Jacobi’s Bound: Jacobi’s results translated in Kőnig’s, Egerváry’s and Ritt’s mathematical languages

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 P₁, … , P_n is not greater than the maximum O of the sums ∑i=1nai,σ(i), for all permutations σ of the indices, where ai,j:=ordxjPi, 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.