Carlitz–Wan conjecture

In mathematics, the Carlitz–Wan conjecture classifies the possible degrees of exceptional polynomials over a finite field F_q of q elements. 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) or (f(x) − f(y))/(x − y)) over F_q becomes 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 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]