Noncommutative polynomial maps

Polynomial maps attached to polynomials of an Ore extension are naturally defi ned. In this setting we show the importance of pseudo-linear transformations and give some applications. In particular, factorizations of polynomials in an Ore extension over a fi nite fi eld F_q[t;S ], where S is the Frobenius automorphism, are translated into factorizations in the usual polynomial ring F_q[x].