2020/12/15 by Boris Adamczewski, Adamczewski, Boris, Colin Faverjon +1 · 5 citations
Computer Science · Mathematics · #11J81 11J85 11B85 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2012.08283
openalex publication_date 2020/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
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 Our main results go far beyond the existing literature, also surpassing those of two unpublished preprints the authors made available on the arXiv in 2018.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.