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

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 “Button Madness”.",

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}",

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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

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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

Keywords

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