By Jean Dieudonné

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1 Introduction Differential equations are of central importance in all applied sciences. Whenever a continuous process is modelled mathematically, chances are high that the model is built upon them. Thus it is not surprising that almost any computer algebra system provides functions for analysing differential equations. Within MuPAD, this is largely the task of the library DETools. 0). We describe briefly the main components of the library in its current state: a fairly general package for Lie symmetry analysis and a algebraicgeometric completion package for overdetermined systems.

The main difficulty is still to find out the strata where P has to assume minima, and, of course, the construction of the function P which assumes the minima on the 'right' strata. , in the determination of minima of Higgs potentials. Acknowledgements The author is very grateful to Johannes Zimmer from technical university munich for showing me the connection of shape memory alloys to invariant theory and for the numerous useful discussions. References 1. , The Geometry of Spontaneous Symmetry Breaking.

Our new algorithm avoids these using some algebraic concepts. An open question is its relative efficiency compared with approaches based on differential forms [9]. 1 The Algorithm The general framework for our algorithm is the Cartan-Kuranishi completion in formal theory. A system of differential equations of order q is interpreted as a fibred submanifold Rq of a jet bundle J q £. Two natural operations are prolongation to higher order jet bundles and projection to lower order bundles which may produce integrability conditions.

### Cours de géométrie algébrique. / 1, Aperçu historique sur le développement de la géométrie algébrique by Jean Dieudonné

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