A class of exceptional polynomials

Cohen, Stephen D.; Matthews, Rex W.

doi:10.1090/S0002-9947-1994-1272675-0

A class of exceptional polynomials
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by Stephen D. Cohen and Rex W. Matthews PDF

Trans. Amer. Math. Soc. 345 (1994), 897-909 Request permission

Abstract:

We present a class of indecomposable polynomials of non prime-power degree over the finite field of two elements which are permutation polynomials on infinitely many finite extensions of the field. The associated geometric monodromy groups are the simple groups $PS{L_2}({2^k})$, where $k \geq 3$ and odd. (The first member of this class was previously found by P. Müller [17]. This realises one of only two possibilities for such a class which remain following deep work of Fried, Guralnick and Saxl [7]. The other is associated with $PS{L_2}({3^k})$, $k \geq 3$ , and odd in fields of characteristic 3.

References

J. V. Brawley, L. Carlitz, and Jack Levine, Scalar polynomial functions on the $n\times n$ matrices over a finite field, Linear Algebra Appl. 10 (1975), 199–217. MR 376625, DOI 10.1016/0024-3795(75)90069-5
Leonard Carlitz, On factorable polynomials in several indeterminates, Duke Math. J. 2 (1936), no. 4, 660–670. MR 1545956, DOI 10.1215/S0012-7094-36-00256-9
Stephen D. Cohen, The distribution of polynomials over finite fields, Acta Arith. 17 (1970), 255–271. MR 277501, DOI 10.4064/aa-17-3-255-271
Stephen D. Cohen, Exceptional polynomials and the reducibility of substitution polynomials, Enseign. Math. (2) 36 (1990), no. 1-2, 53–65. MR 1071414
Stephen D. Cohen, Proof of a conjecture of Chowla and Zassenhaus on permutation polynomials, Canad. Math. Bull. 33 (1990), no. 2, 230–234. MR 1060378, DOI 10.4153/CMB-1990-036-3
Stephen D. Cohen, Permutation polynomials and primitive permutation groups, Arch. Math. (Basel) 57 (1991), no. 5, 417–423. MR 1129514, DOI 10.1007/BF01246737
Michael D. Fried, Robert Guralnick, and Jan Saxl, Schur covers and Carlitz’s conjecture, Israel J. Math. 82 (1993), no. 1-3, 157–225. MR 1239049, DOI 10.1007/BF02808112
Marie Henderson and Rex Matthews, Permutation properties of Chebyshev polynomials of the second kind over a finite field, Finite Fields Appl. 1 (1995), no. 1, 115–125. MR 1334629, DOI 10.1006/ffta.1995.1008
N. S. James and R. Lidl, Permutation polynomials on matrices, Linear Algebra Appl. 96 (1987), 181–190. MR 910993, DOI 10.1016/0024-3795(87)90343-0
R. Lidl and R. W. Matthews, GALOIS: an algebra microcomputer package, Congr. Numer. 66 (1988), 145–156. Nineteenth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Baton Rouge, LA, 1988). MR 992897
Rudolf Lidl and Gary L. Mullen, Unsolved Problems: When Does a Polynomial Over a Finite Field Permute the Elements of the Field?, Amer. Math. Monthly 95 (1988), no. 3, 243–246. MR 1541277, DOI 10.2307/2323626
R. Lidl, G. L. Mullen, and G. Turnwald, 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, 1993. MR 1237403
Rudolf Lidl and Harald Niederreiter, Finite fields, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997. With a foreword by P. M. Cohn. MR 1429394

Permutation polynomials in one and several variables

W. H. Mills, The degrees of the factors of certain polynomials over finite fields, Proc. Amer. Math. Soc. 25 (1970), 860–863. MR 263783, DOI 10.1090/S0002-9939-1970-0263783-0
Gary L. Mullen, Permutation polynomials over finite fields, Finite fields, coding theory, and advances in communications and computing (Las Vegas, NV, 1991) Lecture Notes in Pure and Appl. Math., vol. 141, Dekker, New York, 1993, pp. 131–151. MR 1199828
Peter Müller, New examples of exceptional polynomials, Finite fields: theory, applications, and algorithms (Las Vegas, NV, 1993) Contemp. Math., vol. 168, Amer. Math. Soc., Providence, RI, 1994, pp. 245–249. MR 1291433, DOI 10.1090/conm/168/01704
T. Tsuzuku, Finite groups and finite geometries, Cambridge Tracts in Mathematics, vol. 70, Cambridge University Press, Cambridge-New York, 1982. Translated from the Japanese by A. Sevenster and T. Okuyama [Tetsuro Okuyama]. MR 645725
Da Qing Wan, A $p$-adic lifting lemma and its applications to permutation polynomials, Finite fields, coding theory, and advances in communications and computing (Las Vegas, NV, 1991) Lecture Notes in Pure and Appl. Math., vol. 141, Dekker, New York, 1993, pp. 209–216. MR 1199834