A theorem of Chevalley and Warning states that the number of solutions of a
system of polynomial equations over a finite field is divisible by its characteristic if the number of variables is strictly larger than the sum of the total degrees. We show
a generalisation of this theorem to functions between abelian p-groups. To describe the degree of a function between abelian groups we use a concept of functional degree, which is based on similar concepts used, e.g., by M. Vaughan-Lee and P. Mayr. We apply this theorem to functions on not necessarily commutative rings, finite fields and additive subgroups of finite fields to obtain new results and to retrieve some
already known improvements of the Chevalley Warning theorem.
This is joint research with Erhard Aichinger