  In this paper we introduce a new approach and obtain new results for the
problem of studying polynomial images of affine subspaces of finite fields. We
improve and generalise several previous known results, and also extend the
range of such results to polynomials of degrees higher than the characteristic
of the field. Such results have a wide scope of applications similar to those
associated with their counterparts studying consecutive intervals over prime
fields instead of affine subspaces. Here we give only two immediate
consequences: to a bound on the size of the intersection of orbits of
polynomial dynamical systems with affine subspaces and to the Waring problem in
affine subspaces. These results are based on estimates for a certain new type
of exponential sums.
