Complete mappings and Carlitz rank

摘要

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any d >= 2 and any prime p > (d(2) - 3d + 4)(2) there is no complete mapping polynomial in F-p[x] of degree d. For arbitrary finite fields F-q, we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if n < left perpendicular q/2 right perpendicular, then there is no complete mapping in F-q [x] of Carlitz rank n of small linearity. We also determine how far permutation polynomials f of Carlitz rank n < left perpendicular q/2 right perpendicular are from being complete, by studying value sets of f + x. We provide examples of complete mappings if n = left perpendicular q/2 right perpendicular, which shows that the above bound cannot be improved in general.