Jacobi's Bound. Jacobi's results translated in K{\"O}nig's, Egerv{\'a}ry's and Ritt's mathematical languages

TL;DR

This paper translates Jacobi's classical results on differential systems into modern algebraic language, proving Jacobi's bound for the order of such systems and providing algorithms for their analysis.

Contribution

It reformulates Jacobi's results in differential algebra, proves Jacobi's bound in the quasi-regular case, and offers polynomial-time algorithms for computing system order and normal forms.

Findings

Jacobi's bound for the order of differential systems is proven in the quasi-regular case.

A polynomial-time algorithm for computing the bound is provided.

Fundamental results on system orderings and normal forms are established.

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 {\it 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 $\cal O$ of the sums $\sum_{i=1}^{n} a_{i,\sigma(i)}$, for all permutations $\sigma$ of the indices, where $a_{i,j}:={\rm ord}_{x_{j}}P_{i}$, \emph{viz.}\ the \emph{tropical determinant of the matrix $(a_{i,j})$}. The order is precisely equal to $\cal O$ iff Jacobi's \emph{truncated determinant} does not vanish. Jacobi also gave a polynomial time algorithm to compute $\cal O$, 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.