Carlitz–Wan conjecture

In the field of mathematics, the Carlitz–Wan conjecture classifies the possible degrees of exceptional polynomials over a finite field F_q of q elements. Recall that a polynomial f(x) in F_q[x] of degree d is called exceptional over F_q if every irreducible factor (differing from x − y) of (f(x) − f(y))/(x − y) over F_q will become reducible over the algebraic closure of F_q. If q > d⁴, then f(x) is exceptional if and only if f(x) is a permutation polynomial over F_q. The Carlitz–Wan conjecture states that there are no exceptional polynomials of degree d over F_q 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

^ Fried, M.; Guralnick, R.; Saxl, J. (1993), "Schur covers and Carlitz's conjecture", Israel J. Math., 82: 157–225.
^ Wan, Daqing (1993), "A generalization of the Carlitz conjecture", in Finite Fields, Coding Theory and Advances in Communications and Computing: 431–432 {{citation}}: Italic or bold markup not allowed in: |journal= (help).
^ 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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[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 {{citation}}: Italic or bold markup not allowed in: |journal= (help).

[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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