Languages, groups and equations
Languages, groups and equations
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语言、群和方程
DOI：
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发表时间：
2023
期刊：
影响因子：
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通讯作者：
A. Levine
中科院分区：
文献类型：
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作者：
L. Ciobanu;A. Levine
该调查概述了在过去十年中所做的工作，以形式语言描述方程组的解。我们从Ciobanu， Diekert和Elder的工作开始，他们证明了自由群中方程系统的解可以用约简词表示为EDT0L语言。我们提供了他们的算法的草图，并描述了自由群的结果如何推广到双曲群。作为EDT0L语言的解决方案的特征是非常健壮的，并且许多组结构保留了这一点，正如Levine所示。该领域的最新进展是针对没有负曲率的群，如虚拟阿贝尔群、积分海森堡群或可溶的鲍姆slag- solitar群，其中描述解的方法与负曲率群不同。在虚阿贝尔群中，解集实际上是有理的，我们可以得到它们为$m$正则集。在海森堡群中，求解单个方程简化为理解二次丢番图方程的解，并使用数论技术。在Baumslag-Solitar群中，方法是组合的，并且着重于范式的相互作用来解决特定类型的方程。总之，EDT0L语言对许多重要的类群中看似高度复杂的集合给出了有效而简单的组合表征。
The survey provides an overview of the work done in the last 10 years to characterise solutions to equations in groups in terms of formal languages. We begin with the work of Ciobanu, Diekert and Elder, who showed that solutions to systems of equations in free groups in terms of reduced words are expressible as EDT0L languages. We provide a sketch of their algorithm, and describe how the free group results extend to hyperbolic groups. The characterisation of solutions as EDT0L languages is very robust, and many group constructions preserve this, as shown by Levine. The most recent progress in the area has been made for groups without negative curvature, such as virtually abelian, the integral Heisenberg group, or the soluble Baumslag-Solitar groups, where the approaches to describing the solutions are different from the negative curvature groups. In virtually abelian groups the solutions sets are in fact rational, and one can obtain them as $m$-regular sets. In the Heisenberg group producing the solutions to a single equation reduces to understanding the solutions to quadratic Diophantine equations and uses number theoretic techniques. In the Baumslag-Solitar groups the methods are combinatorial, and focus on the interplay of normal forms to solve particular classes of equations. In conclusion, EDT0L languages give an effective and simple combinatorial characterisation of sets of seemingly high complexity in many important classes of groups.
几乎阿贝尔群中的有理集：语言和增长
DOI：
10.48550/arxiv.2205.05621
发表时间：
2022
期刊：
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影响因子：
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作者：
Ciobanu L
通讯作者：
Ciobanu L
影响因子：
1
作者：
Ciobanu L
通讯作者：
Ciobanu L