Dickson polynomial

In mathematics, the Dickson polynomials, denoted $D n (x, α)$ , form a polynomial sequence introduced by L. E. Dickson (1897). They were rediscovered by Brewer (1961) in his study of Brewer sums and have at times, although rarely, been referred to as Brewer polynomials.

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.

Definition

First kind

For integer $n > 0$ and $α$ in a commutative ring $R$ with identity (often chosen to be the finite field $F q = GF(q)$ ) the Dickson polynomials (of the first kind) over $R$ are given by^[1]

D_{n}(x,\alpha )=\sum _{i=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\frac {n}{n-i}}{\binom {n-i}{i}}(-\alpha )^{i}x^{n-2i}\,.

The first few Dickson polynomials are

{\begin{aligned}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}\\D_{5}(x,\alpha )&=x^{5}-5x^{3}\alpha +5x\alpha ^{2}\,.\end{aligned}}

They may also be generated by the recurrence relation for $n \geq 2$ ,

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

with the initial conditions $D 0 (x, α) = 2$ and $D 1 (x, α) = x$ .

Second kind

The Dickson polynomials of the second kind, $E n (x, α)$ , are defined by

E_{n}(x,\alpha )=\sum _{i=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\binom {n-i}{i}}(-\alpha )^{i}x^{n-2i}.

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

{\begin{aligned}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}\,.\end{aligned}}

They may also be generated by the recurrence relation for $n \geq 2$ ,

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

with the initial conditions $E 0 (x, α) = 1$ and $E 1 (x, α) = x$ .

Properties

The $D n$ are the unique monic polynomials satisfying the functional equation

D_{n}\left(u+{\frac {\alpha }{u}},\alpha \right)=u^{n}+\left({\frac {\alpha }{u}}\right)^{n},

where $α \in F q$ and $u \neq 0 \in F q 2$ .^[2]

They also satisfy a composition rule,^[2]

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

The $E n$ also satisfy a functional equation^[2]

E_{n}\left(y+{\frac {\alpha }{y}},\alpha \right)={\frac {y^{n+1}-\left({\frac {\alpha }{y}}\right)^{n+1}}{y-{\frac {\alpha }{y}}}}\,,

for $y \neq 0$ , $y 2 \neq α$ , with $α \in F q$ and $y \in F q 2$ .

The Dickson polynomial $y = D n$ is a solution of the ordinary differential equation

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

and the Dickson polynomial $y = E n$ is a solution of the differential equation

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

Their ordinary generating functions are

{\begin{aligned}\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}}}\,.\end{aligned}}

Links to other polynomials

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.

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

The Dickson polynomials with parameter $α = 1$ are related to Chebyshev polynomials $T n (x) = cos (n arccos x)$ of the first kind by^[1]

$D_{n}(2x,1)=2T_{n}(x)\,.$

Since the Dickson polynomial $D n (x, α)$ can be defined over rings with additional idempotents, $D n (x, α)$ is often not related to a Chebyshev polynomial.

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 $q 2 - 1$ .^[3]

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 $F q$ whose degree is simultaneously coprime to $q$ and less than $q.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/4⁠$ must be a composition of Dickson polynomials and linear polynomials.

Generalization

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.^[4] Specifically, for $α \neq 0 \in F q$ with $q = p e$ for some prime $p$ and any integers $n \geq 0$ and $0 \leq k < p$ , the $n$ th Dickson polynomial of the $(k + 1)$ th kind over $F q$ , denoted by $D n, k (x, α)$ , is defined by^[5]

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

and

D_{n,k}(x,\alpha )=\sum _{i=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\frac {n-ki}{n-i}}{\binom {n-i}{i}}(-\alpha )^{i}x^{n-2i}\,.

$D n,0 (x, α) = D n (x, α)$ and $D n,1 (x, α) = E n (x, α)$ , showing that this definition unifies and generalizes the original polynomials of Dickson.

The significant properties of the Dickson polynomials also generalize:^[6]

Recurrence relation: For $n \geq 2$ ,

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

with the initial conditions

D 0, k (x, α) = 2 - k

and

D 1, k (x, α) = x

.

Functional equation:

D_{n,k}\left(y+\alpha y^{-1},\alpha \right)={\frac {y^{2n}+k\alpha y^{2n-2}+\cdots +k\alpha ^{n-1}y^{2}+\alpha ^{n}}{y^{n}}}={\frac {y^{2n}+{\alpha }^{n}}{y^{n}}}+\left({\frac {k\alpha }{y^{n}}}\right){\frac {y^{2n}-{\alpha }^{n-1}y^{2}}{y^{2}-\alpha }}\,,

where

y \neq 0

,

y 2 \neq α

.

Generating function:

\sum _{n=0}^{\infty }D_{n,k}(x,\alpha )z^{n}={\frac {2-k+(k-1)xz}{1-xz+\alpha z^{2}}}\,.

Notes

^ ^a ^b Lidl & Niederreiter 1983, p. 355
^ ^a ^b ^c Mullen & Panario 2013, p. 283
^ Lidl & Niederreitter 1983, p. 356
^ 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
^ Mullen & Panario 2013, p. 287
^ Mullen & Panario 2013, p. 288

References

Brewer, B. W. (1961), "On certain character sums", Transactions of the American Mathematical Society, 99 (2): 241–245, doi:10.2307/1993392, ISSN 0002-9947, JSTOR 1993392, 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. 11 (1/6). The Annals of Mathematics: 65–120, 161–183. doi:10.2307/1967217. ISSN 0003-486X. JFM 28.0135.03. JSTOR 1967217. {{cite journal}}: Invalid |ref=harv (help)
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. {{cite journal}}: Invalid |ref=harv (help)
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. {{cite book}}: Invalid |ref=harv (help)
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. (2001) [1994], "Dickson polynomials", Encyclopedia of Mathematics, EMS Press
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. {{cite journal}}: Invalid |ref=harv (help)
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. {{cite journal}}: Invalid |ref=harv (help)
Young, Paul Thomas (2002). "On modified Dickson polynomials" (PDF). Fibonacci Quarterly. 40 (1): 33–40.

[LN355-1] Lidl & Niederreiter 1983, p. 355

[MP283-2] Mullen & Panario 2013, p. 283

[LN356-3] Lidl & Niederreitter 1983, p. 356

[4] 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

[5] Mullen & Panario 2013, p. 287

[6] Mullen & Panario 2013, p. 288

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