Enumerating permutation polynomials II: k-cycles with minimal degree

We consider the function m[k](q) that counts the number of cycle permutations of a finite field Fq of fixed length k such that their permutation polynomial has the smallest possible degree. We prove the upper–bound m[k](q) ≤ (k−1)!(q(q−1))/k for char(Fq) > e(k−3)/e and the lower–bound m[k](q) ≥ φ(k)(q(q−1))/k for q ≡ 1 (mod k). This is done by establishing a connection with the Fq–solutions of a system of equations Ak defined over Z. As example, we give complete formulas for m[k](q) when k = 4, 5 and partial formulas for k = 6. Finally we analyze the Galois structure of the algebraic set Ak