The Carlitz rank of permutations of finite fields: A survey

L. Carlitz proved that any permutation polynomial f of a finite field F-q is a composition of linear polynomials and the monomials x(q-2). This result motivated the study of Carlitz rank of f, which is defined in 2009 to be the minimum number of inversions x(q-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. (C) 2013 Elsevier B.V. All rights reserved.