Additive functions for number systems in function fields

Let Fq be a finite field with q elements and p ∈ Fq [X, Y ]. In this paper we study properties of additive functions with respect to number systems which are defined in the ring Fq [X, Y ]/p(X, Y )Fq [X, Y ]. Our results comprise distribution results, exponential sum estimations as well as a version of Waring’s Problem restricted by such additive functions. Similar results have been shown for q-are number systems as well as number systems in finite fields in the sense of Kovács and Pethő. In the proofs of the results contained in the present paper new difficulties occur because the “fundamental domains” associated to the number systems studied here have a complicated structure.