|
|
| Line 1: |
Line 1: |
| In [[mathematics]], the '''Dickson polynomials''' (or '''Brewer polynomials'''), denoted ''D''<sub>''n''</sub>(''x'',α), form a [[polynomial sequence]] introduced by {{harvs|txt=yes|authorlink=Leonard Eugene Dickson|first=L. E. |last=Dickson|year= 1897}} and rediscovered by {{harvtxt|Brewer|1961}} in his study of [[Brewer sum]]s.
| | Hi there! :) My name is Candra, I'm a student studying Arts and Sciences from Sainte-Anne, France.<br><br>Take a look at my web blog [http://thaawakening.com/index.php?mod=users&action=view&id=18649 wordpress backup plugin] |
| | |
| Over the complex numbers, Dickson polynomials are essentially equivalent to [[Chebyshev polynomial]]s 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 polynomial]]s''': polynomials acting as permutations of finite fields.
| |
| | |
| ==Definition==
| |
| ''D''<sub>0</sub>(''x'',α) = 2, and for ''n'' > 0 Dickson polynomials (of the first kind) are given by
| |
| | |
| :<math>D_n(x,\alpha)=\sum_{p=0}^{\lfloor n/2\rfloor}\frac{n}{n-p} \binom{n-p}{p} (-\alpha)^p x^{n-2p}. </math>
| |
| | |
| The first few Dickson polynomials are
| |
| | |
| :<math> D_0(x,\alpha) = 2 \,</math>
| |
| | |
| :<math> D_1(x,\alpha) = x \,</math>
| |
| | |
| :<math> D_2(x,\alpha) = x^2 - 2\alpha \,</math>
| |
| | |
| :<math> D_3(x,\alpha) = x^3 - 3x\alpha \,</math>
| |
| | |
| :<math> D_4(x,\alpha) = x^4 - 4x^2\alpha + 2\alpha^2. \,</math>
| |
| | |
| The Dickson polynomials of the second kind ''E''<sub>''n''</sub> are defined by
| |
| | |
| :<math>E_n(x,\alpha)=\sum_{p=0}^{\lfloor n/2\rfloor}\binom{n-p}{p} (-\alpha)^p x^{n-2p}. </math>
| |
| 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
| |
| | |
| :<math> E_0(x,\alpha) = 1 \,</math>
| |
| | |
| :<math> E_1(x,\alpha) = x \,</math>
| |
| | |
| :<math> E_2(x,\alpha) = x^2 - \alpha \,</math>
| |
| | |
| :<math> E_3(x,\alpha) = x^3 - 2x\alpha \,</math>
| |
| | |
| :<math> E_4(x,\alpha) = x^4 - 3x^2\alpha + \alpha^2. \,</math>
| |
| | |
| ==Properties==
| |
| The ''D''<sub>''n''</sub> satisfy the identities
| |
| | |
| :<math>D_n(u + \alpha/u,\alpha) = u^n + (\alpha/u)^n \, ; </math>
| |
| :<math>D_{mn}(x,\alpha) = D_m(D_n(x,\alpha),\alpha^n) \, . </math>
| |
| | |
| For ''n''≥2 the Dickson polynomials satisfy the [[recurrence relation]]
| |
| | |
| :<math>D_n(x,\alpha) = xD_{n-1}(x,\alpha)-\alpha D_{n-2}(x,\alpha) \, </math>
| |
| :<math>E_n(x,\alpha) = xE_{n-1}(x,\alpha)-\alpha E_{n-2}(x,\alpha). \, </math>
| |
| | |
| The Dickson polynomial ''D''<sub>''n''</sub> = ''y'' is a solution of the [[ordinary differential equation]]
| |
| :<math>(x^2-4\alpha)y'' + xy' - n^2y=0 \, </math>
| |
| and the Dickson polynomial ''E''<sub>''n''</sub> = ''y'' is a solution of the differential equation
| |
| :<math>(x^2-4\alpha)y'' + 3xy' - n(n+2)y=0. \, </math>
| |
| Their [[Generating function#Ordinary gnerating function|ordinary generating function]]s are
| |
| :<math>\sum_nD_n(x,\alpha)z^n = \frac{2-xz}{1-xz+\alpha z^2} \, </math>
| |
| :<math>\sum_nE_n(x,\alpha)z^n = \frac{1}{1-xz+\alpha z^2}. \, </math>
| |
| | |
| == Links to other polynomials ==
| |
| * Dickson polynomials over the complex numbers are related to [[Chebyshev polynomial]]s ''T''<sub>''n''</sub> and ''U''<sub>''n''</sub> by
| |
| :<math>D_n(2xa,a^2)= 2a^{n}T_n(x) \, </math>
| |
| :<math>E_n(2xa,a^2)= a^{n}U_n(x). \, </math>
| |
| Crucially, the Dickson polynomial ''D''<sub>''n''</sub>(''x'',''a'') can be defined over rings in which ''a'' is not a square, and over rings of characteristic 2; in these cases, ''D''<sub>''n''</sub>(''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 polynomials|Fibonacci]] and [[Lucas polynomials]].
