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 $P_{1}, \ldots, P_{n}$ is not greater than the maximum $\cO$ of the sums $\sum_{i=1}^{n} a_{i,\sigma(i)}$, for all permutations $\sigma$ of the indices, where $a_{i,j}:={\rm ord}_{x_{j}}P_{i}$, \emph{viz.}\ the \emph{tropical determinant of the matrix $(a_{i,j})$}. The order is precisely equal to $\cO$ iff Jacobi's \emph{truncated determinant} does not vanish. Jacobi also gave a polynomial time algorithm to compute $\cO$, similar to Kuhn's \index{Hungarian method}``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.


翻译:Jacobi 计算差异系统顺序和正常形式的计算结果不大于 $(cO) $(sum) =1@n} a,\\\ sigma(i)}$(l) $) 。在准常规情况下,我们根据 Jacobi 的论点提供完整的证明。主要结果为 $(i) cocobi 的框框 }, 在一般情况下, 仍然推测 : 差分系统的顺序 $(P) 1},\ ldots, P ⁇ n} $(美元) 不大于 $(c) 的总额上限 $(c) 。 如果 i,\\ squm(i) =quality(i) $(i) $(lgma) 美元), 对于所有指数的变形($(sgma) ) 美元, 我们给出了完整的证明。 $(rm) rm=(d)\\\\\\\ j) P ⁇ (i) commax(ral) ralal) ral) ral ral-x(x) (a) ) (x) (a) (a) (a) (a) (a) (a) (ax) (a) (a) (a) (a) (a) (a) (c) (a) (c) (c) (c) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a (a) (a) (a) (a) (a) (a) (a(a) (a) (a) (a) (a) (a) (a) (a) (a)) (a) (a) (a) (a) (a) (a) (a) (a) (a(a(a))) (a) (a(a)))) (a) (a(a(a(a(a(a(a(a(a(a(a(a))))

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