The paper generalizes Lazarus Fuchs' theorem on the solutions of complex ordinary linear differential equations with regular singularities to the case of ground fields of arbitrary characteristic, giving a precise description of the shape of each solution. This completes partial investigations started by Taira Honda and Bernard Dwork. The main features are the introduction of a differential ring $\mathcal{R}$ in infinitely many variables mimicking the role of the (complex) iterated logarithms, and the proof that adding these "logarithms" already provides sufficiently many primitives so as to solve any differential equation with regular singularity in $\mathcal{R}$. A key step in the proof is the reduction of the involved differential operator to an Euler operator, its normal form, to solve Euler equations in $\mathcal{R}$ and to lift their (monomial) solutions to solutions of the original equation. The first (and already very striking) example of this outset is the exponential function $\exp_p$ in positive characteristic, solution of $y' = y$. We prove that it necessarily involves all variables and we construct its explicit (and quite mysterious) power series expansion. Additionally, relations of our results to the Grothendieck-Katz $p$-curvature conjecture and related conjectures will be discussed.

翻译：本文将此论文推广了拉撒路·Fuchs关于具有正则奇点的复常微分线性方程解的定理，将其推广到任意特征基域的情形，并精确描述了每个解的形状。这完成了由Tai Honda和Bernard Dwork开启的部分研究。主要创新在于引入了一个含无穷多个变量的微分环$\mathcal{R}$，用以模拟（复数域上的）迭代对数，并证明添加这些“对数”已能提供足够多的原函数，从而在$\mathcal{R}$中求解任何具有正则奇点的微分方程。证明的关键步骤是将涉及到的微分算子化为欧拉算子（即正规型），在$\mathcal{R}$中求解欧拉方程，并将其（单项式）解提升为原方程的解。这一框架的首个（且已极具启发性的）例子是正特征下的指数函数$\exp_p$，其为方程$y' = y$的解。我们证明该解必然涉及所有变量，并给出了其显式（且相当神秘的）幂级数展开。此外，还将讨论我们的结果与Grothendieck-Katz $p$-曲率猜想及相关猜想之间的联系。