The Carlitz rank of permutations of finite fields: a survey

Official URL: http://dx.doi.org/10.1016/j.jsc.2013.07.004

L. Carlitz proved that any permutation polynomial f of a finite field Fq is a composition of linear polynomials and the monomials xq−2. This result motivated the study of Carlitz rank of f, which is defined in 2009 to be the minimum number of inversions xq−2, needed to obtain f, by E. Aksoy et al. We give a survey of results obtained so far on natural questions related to this concept and indicate a variety of applications, which emerged recently.