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Dimension of the solutions space of PDEs. (English) Zbl 1200.35194

Calmet, J. (ed.) et al., Global integrability of field theories. Proceedings of GIFT 2006, Cockcroft Institute, Daresbury, UK, November 1–3, 2006. Karlsruhe: Universitätsverlag Karlsruhe (ISBN 3-86644-035-9/pbk). 5-25 (2006).
The authors discuss the dimensional characterization of the solutions space of a system of partial differential equations. The first question is about what should be called a solution. A choice of category, for example finitely differentiable, smooth, analytic and others, is basic. In this paper, the authors restrict the discussion to local or even formal solutions. They impose a topology on the solutions space and ignore, in the dimensional count, isolated and special solutions or their families. They take those of connected components, that have more parameters in. The number of parameters, on which a general solution depends, is a dimensional characteristic. Only (over)determined systems are considered. They calculate the dimensional characteristics of involutive systems and Cohen-Macaulay systems. The paper ends with a number of examples.
For the entire collection see [Zbl 1170.37001].

MSC:

35N10 Overdetermined systems of PDEs with variable coefficients
58A20 Jets in global analysis
58H10 Cohomology of classifying spaces for pseudogroup structures (Spencer, Gelfand-Fuks, etc.)
35A30 Geometric theory, characteristics, transformations in context of PDEs
13C14 Cohen-Macaulay modules
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