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 \(P_{1}, \ldots , P_{n}\) 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 \(a_{i,j}:=\mathrm{ord}_{x_{j}}P_{i}\), viz. the tropical determinant of the matrix \((a_{i,j})\). The order is precisely equal to \({{\mathcal {O}}}\) iff Jacobi’s 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.

Jacobi 关于微分系统的阶数和范式计算的结果被转化为微分代数的形式。在准正则情况下，我们根据 Jacobi 的论证给出完整的证明。主要结果是Jacobi 界，在一般情况下仍然是推测性的：微分系统的阶\(P_{1}, \ldots , P_{n}\)不大于最大值\({{\mathcal {O }}}\)的总和\(\sum _{i=1}^{n} a_{i,\sigma (i)}\)，对于索引的所有排列\(\sigma \)，其中\ (a_{i,j}:=\mathrm{ord}_{x_{j}}P_{i}\)，即。矩阵的热带行列式 \((a_{i,j})\). 如果 Jacobi 的截断行列式不消失，则该顺序正好等于\({{\mathcal {O}}}\) 。Jacobi 还给出了一个多项式时间算法来计算\({{\mathcal {O}}}\)，类似于库恩的“匈牙利方法”和一些最短路径算法的变体，与整数的计算有关\(\ell _{ i}\)使得在一般情况下，可以通过对\(\ell _{i}\)时间方程\(P_{i}\)进行微分来获得范式。还提供了关于顺序变化和系统可能具有的各种范式（包括差分解析器）的基本结果。