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Jacobi's bound and normal forms computations. A historical survey

Jacobi is one of the most famous mathematicians of his century. His name is attached to many results in various fields of mathematics and his complete works in seven volumes have been available since the end of the XIXth century and are very often quoted in many papers. It is then surprising that some of his results may have fallen into oblivion, at least in part. We will try to describe some of Jacobi's results on ordinary differential equations and the available, published or unpublished material he left. We will then expose the selective interests of his followers and their own contributions. There are in fact many interrelated results: a bound on the order of a differential system, a necessary and sufficient condition, given by a determinant, for the bound to be reached, an algorithm to compute the bound in polynomial time, and processes for computing normal forms using as few derivatives as possible. We give for all of them the form under which they could have been proved or rediscovered, sometimes independently of Jacobi's findings. In conclusion, we give the state of the art and suggest some possible applications of Jacobi's bound to improve some algorithms in differential algebra.