论文标题
Mahler的方法中的几个变量和有限自动机
Mahler's method in several variables and finite automata
论文作者
论文摘要
我们从超越和代数独立性的角度从几个变量中开发了线性Mahler系统的理论，其中还包括处理与足够独立的矩阵变换相关的几种系统。我们的主要结果远远超出了现有文献，还超过了作者在2018年在Arxiv上提供的两个未发表的预印象的预印象。主要的新功能是，它们现在不使用定义相应的Mahler系统的矩阵而没有任何限制。结果，我们解决了有关在多个独立碱中数字扩展的几个问题。例如，我们证明，在两个多重独立的整数碱基中，没有非理性的实际数字可以自动，并且我们在自动机理论中对Cobham的定理提供了新的证明和广泛的代数概括。我们还提供了Nishioka定理的新证明和多元概括，这是Mahler方法的里程碑。
We develop a theory of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence, which also includes the possibility of dealing with several systems associated with sufficiently independent matrix transformations. Our main results go far beyond the existing literature, also surpassing those of two unpublished preprints the authors made available on the arXiv in 2018. The main new feature is that they apply now without any restriction on the matrices defining the corresponding Mahler systems. As a consequence, we settle several problems concerning expansions of numbers in multiplicatively independent bases. For instance, we prove that no irrational real number can be automatic in two multiplicatively independent integer bases, and we give a new proof and a broad algebraic generalization of Cobham's theorem in automata theory. We also provide a new proof and a multivariate generalization of Nishioka's theorem, a landmark result in Mahler's method.
