® Translations

In two posthumous papers, Jacobi proposed a sharp bound on the order of a system of n ordinary differential equations in n variables

u_i(x)=0.

If a_i,j is the order of the i^th equation in the j^th variable, he claims that the order of the system is bounded by

H=max_σÎSn Σ_i a_i,σ(i).

We provide an English translation of the first of these texts, written in Latin. A French translation is available for the second one. The work of translation in English is in progress… One will also find above the original papers and the transcription of some related document from Jacobis Nachlaß.

It must be noticed that these papers contain many interesting and forgotten results. The first one contains, besides Jacobi’s bound, the first mention of what Ritt called the “differential analog of Bézout’s theorem”, that bounds the order of the system by Σ_i max_j a_i,j, but it seems to have been “well known” at that time for Jacobi does just recall it in the introduction. The second proves a bound on the minimal order one must differentiate the equations in order to compute a normal form by elimination. He also shows how to compute the orders of derivation of the system equations in order to produce an auxiliary system, generically allowing to compute an equation depending only of one chosen variable, that is a differential resolvent.

These two papers also show that the bound is reached if and only if an expression that Jacobi calls the truncated (determinans mancum ou determinans mutilatum) does not vanish.

Last, but not least, Jacobi exposes a polynomial time algorithm to solve the following problem: let (a_i,j) be a square matrix n x n,  find a permutation σ such that  Σ a_i,σ(i) be maximal. This algorithm relies on the computation of a canon. Such an algorithm was only rediscovered in 1955 par Kuhn, who called it “Hungarian method”, having obtained it from results Kőnig and Egerváry that relies on the notion of minimal cover. It is a well-known problem in mathematical economy, under the name of  “assignment problem”. Working with canon or minimal covers turn to be equivalent.

You can also consult our translations published in a special issue of AAECC  in 2009:

Carl Gustav Jacob  Jacobi,  “Looking for the order of a system of arbitrary ordinary differential equations. De investigando ordine systematis aequationum differentialium vulgarium cujuscunque”, translation from the Latin by F.O., AAECC 20, n° 1, 7-32, 2009. DOI Author's version ;

Carl Gustav Jacob  Jacobi,  “The reduction to normal form of a non-normal system of  differential equations. De aequationum differentialium systemate non normali ad formam normalem revocando”, translation from the Latin by F.O., AAECC 20, n° 1, 33-64, 2009. DOI Author's version ;

or an unpublished paper devoted to the story of Jacobi's bound:

F. O., “Jacobi's Bound and Normal Forms Computations”, Differential Algebra and Related Topics. Author's version ;

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