The Carlitz rank of permutations of finite fields: A survey

L. Carlitz proved that any permutation polynomial f   of a finite field FqFq is a composition of linear polynomials and the monomials xq−2xq−2. This result motivated the study of Carlitz rank of f  , which is defined in 2009 to be the minimum number of inversions xq−2xq−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.