In mathematics, the Dickson polynomials (or Brewer polynomials), denoted Dn(x,α), form a polynomial sequence introduced by L. E. Dickson (1897) and rediscovered by Brewer (1961) in his study of Brewer sums.
Over the complex numbers, Dickson polynomials are essentially equivalent to Chebyshev polynomials with a change of variable, and in fact Dickson polynomials are sometimes called Chebyshev polynomials. Dickson polynomials are mainly studied over finite fields, when they are not equivalent to Chebyshev polynomials. One of the main reasons for interest in them is that for fixed α, they give many examples of permutation polynomials: polynomials acting as permutations of finite fields.
D0(x,α) = 2, and for n > 0 Dickson polynomials (of the first kind) are given by
The first few Dickson polynomials are
The Dickson polynomials of the second kind En are defined by
They have not been studied much, and have properties similar to those of Dickson polynomials of the first kind. The first few Dickson polynomials of the second kind are
The Dn satisfy the identities
For n≥2 the Dickson polynomials satisfy the recurrence relation
The Dickson polynomial Dn = y is a solution of the ordinary differential equation
and the Dickson polynomial En = y is a solution of the differential equation
Their ordinary generating functions are
Links to other polynomials
- Dickson polynomials over the complex numbers are related to Chebyshev polynomials Tn and Un by
Crucially, the Dickson polynomial Dn(x,a) can be defined over rings in which a is not a square, and over rings of characteristic 2; in these cases, Dn(x,a) is often not related to a Chebyshev polynomial.
- The Dickson polynomials with parameter α = 1 or α = -1 are related to the Fibonacci and Lucas polynomials.
- The Dickson polynomials with parameter α = 0 give monomials:
Permutation polynomials and Dickson polynomials
A permutation polynomial (for a given finite field) is one that acts as a permutation of the elements of the finite field.
The Dickson polynomial Dn(x,α) (considered as a function of x with α fixed) is a permutation polynomial for the field with q elements if and only if n is coprime to q2−1.
M. Fried (1970) proved that any integral polynomial that is a permutation polynomial for infinitely many prime fields is a composition of Dickson polynomials and linear polynomials (with rational coefficients). This assertion has become known as Schur's conjecture, although in fact Schur did not make this conjecture. Since Fried's paper contained numerous errors, a corrected account was given by G. Turnwald (1995), and subsequently P. Müller (1997) gave a simpler proof along the lines of an argument due to Schur.
Further, P. Müller (1997) proved that any permutation polynomial over the finite field Fq whose degree is simultaneously coprime to q−1 and less than q1/4 must be a composition of Dickson polynomials and linear polynomials.
- Lidl & Niederreiter (1997) p.356
- Brewer, B. W. (1961), "On certain character sums", Transactions of the American Mathematical Society 99: 241–245, ISSN 0002-9947, MR 0120202, Zbl 0103.03205
- Dickson, L.E. (1897). "The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group I,II". Ann. Of Math. (The Annals of Mathematics) 11 (1/6): 65–120; 161–183. doi:10.2307/1967217. ISSN 0003-486X. JFM 28.0135.03. JSTOR 1967217.
- Fried, Michael (1970). "On a conjecture of Schur". Michigan Math. J. 17: 41–55. doi:10.1307/mmj/1029000374. ISSN 0026-2285. MR 0257033. Zbl 0169.37702.
- Lidl, R.; Mullen, G. L.; Turnwald, G. (1993). Dickson polynomials. Pitman Monographs and Surveys in Pure and Applied Mathematics 65. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York. ISBN 0-582-09119-5. MR 1237403. Zbl 0823.11070.
- Lidl, Rudolf; Niederreiter, Harald (1997). Finite fields. Encyclopedia of Mathematics and Its Applications 20 (2nd ed.). Cambridge University Press. ISBN 0-521-39231-4. Zbl 0866.11069.
- Mullen, Gary L. (2001), "D/d120140", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Müller, Peter (1997). "A Weil-bound free proof of Schur's conjecture". Finite Fields Appl. 3: 25–32. doi:10.1006/ffta.1996.0170. Zbl 0904.11040.
- Rassias, Thermistocles M.; Srivastava, H.M.; Yanushauskas, A. (1991). Topics in Polynomials of One and Several Variables and Their Applications: A Legacy of P.L.Chebyshev. World Scientific. pp. 371–395. ISBN 981-02-0614-3.
- Turnwald, Gerhard (1995). "On Schur's conjecture". J. Austral. Math. Soc. Ser. A 58 (03): 312–357. doi:10.1017/S1446788700038349. MR 1329867. Zbl 0834.11052.
- Young, Paul Thomas (2002). "On modified Dickson polynomials". Fib. Quaterly 40 (1): 33–40.