This paper is a 1st step in the direction of a better understanding of the structure of the so-called Mahler systems: we classify these systems over the field |$\mathscr{H}$| of Hahn series over |$\overline{{\mathbb{Q}}}$| and with value group |${\mathbb{Q}}$|⁠. As an application of (a variant of) our main result, we give an alternative proof of the following fact: if, for almost all primes |$p$|⁠, the reduction modulo |$p$| of a given Mahler equation with coefficients in |${\mathbb{Q}}(z)$| has a full set of algebraic solutions over |$\mathbb{F}_{p}(z)$|⁠, then the given equation has a full set of solutions in |$\overline{{\mathbb{Q}}}(z)$| (this is analogous to Grothendieck’s conjecture for differential equations).

中文翻译：

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论马勒系统的局部结构

本文是朝着更好地理解所谓马勒系统结构的方向迈出的第一步：我们在|$\mathscr{H}$|领域对这些系统进行分类。哈恩系列在|$\overline{{\mathbb{Q}}}$| 和价值组|${\mathbb{Q}}$|⁠。作为我们主要结果（的变体）的应用，我们给出了以下事实的替代证明：如果对于几乎所有素数|$p$|⁠，约减模|$p$| 系数在|${\mathbb{Q}}(z)$|中的给定马勒方程的 在|$\mathbb{F}_{p}(z)$|⁠ 上有一套完整的代数解，那么给定的方程在|$\overline{{\mathbb{Q}}} 上有一套完整的解(z)$| （这类似于格洛腾迪克对微分方程的猜想）。