Free differential Galois groups

Bachmayr, Annette; Harbater, David; Hartmann, Julia; Wibmer, Michael

doi:10.1090/tran/8352

Free differential Galois groups
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by Annette Bachmayr, David Harbater, Julia Hartmann and Michael Wibmer PDF

Trans. Amer. Math. Soc. 374 (2021), 4293-4308 Request permission

Abstract:

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 $\mathbb {Q}$. This is the first known case of the twenty year old conjecture.

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