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.

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