Carlitz–Wan conjecture

In the field of mathematics, the Carlitz–Wan conjecture classifies the possible degrees of exceptional polynomials over a finite field Fq of q elements. Recall that a polynomial f(x) in Fq[x] of degree d is called exceptional over Fq if every irreducible factor (differing from x − y) of (f(x) − f(y))/(x − y) over Fq will become 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 (d, q − 1) >  1. In the special case that q is odd and d is even, this conjecture was proposed by Carlitz (1966) and proved by Fried–Guralnick–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

  1. ↑ Fried, M.; Guralnick, R.; Saxl, J. (1993), "Schur covers and Carlitz's conjecture", Israel J. Math., 82: 157–225.
  2. ↑ Wan, Daqing (1993), "A generalization of the Carlitz conjecture", in Finite Fields, Coding Theory and Advances in Communications and Computing: 431–432.
  3. ↑ Cohen, S.; Fried, M. (1995), "Lenstra's proof of the Carlitz–Wan conjecture on exceptional polynomials: an elementary version", Finite Fields Appl., 1 (3): 372–375.


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