In mathematics, the Carlitz–Wan conjecture classifies the possible degrees of exceptional polynomials over a finite field Fq of q elements. A polynomial f(x) in Fq[x] of degree d is called exceptional over Fq if every irreducible factor (differing from x − y) or (f(x) − f(y))/(x − y)) over Fq becomes reducible over the algebraic closure of Fq. If q > d4, then f(x) is exceptional if and only if f(x) is a permutation polynomial over Fq.
The Carlitz–Wan conjecture states that there are no exceptional polynomials of degree d over Fq if gcd(d, q − 1) > 1.
In the special case that q is odd and d is even, this conjecture was proposed by Leonard Carlitz (1966) and proved by Fried, Guralnick, and Saxl (1993).[1] The general form of the Carlitz–Wan conjecture was proposed by Daqing Wan (1993)[2] and later proved by Hendrik Lenstra (1995)[3]
References
Fried, Michael D.; Guralnick, Robert; Saxl, Jan (1993), "Schur covers and Carlitz's conjecture", Israel Journal of Mathematics, 82 (1–3): 157–225, doi:10.1007/BF02808112, MR 1239049, S2CID 18446871
Wan, Daqing (1993), "A generalization of the Carlitz conjecture", in Mullen, Gary L.; Shiue, Peter Jau-Shyong (eds.), Finite fields, Coding Theory, and Advances in Communications and Computing: Proceedings of the International Conference held at the University of Nevada, Las Vegas, Nevada, August 7–10, 1991, Lecture Notes in Pure and Applied Mathematics, 141, Marcel Dekker, Inc., New York, pp. 431–432, ISBN 0-8247-8805-2, MR 1199817
Cohen, Stephen D.; Fried, Michael D. (1995), "Lenstra's proof of the Carlitz–Wan conjecture on exceptional polynomials: an elementary version", Finite Fields and Their Applications, 1 (3): 372–375, doi:10.1006/ffta.1995.1027, MR 1341953
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