PERMUTATION POLYNOMIALS OF DEGREE 8 OVER FINITE FIELDS OF ODD CHARACTERISTIC

We give an algorithmic generalisation of Dickson's method of classifying permutation polynomials (PPs) of a given degree d over finite fields. Dickson's idea is to formulate from Hermite's criterion several polynomial equations satisfied by the coe fficients of an arbitrary PP of degree d. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for d > 6. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order q > 8. Such PPs exist if and only if q 2 f11; 13; 19; 23; 27; 29; 31 g and are explicitly listed in normalised form.