Mahler's method in several variables and finite automata

Abstract

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 Our main results go far beyond the existing literature, also surpassing those of two unpublished preprints the authors made available on the arXiv in 2018.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.

Addendum to: Mahler's method in several variables and finite automata
Boris Adamczewski, Colin Faverjon
Annals of Mathematics1 Sep 2026Peer-reviewedCC BY-NC-ND 4.0Full text1 citedAlgebra & number theory Combinatorics & discrete mathematicsHAL
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BibTeX RIS

BibTeX

@article{adamczewski2026mahlers,
  title = {{Mahler's method in several variables and finite automata}},
  author = {Boris Adamczewski and Colin Faverjon},
  journal = {Annals of Mathematics},
  year = {2026},
  month = oct,
  doi = {10.4007/annals.2026.204.2.1},
  url = {https://hal.science/hal-04763287v2},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - Mahler's method in several variables and finite automata
AU  - Boris Adamczewski
AU  - Colin Faverjon
PY  - 2026
DA  - 2026/10/21
JO  - Annals of Mathematics
DO  - 10.4007/annals.2026.204.2.1
UR  - https://hal.science/hal-04763287v2
AB  - 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 Our main results go far beyond the existing literature, also surpassing those of two unpublished preprints the authors made available on the arXiv in 2018.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.
SN  - 0003-486X
ER  -

Abstract

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