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 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 and the lower-bound m[k](q) (k)(q(q−1))/k for q≡1 (mod k). This is done by establishing a connection with the -solutions of a system of equations defined over . 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 .