On the cycle structure of permutation polynomials
Çeşmelioğlu, Ayça (2008) On the cycle structure of permutation polynomials. [Thesis]
Official URL: http://192.168.1.20/record=b1225695 (Table of Contents)
L. Carlitz observed in 1953 that for any a € F*q, the transposition (0 a) can be represented by the polynomial Pa(x) = -a(((x - a)[q-2] + a-)[q-2] - a)[q-2] which shows that the group of permutation polynomials over Fq is generated by the linear polynomials ax + b, a, b € Fq, a≠0, and x[q-2]. Therefore any permutation polynomial over Fq can be represented as Pn = (...((ax + a)[q-2] +a) [q-2] ... + a[n])[q-2] + a[n+1], for some n ≥ 0. In this thesis we study the cycle structure of permutation polynomials Pn, and we count the permutations Pn, n ≤ 3, with a full cycle. We present some constructions of permutations of the form Pn with a full cycle for arbitrary n. These constructions are based on the so called binary symplectic matrices. The use of generalized Fibonacci sequences over Fq enables us to investigate a particular subgroup of Sq, the group of permutations on Fq. In the last chapter we present results on this special group of permutations.
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