Thanks, Joerg! This is a terrific set of links on this subject. At 12:57 AM 3/26/2016, Joerg Arndt wrote:
From my bib file. You may want to start with Lidl. Can send files if hard to get (personal email).
%\bibitem{}{James Alexander, Paul Hearding: %\jjbibtitle{A graph theoretic encoding of Lucas sequences}, %arXiv:1501.00061 [math.CO], \bdate{31-December-2014}, %URL: \url{http://arxiv.org/abs/1501.00061}.} %% comb. interpretation of Dickson polynomials
%\bibitem{}{Bijan Ansari, M.\ Anwar Hasan: %\jjbibtitle{Revisiting Finite Field Multiplication Using Dickson Bases}, % CACR 2007-06, \bdate{2007}. %URL: \url{http://www.cacr.math.uwaterloo.ca/}.}
%\bibitem{}{Cunsheng Ding: %\jjbibtitle{Cyclic Codes from Dickson Polynomials}, %arXiv:1206.4370v1 [cs.IT], \bdate{20-June-2012}. %URL: \url{http://arxiv.org/abs/1206.4370}.}
%\bibitem{}{Robert W.\ Fitzgerald, Joseph L.\ Yucas: %\jjbibtitle{Generalized reciprocals, factors of Dickson polynomials and generalized cyclotomic polynomials over finite fields}, %Finite Fields and Their Applications, vol.~13, no.~3, pp.~492-515, \bdate{July-2007}. %URL: \url{http://dx.doi.org/10.1016/j.ffa.2006.03.002}.}
%\bibitem{}{Robert W.\ Fitzgerald, Joseph L.\ Yucas: %\jjbibtitle{Explicit Factorizations of Cyclotomic and Dickson Polynomials over Finite Fields}, %Lecture Notes in Computer Science, vol.~4547, Springer-Verlag, \bdate{2007}. %URL: \url{}.}
%\bibitem{}{Joachim von zur Gathen, Jos\'{e} Luis Imana, \c{C}etin Kaya Ko\c{c} (eds.): %\jjbibtitle{Arithmetic of finite fields}, %second international workshop, WAIFI 2008, Siena, Italy, % Lecture Notes in Computer Science, vol.~5130, \bdate{2008}.
%\bibitem{}{M.\ Anwar Hasan, Christophe Negre: %\jjbibtitle{Subquadratic space complexity multiplication over binary fields with Dickson polynomial representation}, %Lecture Notes in Computer Science, vol.~5130, WAIFI 2008, Springer-Verlag, \bdate{2008}.}
%% ***** SUGGEST TO LOOK AT THIS ONE FIRST: ***** %\bibitem{}{Rudolf Lidl, G.\ L.\ Mullen, G.\ Turnwald: %\jjbibtitle{Dickson Polynomials}, %Longman, \bdate{1993}.
%% ***** SUGGEST TO LOOK AT THIS ONE AS WELL: ***** \bibitem{Intr2FF}{Rudolf Lidl, Harald Niederreiter: \jjbibtitle{Introduction to finite fields and their applications}, Cambridge University Press, revised edition, \bdate{1994}.}
%\bibitem{}{Rudolf Lidl, Gary L.\ Mullen: %\jjbibtitle{When does a polynomial over a finite field permute the elements of the field?}, %The American Mathematical Monthly, vol.~95, no.~3, pp.~243-246, \bdate{March-1988}.
%%: IRRED %\bibitem{}{Rudolf Lidl, Gary L.\ Mullen: %\jjbibtitle{When does a polynomial over a finite field permute the elements of the field?, II}, %The American Mathematical Monthly, vol.~100, no.~1, pp.~71-74, \bdate{January-1993}.
%\bibitem{dicksonbases}{Ronald C.\ Mullin, Ayan Mahalanobis: %\jjbibtitle{Dickson Bases and Finite Fields}, %URL: \url{http://www.cacr.math.uwaterloo.ca/techreports/2005/cacr2005-03.pdf}.
%\bibitem{dicksonnormal}{Alfred Scheerhorn: %\jjbibtitle{Dickson Polynomials, Completely Normal Polynomials and the Cyclic Module Structure of Specific Extensions of Finite Fields}, %Designs, Codes and Cryptography, vol.~9, pp.~193-202, \bdate{1996}.}
%\bibitem{}{Thomas Stoll: %\jjbibtitle{Complete decomposition of Dickson-type recursive polynomials and related Diophantine equations}, %J. Number Theory, vol.~128, 1157-1181, \bdate{2008}. %URL: \url{http://iecl.univ-lorraine.fr/~Thomas.Stoll/publ.html}.}
* Henry Baker <hbaker1@pipeline.com> [Mar 24. 2016 12:29]:
The really cool Chebfun computer algebra system represents polynomials in a Chebyshev basis instead of a "monomial" (power) basis:
www.chebfun.org
Among other things, there are "fast" algorithms for computing products of polys in a Chebyshev basis.
The recent discussions of "permutation polynomials" led to finding that some of these polynomials are *Dickson* polynomials -- i.e., Chebyshev polynomials!
Soooo, inspired by Chebfun, perhaps polynomials over finite fields should also be represented by Chebyshev/Dickson polynomials.
Such a representation inverts the discussion, because we are now representing (at least in some cases) non-permutation polynomials in terms of permutation polynomials.
Perhaps someone has already done this?