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 communication and computing, 2023-09, Vol.34 (5), p.793-885
• Main Author: Ollivier, François
• Format: Article
• Language: English
• Subjects: Algorithms; Artificial Intelligence; Canonical forms; Computation; Computer Hardware; Computer Science; History and Overview; Mathematical analysis; Mathematics; Original Paper; Permutations; Polynomials; Symbolic and Algebraic Manipulation; Symbolic Computation; Theory of Computation
• ISSN: 0938-1279; 1432-0622
• DOI: 10.1007/s00200-022-00547-6

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 , … , P n is not greater than the maximum O of the sums ∑ i = 1 n a i , σ ( i ) , for all permutations σ of the indices, where a i , j : = ord x j P i , 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.