Taiwanese Journal of Mathematics

ON THE NUMBER OF SOLUTIONS OF EQUATIONS OF DICKSON POLYNOMIALS OVER FINITE FIELDS

Wun-Seng Chou, Gary L. Mullen, and Bertram Wassermann

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Abstract

Let $k, n_1, \dots, n_k$ be fixed positive integers, $c_1, \dots, c_k \in GF(q)^*$, and $a_1, \dots, a_k, c \in GF(q)$. We study the number of solutions in $GF(q)$ of the equation $c_1D_{n_1}(x_1, a_1) + c_2D_{n_2}(x_2, a_2) + \cdots + c_kD_{n_k}(x_k, a_k) = c$, where each $D_{n_i}(x_i, a_i)$, $1 \leq i \leq k$, 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].

Article information

Source
Taiwanese J. Math., Volume 12, Number 4 (2008), 917-931.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500404986

Digital Object Identifier
doi:10.11650/twjm/1500404986

Mathematical Reviews number (MathSciNet)
MR2426536

Zentralblatt MATH identifier
1154.11044

Subjects
Primary: 11T06: Polynomials

Keywords
finite field Dickson polynomial character Gauss sum trace

Citation

Chou, Wun-Seng; Mullen, Gary L.; Wassermann, Bertram. ON THE NUMBER OF SOLUTIONS OF EQUATIONS OF DICKSON POLYNOMIALS OVER FINITE FIELDS. Taiwanese J. Math. 12 (2008), no. 4, 917--931. doi:10.11650/twjm/1500404986. https://projecteuclid.org/euclid.twjm/1500404986


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References

  • W.-S. Chou, J. Gomez-Calderon and G. L. Mullen, Value sets of Dickson polynomials over finite fields, J. Number Theory, 30 (1988), 334-344.
  • W.-C. W. Li, Number Theory With Applications, Series on University Mathematics, Vol. 7, World Scientific, Singapore, 1996.
  • R. Lidl, G. L. Mullen and G. Turnwald, Dickson Polynomials, Longman Scientific and Technical, Essex, United Kingdom, 1993.
  • R. Lidl and H. Niederreiter, Finite Fields, Encyclo. of Math. & Its Appls, Second Ed., Vol. 20, Cambridge University Press, Cambridge, 1997.