| |
| * The Dickson polynomials with parameter α = 0 give monomials:
| |
| :<math>D_n(x,0) = x^n \, . </math>
| |
| | |
| ==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''<sub>''n''</sub>(''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''<sup>2</sup>−1.<ref name=LN356>Lidl & Niederreiter (1997) p.356</ref>
| |
| | |
| M. {{harvtxt|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). <!-- See talk page.---> This assertion has become known as Schur's conjecture, although in fact Schur did not make this conjecture. <!-- See talk page ---> Since Fried's paper contained numerous errors, a corrected account was given by G. {{harvtxt|Turnwald|1995}}, and subsequently P. {{harvtxt|Müller|1997}} gave a simpler proof along the lines of an argument due to Schur.
| |
| | |
| Further, P. {{harvtxt|Müller|1997}} proved that any permutation polynomial over the finite field '''F'''<sub>''q''</sub> whose degree is simultaneously coprime to ''q''−1 and less than ''q''<sup>1/4</sup> must be a composition of Dickson polynomials and linear polynomials.
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| *{{Citation | last1=Brewer | first1=B. W. | title=On certain character sums | url=http://www.jstor.org/stable/1993392 | mr=0120202 | zbl=0103.03205 | year=1961 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=99 | pages=241–245}}
| |
| *{{cite journal | first=L.E. | last=Dickson | authorlink=Leonard Eugene Dickson | title=The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group I,II | journal=Ann. Of Math. | volume=11 | year=1897 | pages=65–120; 161–183 | doi=10.2307/1967217 | issue=1/6 | publisher=The Annals of Mathematics | ref=harv | jstor=1967217 | jfm=28.0135.03 | issn=0003-486X}}
| |
| *{{cite journal | first=Michael | last=Fried | title=On a conjecture of Schur | journal=Michigan Math. J. | volume= 17 | year=1970 | pages=41–55 | url=http://projecteuclid.org/euclid.mmj/1029000374 | doi=10.1307/mmj/1029000374 | ref=harv | mr=0257033 | zbl=0169.37702 | issn=0026-2285}}
| |
| *{{Cite book
| |
| |last=Lidl|first= R.|last2=Mullen|first2= G. L.|last3= Turnwald|first3= G.
| |
| |title=Dickson polynomials
| |
| |series=Pitman Monographs and Surveys in Pure and Applied Mathematics|volume= 65|publisher= Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York |year=1993 | isbn=0-582-09119-5 | ref=harv | zbl=0823.11070 | mr=1237403}}
| |
| * {{cite book | zbl=0866.11069 | last1=Lidl | first1=Rudolf | last2=Niederreiter | first2=Harald | title=Finite fields | edition=2nd | series=Encyclopedia of Mathematics and Its Applications | volume=20 | publisher=[[Cambridge University Press]] | year=1997 | isbn=0-521-39231-4 }}
| |
| *{{springer|id=D/d120140|first=Gary L.|last= Mullen}}
| |
| *{{cite journal | first=Peter | last=Müller | title=A Weil-bound free proof of Schur's conjecture | journal = Finite Fields Appl. |volume = 3 | year=1997 | pages=25–32 | url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WFM-45M8VYC-J&_user=99318&_coverDate=01%2F31%2F1997&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000007678&_version=1&_urlVersion=0&_userid=99318&md5=b40a5d38475bafcd5b0df1826f48680f&searchtype=a| zbl=0904.11040 | doi=10.1006/ffta.1996.0170 | ref=harv}}
| |
| * {{cite book | first1=Thermistocles M. | last1=Rassias | first2=H.M. | last2=Srivastava | first3=A. | last3=Yanushauskas | title=Topics in Polynomials of One and Several Variables and Their Applications: A Legacy of P.L.Chebyshev | publisher=World Scientific | year=1991 | isbn=981-02-0614-3 | pages=371–395 }}
| |
| *{{cite journal|first=Gerhard |last=Turnwald|title=On Schur's conjecture|journal= J. Austral. Math. Soc. Ser. A |volume= 58 |year=1995|pages=312–357|url=http://journals.cambridge.org./action/displayFulltext?type=1&pdftype=1&fid=4986396&jid=JAZ&volumeId=58&issueId=&aid=4986388|ref=harv|mr=1329867|doi=10.1017/S1446788700038349|issue=03|zbl=0834.11052}}
| |
| *{{cite journal|first1=Paul Thomas|last1=Young
| |
| |title=On modified Dickson polynomials |year=2002
| |
| |journal=Fib. Quaterly | volume=40 | pages=33-40 | number=1
| |
| |url=http://www.fq.math.ca/Scanned/40-1/young.pdf}}
| |
| | |
| [[Category:Polynomials]]
| |