Dickson polynomial

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In mathematics, the Dickson polynomials, denoted Dn(x,α), are polynomials studied by L. E. Dickson (1897).

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.

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[edit] 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 En 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 \,

[edit] Properties

For n≥2 Dickson polynomials satisfy the recursion relations

Dn(x,α) = xDn − 1(x.α) − αDn − 2(x,α)
En(x,α) = xEn − 1(x.α) − αEn − 2(x,α)

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

(x2 − 4α)y'' + xy' − n2y = 0

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

(x2 − 4α)y'' + 3xy' − n(n + 2)y = 0

Their generating functions are

\sum_nD_n(x,\alpha)z^n = \frac{2-xz}{1-xz+\alpha z^2}
\sum_nE_n(x,\alpha)z^n = \frac{1-xz}{1-xz+\alpha z^2}

Dickson polynomials are related to Chebyshev polynomials Tn and Un by

Dn(2xa,a2) = 2anTn(x)
En(2xa,a2) = anUn(x)

[edit] 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 whenever n is coprime to q2−1.

M. Fried (1970) proved Schur's conjecture that any integral polynomial that is a permutation polynomial for infinitely many prime fields is a composition of monomials, (rational) linear polynomials, and Dickson polynomials. (The article uses Chebyshev polynomials rather than Dickson polynomials, but Chebyshev polynomials and Dickson polynomials over the rationals can be converted into each other by composition with rational linear maps.)

[edit] References