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Transactions of the American Mathematical Society

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Regular extensions and algebraic relations between values of Mahler functions in positive characteristic


Author: Gwladys Fernandes
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11J81, 11J72, 11J85
DOI: https://doi.org/10.1090/tran/7798
Published electronically: June 10, 2019
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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.


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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

DOI: https://doi.org/10.1090/tran/7798
Keywords: Mahler's method, algebraic independence, linear independence, transcendence, regular extensions, function fields.
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.
Article copyright: © Copyright 2019 American Mathematical Society