Involution - The Formal Theory of Differential Equations and its Applications in Computer Algebra
Reference: Springer-Verlag, Berlin/Heidelberg 2010, Algorithms and Computation in Mathematics, Vol. 24 (ISBN 3-642-01286-0)
Description: Cover

This book grew out of my habilitation thesis; however, it is not identical with it. Actually, the number of pages has doubled and the book covers topics not treated in my thesis (most notably the homological theory). On the other hand, one or two topics of the thesis have been omitted (in particular, numerical analysis).

The book tries to combine the formal theory of general systems of differential equations, i.e. including under- and overdetermined systems, with the algebraic theory of involutive bases. Thus some parts of it are written in a very geometric manner using the language of fibred manifolds and jet bundles, whereas other parts are of a purely algebraic nature with strong emphasis on computational commutative algebra. A number of appendices give at least an introduction to the basic notions and concepts of the various fields like differential geometry or commutative and homological algebra, so that also non-experts should have a chance to understand the parts in which they are not specialised.

Special highlights are the first presentation of the theory of involutive bases (with special emphasis on Pommaret bases and their application in algebraic geometry) in book form and a rigorous treatment of Vessiot's vector field approach to partial differential equations. More details can be found in the table of contents. Of course, nobody is perfect and thus there exists an errata file (containing not only corrections of errors but also some additional comments on various topics in the book).

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