The origin of the theory of involutive systems of partial differential equations goes back to Élie Cartan, who studied them in terms of exterior differential systems. During the last fifty years, methods of homological algebra were applied with success to study such systems. In the first part of the book, the author reviews the theory of involutive systems from the point of view of partial differential equations (a relation to the approach by exterior differential systems is explained in appendix B). In the main part of the book (chapters 4 and 5) the author introduces a notion of D-analytic spaces, proves the finiteness theorem and proves the involutiveness of the system for a generic case. In such a way, the author gives a precise sense to the old statement of Cartan that “by a prolongation, a differential system becomes eventually involutive”. This is a small booklet with a very interesting content related to many questions studied recently in the realm of partial differential equations with algebraic or analytic coefficients.

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