Mahler Method and Vanishing Theorem

@misc{nishioka2026mahler,
  title = {{Mahler Method and Vanishing Theorem}},
  author = {Kumiko Nishioka and Taka-aki Tanaka and Yusuke Tanuma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23435},
  eprint = {2610.23435},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23435},
}

TY  - UNPB
TI  - Mahler Method and Vanishing Theorem
AU  - Kumiko Nishioka
AU  - Taka-aki Tanaka
AU  - Yusuke Tanuma
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.23435
UR  - https://arxiv.org/abs/2610.23435
AB  - This paper is motivated by the recent work of Adamczewski and Faverjon [5], who established the algebraic independence of values of Mahler functions satisfying linear functional equations. While their work is a landmark result, some proofs are long and complex. The primary aim of this paper is to simplify those proofs and provide new applications. In their proofs, a vanishing theorem for polynomials, rather than power series, is sufficient. We give a proof of the vanishing theorem without using Szemerédi's theorem or Corvaja-Zannier's theorem. In the case of a single transformation, we present a refined vanishing theorem, which allows us to simplify the proof of algebraic independence. We also provide an elementary proof of a vanishing theorem for q_i-Mahler functions with various q_i. As an application, we prove the algebraic independence of certain continued fractions, extending previous results by Hančl, Adamczewski, and Bundschuh and Väänänen in which only linear independence and transcendence had been established.
ER  -