On the Carlitz rank of permutation polynomials
Aksoy, Esen and Çeşmelioğlu, Ayça and Meidl, Wilfried and Topuzoğlu, Alev (2009) On the Carlitz rank of permutation polynomials. Finite Fields and Their Applications, 15 (4). pp. 428-440. ISSN 1071-5797
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Official URL: http://dx.doi.org/10.1016/j.ffa.2009.02.006
A well-known result of Carlitz, that any permutation polynomial p(x) of a finite field F-q is a composition of linear polynomials and the monomial x(q-2). implies that V(x) can be represented by a polynomial P-n(x) = (...((a(0)x + a(1))(q-2) + a(2))(q-2)...+ a(n))(q-2) + a(n+1). for some n >= 0. The smallest integer n, such that P,,(x) represents p(x) is of interest since it is the least number of "inversions" x(q-2), needed to obtain p(x). We define the Carlitzrank of p(x) as n, and focus here on the problem of evaluating it. We also obtain results on the enumeration of permutations of F-q with a fixed Carlitz rank.
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