Jacobi's Bound. Jacobi's results translated in KÖnig's, Egerváry's and Ritt's mathematical languages
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Jacobi's results on the computation of the order and of the normal forms of a differential system are translated in the formalism of differential algebra. In the quasi-regular case, we give complete proofs according to Jacobi's arguments. The main result is Jacobi's bound, still conjectural in the general case: the order of a differential system P_1, …, P_n is not greater than the maximum O of the sums ∑_i=1^n a_i,σ(i), for all permutations σ of the indices, where a_i,j:= ord_x_jP_i, viz. the tropical determinant of the matrix (a_i,j). The order is precisely equal to O iff Jacobi's truncated determinant does not vanish. Jacobi also gave a polynomial time algorithm to compute O, similar to Kuhn's "Hungarian method" and some variants of shortest path algorithms, related to the computation of integers ℓ_i such that a normal form may be obtained, in the generic case, by differentiating ℓ_i times equation P_i. Fundamental results about changes of orderings and the various normal forms a system may have, including differential resolvents, are also provided.
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