Involution : the formal theory of differential equations and its applications in computer algebra
The book provides a self-contained account of the formal theory of general, i.e. also under- and overdetermined, systems of differential equations which in its central notion of involution combines geometric, algebraic, homological and combinatorial ideas. It presents for the first time in book form the theory of Pommaret bases, a special kind of Gröbner bases closely related to Koszul homology, and contains an extensive discussion of the existence and uniqueness of solutions of formally well-posed initial value problems and a novel presentation of Vessiot's dual version of the Cartan-Kähler theory. Special emphasis is put on a constructive approach leading to effective algorithms
1 online resource (xxii, 650 pages) : illustrations (some color)
9783642012877, 3642012876
663094102
1 Introduction
2 Formal Geometry of Differential Equations
3 Involution I: Algebraic Theory
4 Completion to Involution
5 Structure Analysis of Polynomial Modules
6 Involution II: Homological Theory
7 Involution III: Differential Theory
8 The Size of the Formal Solution Space
9 Existence and Uniqueness of Solutions
10 Linear Differential Equations
A Miscellaneous
B Algebra
C Differential Geometry
References
Glossary
Index
English