Given a permutation sigma of the element of a finite field F_q, the permutation polynomial f_sigma in F_q[x] is the unique polynomial with degree less than q-1 that has the property that f_sigma(t)=sigma(t) for every t in F_q. We consider the natural question of enumerating the permutations in a given conjugacy class for which the permutation polynomial has degree less than q-2. we give formulas that ewxted existing ones. Furthermore for the case of k-cycles, we consider the harder problem of enumerating the permutations within a given conjugacy class for which the permutation polynomial has minimal degree. After giving an upper bound and a lower bound (for q congruent to 1 mod k) we consider various examples in which interesting Galois properties arise.
On the Enumeration of Permutation Polynomials / Malvenuto, Claudia; Pappalardi, Francesco. - STAMPA. - 20(2000), pp. 233-240. [E233232].
On the Enumeration of Permutation Polynomials
MALVENUTO, Claudia;
2000
Abstract
Given a permutation sigma of the element of a finite field F_q, the permutation polynomial f_sigma in F_q[x] is the unique polynomial with degree less than q-1 that has the property that f_sigma(t)=sigma(t) for every t in F_q. We consider the natural question of enumerating the permutations in a given conjugacy class for which the permutation polynomial has degree less than q-2. we give formulas that ewxted existing ones. Furthermore for the case of k-cycles, we consider the harder problem of enumerating the permutations within a given conjugacy class for which the permutation polynomial has minimal degree. After giving an upper bound and a lower bound (for q congruent to 1 mod k) we consider various examples in which interesting Galois properties arise.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.