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.

中文翻译：

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几乎阿贝尔群中的方程：语言和增长

本文探讨了虚拟阿贝尔群中方程组的解集的性质。我们从两个角度来看这个问题。从形式语言的角度来看，我们证明方程组的解集形成了 EDT0L 语言，相对于自然范式。从增长来看，我们表明解决方案语言的增长系列是合理的。此外，将解集视为一组元组元组，我们证明它对于任何有限生成集具有合理的相对增长序列。