Equations in virtually abelian groups: languages and growth

TL;DR

This paper investigates the solution sets of systems of equations in virtually abelian groups, demonstrating they form EDT0L languages and have rational growth series, thus linking formal language theory and growth properties.

Contribution

It proves that solution sets are EDT0L languages and their growth series are rational, providing new insights into the algebraic and language-theoretic structure of virtually abelian groups.

Findings

Solution sets form EDT0L languages

Growth series of solutions are rational

Solution sets have rational relative growth series

Abstract

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 finite generating set.