Equations in virtually abelian groups: Languages and growth

This paper explores the nature of the solution sets of systems of equations in virtually
abelian groups. We view this question from two angles. From a formal language perspective, we
prove that the set of solutions to a system of equations forms an EDT0L language, with respect
to a natural normal form. Looking at growth, we show that the growth series of the language
of solutions is rational. Furthermore, considering the set of solutions as a set of tuples of group
elements, we show that it has rational relative growth series with respect to any nite generating
set.