We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of differential embedding problems to the condition that the absolute differential Galois group is free as a proalgebraic group. Building on this, we prove Matzat's freeness conjecture in the case that the field of constants is algebraically closed of countably infinite transcendence degree over the rationals. This parallels results that have been shown in usual Galois theory, and it strengthens known results in differential Galois theory.

中文翻译：

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自由微分伽罗华群

我们研究了在特征为零的代数闭域上的有理函数域的绝对微分伽罗瓦群的结构。特别地，我们将微分嵌入问题的行为与绝对微分伽罗瓦群作为前代数群自由的条件联系起来。在此基础上，我们证明了 Matzat 的自由度猜想，即常数域在有理数上具有可数无限超越度的代数封闭。这与通常的伽罗瓦理论中显示的结果相似，并且加强了微分伽罗瓦理论中的已知结果。