The Number of Permutation Polynomials of a Given Degree Over a Finite Field

Abstract

We relate the number of permutation polynomials in F q [ x] of degree d≤ q−2 to the solutions ( x 1, x 2,…, x q) of a system of linear equations over F q , with the added restriction that x i ≠0 and x i ≠ x j whenever i≠ j. Using this we find an expression for the number of permutation polynomials of degree p−2 in F p [ x] in terms of the permanent of a Vandermonde matrix whose entries are the primitive pth roots of unity. This leads to nontrivial bounds for the number of such permutation polynomials. We provide numerical examples to illustrate our method and indicate how our results can be generalised to polynomials of other degrees.