25-03-2022 | Original Paper
Jacobi’s Bound: Jacobi’s results translated in Kőnig’s, Egerváry’s and Ritt’s mathematical languages
Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 5/2023
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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 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.