Jacobi's Bound. Jacobi's results translated in K{Ö}nig's, Egerv{á}ry's and Ritt's mathematical languages

Abstract

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 {\it Jacobi's bound}, still conjectural in the general case: the order of a differential system $P_1,\dots ,P_n$ is not greater than the maximum $\mathcal{O}$ of the sums ${\sum }_{i=1}^n{a}_{i,\sigma (i)}$, for all permutations $\sigma$ of the indices, where $a_{i,j}:={\mathrm{ord}}_{x_j}{P}_i$, \emph{viz.}\ the \emph{tropical determinant of the matrix $(a_{i,j})$}. The order is precisely equal to $\mathcal{O}$ iff Jacobi's \emph{truncated determinant} does not vanish. Jacobi also gave a polynomial time algorithm to compute $\mathcal{O}$, similar to Kuhn's "Hungarian method" and some variants of shortest path algorithms, related to the computation of integers ${\ell }_i$ such that a normal form may be obtained, in the generic case, by differentiating ${\ell }_i$ 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.

104 pages, 10 figures, index of words and names, index of notations

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