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 generally studied over finite fields, where they sometimes may not be 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.
Definitionedit
First kindedit
For integer n > 0 and α in a commutative ringR with identity (often chosen to be the finite field Fq = GF(q)) the Dickson polynomials (of the first kind) over R are given by[1]
with the initial conditions D0(x,α) = 2 and D1(x,α) = x.
The coefficients are given at several places in the OEIS[2][3][4][5] with minute differences for the first two terms.
Second kindedit
The Dickson polynomials of the second kind, En(x,α), 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
They may also be generated by the recurrence relation for n ≥ 2,
with the initial conditions E0(x,α) = 1 and E1(x,α) = x.
The coefficients are also given in the OEIS.[6][7]
Propertiesedit
The Dn are the unique monic polynomials satisfying the functional equation
By the recurrence relation above, Dickson polynomials are Lucas sequences. Specifically, for α = −1, the Dickson polynomials of the first kind are Fibonacci polynomials, and Dickson polynomials of the second kind are Lucas polynomials.
By the composition rule above, when α is idempotent, composition of Dickson polynomials of the first kind is commutative.
The Dickson polynomials with parameter α = 0 give monomials.
The Dickson polynomials with parameter α = 1 are related to Chebyshev polynomialsTn(x) = cos (n arccos x) of the first kind by[1]
Since the Dickson polynomial Dn(x,α) can be defined over rings with additional idempotents, Dn(x,α) is often not related to a Chebyshev polynomial.
Permutation polynomials and Dickson polynomialsedit
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.[9]
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 Turnwald (1995), and subsequently Müller (1997) gave a simpler proof along the lines of an argument due to Schur.
Further, Müller (1997) proved that any permutation polynomial over the finite field Fq whose degree is simultaneously coprime to q and less than q1/4 must be a composition of Dickson polynomials and linear polynomials.
Generalizationedit
Dickson polynomials of both kinds over finite fields can be thought of as initial members of a sequence of generalized Dickson polynomials referred to as Dickson polynomials of the (k + 1)th kind.[10] Specifically, for α ≠ 0 ∈ Fq with q = pe for some prime p and any integers n ≥ 0 and 0 ≤ k < p, the nth Dickson polynomial of the (k + 1)th kind over Fq, denoted by Dn,k(x,α), is defined by[11]
and
Dn,0(x,α) = Dn(x,α) and Dn,1(x,α) = En(x,α), showing that this definition unifies and generalizes the original polynomials of Dickson.
The significant properties of the Dickson polynomials also generalize:[12]
Recurrence relation: For n ≥ 2,
with the initial conditions D0,k(x,α) = 2 − k and D1,k(x,α) = x.
^Wang, Q.; Yucas, J. L. (2012), "Dickson polynomials over finite fields", Finite Fields and Their Applications, 18 (4): 814–831, doi:10.1016/j.ffa.2012.02.001
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. 11 (1/6). The Annals of Mathematics: 65–120, 161–183. doi:10.2307/1967217. hdl:2027/uiuo.ark:/13960/t4zh9cw1v. 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. Vol. 65. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York. ISBN 978-0-582-09119-1. MR 1237403. Zbl 0823.11070.
Lidl, Rudolf; Niederreiter, Harald (1983). Finite fields. Encyclopedia of Mathematics and Its Applications. Vol. 20 (1st ed.). Addison-Wesley. ISBN 978-0-201-13519-0. Zbl 0866.11069.
Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, ISBN 978-1-4398-7378-6
Müller, Peter (1997). "A Weil-bound free proof of Schur's conjecture". Finite Fields and Their Applications. 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 978-981-02-0614-7.
Turnwald, Gerhard (1995). "On Schur's conjecture". J. Austral. Math. Soc. Ser. A. 58 (3): 312–357. doi:10.1017/S1446788700038349. MR 1329867. Zbl 0834.11052.
Young, Paul Thomas (2002). "On modified Dickson polynomials" (PDF). Fibonacci Quarterly. 40 (1): 33–40.
Bayad, Abdelmejid; Cangul, Ismail Naci (2012). "The minimal polynomial 2 cos(pi/q) and Dickson polynomials". Appl. Math. Comp. 218: 7014–7022. doi:10.1016/j.amc.2011.12.044.