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On some arithmetic properties of Mahler functions

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Abstract

Mahler functions are power series f(x) with complex coefficients for which there exist a natural number n and an integer ℓ ≥ 2 such that f(x), f(xℓ),..., \(f({x^{{\ell ^{n - 1}}}}),f({x^{{\ell ^n}}})\) are linearly dependent over ℂ(x). The study of the transcendence of their values at algebraic points was initiated by Mahler around the’ 30s and then developed by many authors. This paper is concerned with some arithmetic aspects of these functions. In particular, if f(x) satisfies f(x) = p(x)f(xℓ) with p(x) a polynomial with integer coefficients, we show how the behaviour of f(x) mirrors on the polynomial p(x). We also prove some general results on Mahler functions in analogy with G-functions and E-functions.

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Authors and Affiliations

  1. Institut Fourier, Université Grenoble Alpes, F-38000, Grenoble, France

    Sara Checcoli & Julien Roques

Authors
  1. Sara Checcoli
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  2. Julien Roques
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Correspondence to Julien Roques.

Additional information

The first author’s work has been funded by the ANR project Gardio 14-CE25-0015.

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Checcoli, S., Roques, J. On some arithmetic properties of Mahler functions. Isr. J. Math. 228, 801–833 (2018). https://doi.org/10.1007/s11856-018-1785-y

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  • DOI: https://doi.org/10.1007/s11856-018-1785-y

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