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

Abstract : Jacobi's results on the computation of the order and of the normal forms of a differential systems are expressed in the framework of differential algebra. We give complete proofs according to Jacobi's arguments. The main result is Jacobi's bound: the order of a differential system P_i(x)=0 is not greater than the tropical determinant of the matrix (a_{i,j}), where a_{i,j} is the order of P_i in variable x_j. The order is precisely equal to O if and only if Jacobi's truncated determinant does not vanish. Jacobi also gave an algorithm to compute O in polynomial time, 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, under genericity hypotheses, by differentiating \ell_i times equation P_i. Some fundamental results about changes of orderings and the various normal forms a system may have, including differential resolvents, are also provided.
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https://hal.archives-ouvertes.fr/hal-01363020
Contributor : François Ollivier Connect in order to contact the contributor
Submitted on : Tuesday, September 7, 2021 - 11:54:09 AM
Last modification on : Thursday, September 9, 2021 - 3:34:12 AM

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  • HAL Id : hal-01363020, version 2

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François Ollivier. Jacobi's Bound: Jacobi's results translated in Kőnig's, Egerváry's and Ritt's mathematical languages. 2021. ⟨hal-01363020v2⟩

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