages (from-to)	229-246
Number of pages	18
Journal	Journal of Number Theory
Volume	188
DOIs	https://doi.org/10.1016/j.jnt.2018.01.003
Publication status	Published - 1 Jul 2018

Funding

We are grateful to Professor M. Freedman for the question that motivated this work. X. Cao would like to thank the Institute of Mathematics, Academia Sinica, for the financial support during his visit. A. Blokhuis and X. Hou would like to thank the Institute for Mathematical Sciences of the National University of Singapore for their hospitality and support and for facilitating their collaboration during their visit there. W.-S. Chou would express his gratitude to Dr. Y.-T. Lin of Institute of Mathematics, Academia Sinica, for helping the preparation of this paper. Finally, we are grateful to the anonymous referee for the valuable comments and suggestions which lead to the improvement of this paper.

Keywords

Absolutely irreducible
Button madness
Dickson polynomials
Fermat number
Finite field
Reciprocal polynomial

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10.1016/j.jnt.2018.01.003

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title = "On the roots of certain Dickson polynomials",

abstract = "Let n be a positive integer, q=2n, and let Fq be the finite field with q elements. For each positive integer m, let Dm(X) be the Dickson polynomial of the first kind of degree m with parameter 1. Assume that m>1 is a divisor of q+1. We study the existence of α∈Fq ⁎ such that Dm(α)=Dm(α−1)=0. We also explore the connections of this question to an open question by Wiedemann and a game called {\textquotedblleft}Button Madness{\textquotedblright}.",

keywords = "Absolutely irreducible, Button madness, Dickson polynomials, Fermat number, Finite field, Reciprocal polynomial",

author = "Aart Blokhuis and Xiwang Cao and Chou, \{Wun Seng\} and Hou, \{Xiang Dong\}",

year = "2018",

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doi = "10.1016/j.jnt.2018.01.003",

language = "English",

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T1 - On the roots of certain Dickson polynomials

AU - Blokhuis, Aart

AU - Cao, Xiwang

AU - Chou, Wun Seng

AU - Hou, Xiang Dong

PY - 2018/7/1

Y1 - 2018/7/1

N2 - Let n be a positive integer, q=2n, and let Fq be the finite field with q elements. For each positive integer m, let Dm(X) be the Dickson polynomial of the first kind of degree m with parameter 1. Assume that m>1 is a divisor of q+1. We study the existence of α∈Fq ⁎ such that Dm(α)=Dm(α−1)=0. We also explore the connections of this question to an open question by Wiedemann and a game called “Button Madness”.

AB - Let n be a positive integer, q=2n, and let Fq be the finite field with q elements. For each positive integer m, let Dm(X) be the Dickson polynomial of the first kind of degree m with parameter 1. Assume that m>1 is a divisor of q+1. We study the existence of α∈Fq ⁎ such that Dm(α)=Dm(α−1)=0. We also explore the connections of this question to an open question by Wiedemann and a game called “Button Madness”.

KW - Absolutely irreducible

KW - Button madness

KW - Dickson polynomials

KW - Fermat number

KW - Finite field

KW - Reciprocal polynomial

UR - https://www.scopus.com/pages/publications/85042616259

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DO - 10.1016/j.jnt.2018.01.003

M3 - Article

AN - SCOPUS:85042616259

SN - 0022-314X

VL - 188

SP - 229

EP - 246

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

On the roots of certain Dickson polynomials

Abstract

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Keywords

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