The relationship between the strength of a differential equation as introduced by Einstein, its Cartan characters, and its Hilbert polynomial is studied. Using the framework of formal theory previous results are extended to nonlinear equations of arbitrary order and to overdetermined systems. The problem of computing the number of arbitrary functions in the general solution is treated. Finally, the effect of gauge symmetries is considered.
REFERENCES
1.
G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences 81 (Springer-Verlag, New York, 1989).
2.
P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics 107 (Springer-Verlag, New York, 1986).
3.
4.
5.
6.
A. Einstein, The Meaning of Relativity, 5th ed. (Princeton University, Princeton, 1955).
7.
E. Cartan and A. Einstein, Lettres sur la Parallélisme Absolue 1929–1932, edited by R. Debever (Palais des Académies, Bruxelles, 1979).
8.
9.
10.
11.
12.
13.
E. Cartan, Les Systèmes Différentielles Extérieurs et leurs Applications Géométriques (Hermann, Paris, 1945).
14.
R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior Differential Systems, Mathematical Sciences Research Institute Publications 18 (Springer-Verlag, New York, 1991).
15.
E. Cartan, in Oeuvres Complètes d’Elie Cartan (Gauthier-Villais, Paris, 1953), Partie II, Vol. 2, pp. 1199–1230; [Original: Bull. Soc. Math. France 59, 88 (1931)].
16.
G. Carrà Ferro and S. V. Duzhin, Differential-algebraic and differential-geometric approach to the study of involutive symbols, Preprint University of Catania, Italy, 1992.
17.
J. F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups (Gordon and Breach, London, 1978).
18.
J. Schü, W. M. Seiler, and J. Calmet, Algorithmic methods for Lie pseudogroups, in Proceedings of the International Workshop Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, Acireale (Italy), 1992 (Kluwer, Dordrecht, to published).
19.
20.
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© 1994 American Institute of Physics.
1994
American Institute of Physics
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