Polynomials and polynomial mappings on finite rings

We will investigate polynomials and polynomial functions over finite rings, such as the integers modulo a power of a prime. Concerning polynomial functions, we are interested in the structure of the group of polynomial permutations of a finite ring, and in the structure of projective limits of such groups. These we will investigate with respect to embedding wreath-products of cyclic groups, and eventually, maybe, arbitrary finite p-groups. Concerning the polynomials themselves, we investigate non-unique factorization into irreducibles. According to previous research the sets of lengths show a behaviour different from that in rings that are better known, such as orders in number fields or rings of integer-valued polynomials. Many traditional methods of the theory of non-unique factorization (transfer homomorphisms, conductors) are not applicable to polynomials over finite rings, so we will need to forge new methods.

We have pursued research in the theory of commutative rings, that is, basic research in mathematics, which other disciplines can build upon. We have developed topological methods in ring theory, characterized null ideals of matrices, investigated non-unique factorization in commutative rings, as well as groups of permutation polynomials on finite rings. Thanks to grant, four excellent young researchers have survived for a few years, and have each achieved an important step (doctorate or employment at a mathematics department of a university) in their respective scientific careers.