Polynomial Values in Affine Subspaces of Finite Fields

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.