Dickson polynomial

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In mathematics, the Dickson polynomials (or Brewer polynomials), denoted D_n(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.

Definition

D₀(x,α) = 2, and for n > 0 Dickson polynomials (of the first kind) are given by

$D_{n}(x,\alpha )=\sum _{{p=0}}^{{\lfloor n/2\rfloor }}{\frac {n}{n-p}}{\binom {n-p}{p}}(-\alpha )^{p}x^{{n-2p}}.$

The first few Dickson polynomials are

$D_{0}(x,\alpha )=2\,$

$D_{1}(x,\alpha )=x\,$

$D_{2}(x,\alpha )=x^{2}-2\alpha \,$

$D_{3}(x,\alpha )=x^{3}-3x\alpha \,$

$D_{4}(x,\alpha )=x^{4}-4x^{2}\alpha +2\alpha ^{2}.\,$

The Dickson polynomials of the second kind E_n are defined by

$E_{n}(x,\alpha )=\sum _{{p=0}}^{{\lfloor n/2\rfloor }}{\binom {n-p}{p}}(-\alpha )^{p}x^{{n-2p}}.$

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

$E_{0}(x,\alpha )=1\,$

$E_{1}(x,\alpha )=x\,$

$E_{2}(x,\alpha )=x^{2}-\alpha \,$

$E_{3}(x,\alpha )=x^{3}-2x\alpha \,$

$E_{4}(x,\alpha )=x^{4}-3x^{2}\alpha +\alpha ^{2}.\,$

Properties

The D_n satisfy the identities

$D_{n}(u+\alpha /u,\alpha )=u^{n}+(\alpha /u)^{n}\,;$

$D_{{mn}}(x,\alpha )=D_{m}(D_{n}(x,\alpha ),\alpha ^{n})\,.$

For n≥2 the Dickson polynomials satisfy the recurrence relation

$D_{n}(x,\alpha )=xD_{{n-1}}(x,\alpha )-\alpha D_{{n-2}}(x,\alpha )\,$

$E_{n}(x,\alpha )=xE_{{n-1}}(x,\alpha )-\alpha E_{{n-2}}(x,\alpha ).\,$

The Dickson polynomial D_n = y is a solution of the ordinary differential equation

$(x^{2}-4\alpha )y''+xy'-n^{2}y=0\,$

and the Dickson polynomial E_n = y is a solution of the differential equation

$(x^{2}-4\alpha )y''+3xy'-n(n+2)y=0.\,$

Their ordinary generating functions are

$\sum _{n}D_{n}(x,\alpha )z^{n}={\frac {2-xz}{1-xz+\alpha z^{2}}}\,$

$\sum _{n}E_{n}(x,\alpha )z^{n}={\frac {1}{1-xz+\alpha z^{2}}}.\,$

Links to other polynomials

Dickson polynomials over the complex numbers are related to Chebyshev polynomials T_n and U_n by

$D_{n}(2xa,a^{2})=2a^{{n}}T_{n}(x)\,$

$E_{n}(2xa,a^{2})=a^{{n}}U_{n}(x).\,$

Crucially, the Dickson polynomial D_n(x,a) can be defined over rings in which a is not a square, and over rings of characteristic 2; in these cases, D_n(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:

$D_{n}(x,0)=x^{n}\,.$

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 D_n(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.^[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 F_q whose degree is simultaneously coprime to q−1 and less than q^1/4 must be a composition of Dickson polynomials and linear polynomials.

References

↑ 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.

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