Picard-Vessiot theory and the Jacobian problem (Q1758964)

There are two definitions of Picard-Vessiot extension of partial differential fields, defined respectively in [\textit{E. R. Kolchin}, Proc. Am. Math. Soc. 3, 596--603 (1952; Zbl 0047.33303)] and in [\textit{M. van der Put} and \textit{M. Singer}, Galois theory of linear differential equations. Grundlehren der Mathematischen Wissenschaften 328. Berlin: Springer (2003; Zbl 1036.12008)]. The second one was further developed in [\textit{F. Heiderich}, Picard-Vessiot Theorie für lineare partielle Differentialgleichungen, Diplomarbeit, Fakultät für Mathematik und Informatik, Heidelberg University (2007)]. In the paper under review, these two definitions are proven to be equivalent, and an equivalent statement of the Jacobian conjecture in terms of Picard-Vessiot extensions is also presented.