Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Regular extensions and algebraic relations between values of Mahler functions in positive characteristic
HTML articles powered by AMS MathViewer

by Gwladys Fernandes PDF
Trans. Amer. Math. Soc. 372 (2019), 7111-7140 Request permission

Abstract:

Let $\mathbb {K}$ be a function field of characteristic $p>0$. We have recently established the analogue of a theorem of Ku. Nishioka for linear Mahler systems defined over $\mathbb {K}(z)$. This paper is dedicated to proving the following refinement of this theorem. Let $f_{1}(z),\ldots , f_{n}(z)$ be $d$-Mahler functions such that $\overline {\mathbb {K}}(z)\left (f_{1}(z),\ldots , f_{n}(z)\right )$ is a regular extension over $\overline {\mathbb {K}}(z)$. Then, every homogeneous algebraic relation over $\overline {\mathbb {K}}$ between their values at a regular algebraic point arises as the specialization of a homogeneous algebraic relation over $\overline {\mathbb {K}}(z)$ between these functions themselves. If $\mathbb {K}$ is replaced by a number field, this result is due to B. Adamczewski and C. Faverjon as a consequence of a theorem of P. Philippon. The main difference is that in characteristic zero, every $d$-Mahler extension is regular, whereas in characteristic $p$, non-regular $d$-Mahler extensions do exist. Furthermore, we prove that the regularity of the field extension $\overline {\mathbb {K}}(z)\left (f_{1}(z),\ldots , f_{n}(z)\right )$ is also necessary for our refinement to hold. Besides, we show that when $p\nmid d$, $d$-Mahler extensions over $\overline {\mathbb {K}}(z)$ are always regular. Finally, we describe some consequences of our main result concerning the transcendence of values of $d$-Mahler functions at algebraic points.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11J81, 11J72, 11J85
  • Retrieve articles in all journals with MSC (2010): 11J81, 11J72, 11J85
Additional Information
  • Gwladys Fernandes
  • Affiliation: Université Claude Bernard Lyon 1, Institut Camille Jordan, 43, Boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
  • Email: fernandes@math.univ-lyon1.fr
  • Received by editor(s): August 21, 2018
  • Received by editor(s) in revised form: November 22, 2018
  • Published electronically: June 10, 2019
  • Additional Notes: This project received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under the Grant Agreement No. 48132.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 7111-7140
  • MSC (2010): Primary 11J81, 11J72, 11J85
  • DOI: https://doi.org/10.1090/tran/7798
  • MathSciNet review: 4024549