On the number of solutions of equations of Dickson polynomials over finite fields

Wun Seng Chou; Gary L. Mullen; Bertram Wassermann

doi:10.11650/twjm/1500404986

On the number of solutions of equations of Dickson polynomials over finite fields

Wun Seng Chou, Gary L. Mullen, Bertram Wassermann

Mathematics

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

7 Scopus citations

Abstract

Let k, n₁, ..., n_k be fixed positive integers, c₁, ..., c_k ∈ GF(q)*, and a₁, ..., a_k, c ∈ GF(q). We study the number of solutions in GF(q) of the equation c₁D_n1(x₁, a₁)+c₂D_n2(x₂, a₂)+...+c_kD_nk(x_k, a_k) = c, where each D_ni(x_i, a_i), 1 ≤ i ≤ k, is the Dickson polynomial of degree n_i with parameter a_i. We also employ the results of the k = 1 case to recover the cardinality of preimages and images of Dickson polynomials obtained earlier by Chou, Gomez-Calderon and Mullen [1].

Original language	English (US)
Pages (from-to)	917-931
Number of pages	15
Journal	Taiwanese Journal of Mathematics
Volume	12
Issue number	4
DOIs	https://doi.org/10.11650/twjm/1500404986
State	Published - Jul 2008

All Science Journal Classification (ASJC) codes

General Mathematics

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abstract = "Let k, n1, ..., nk be fixed positive integers, c1, ..., ck ∈ GF(q)*, and a1, ..., ak, c ∈ GF(q). We study the number of solutions in GF(q) of the equation c1Dn1(x1, a1)+c2Dn2(x2, a2)+...+ckDnk(xk, ak) = c, where each Dni(xi, ai), 1 ≤ i ≤ k, is the Dickson polynomial of degree ni with parameter ai. We also employ the results of the k = 1 case to recover the cardinality of preimages and images of Dickson polynomials obtained earlier by Chou, Gomez-Calderon and Mullen [1].",

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AU - Chou, Wun Seng

AU - Mullen, Gary L.

AU - Wassermann, Bertram

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N2 - Let k, n1, ..., nk be fixed positive integers, c1, ..., ck ∈ GF(q)*, and a1, ..., ak, c ∈ GF(q). We study the number of solutions in GF(q) of the equation c1Dn1(x1, a1)+c2Dn2(x2, a2)+...+ckDnk(xk, ak) = c, where each Dni(xi, ai), 1 ≤ i ≤ k, is the Dickson polynomial of degree ni with parameter ai. We also employ the results of the k = 1 case to recover the cardinality of preimages and images of Dickson polynomials obtained earlier by Chou, Gomez-Calderon and Mullen [1].

AB - Let k, n1, ..., nk be fixed positive integers, c1, ..., ck ∈ GF(q)*, and a1, ..., ak, c ∈ GF(q). We study the number of solutions in GF(q) of the equation c1Dn1(x1, a1)+c2Dn2(x2, a2)+...+ckDnk(xk, ak) = c, where each Dni(xi, ai), 1 ≤ i ≤ k, is the Dickson polynomial of degree ni with parameter ai. We also employ the results of the k = 1 case to recover the cardinality of preimages and images of Dickson polynomials obtained earlier by Chou, Gomez-Calderon and Mullen [1].

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On the number of solutions of equations of Dickson polynomials over finite fields

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