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Picard-Vessiot theory and the Jacobian problem. (English) Zbl 1282.12003
There are two definitions of Picard-Vessiot extension of partial differential fields, defined respectively in [E. R. Kolchin, Proc. Am. Math. Soc. 3, 596–603 (1952; Zbl 0047.33303)] and in [M. van der Put and 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 [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.

MSC:
12H05 Differential algebra
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References:
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