Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz $p$-curvature conjecture.

翻译：给定一个系数在$\mathbb{Q}(x)$中的线性微分方程，一个重要的问题是判断其全部解空间是否由代数函数构成，或者至少是否存在某个特定解是代数函数。在呈现来自数学不同分支的启发性例子之后，我们以一种基础的方式介绍一种由Grothendieck在六十年代末期提出的、针对这些问题的优美的局部-全局算术方法。这种方法具有深刻的衍生影响，并导向至今尚未解决的Grothendieck-Katz $p$-曲率猜想。