On components of vectorial permutations of F-q(n)

Official URL: http://dx.doi.org/10.1016/j.ffa.2019.03.006

We consider vectorial maps F(x(1), ..., x(n)) = (f(1) (x(1), ..., x(n)), ..., f(n) (x(1), ..., x(n))) : F-q(n) bar right arrow F-q(n), which induce permutations of F-q(n) . We show that the degrees of the components f(1), f(2), ..., f(n) is an element of F-q [x(1), ..., x(n)] are at least 2 when 2 <= deg(F) = d < root q and d vertical bar (q - 1) Our proof uses an absolutely irreducible curve over F-q and the number of rational points on it that we relate to the cardinality of the value set of a polynomial.