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

ui(x)=0.

If ai,j is the order of the ith equation in the jth variable, he claims that the order of the system is bounded by

H=maxσÎSn Σi ai,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 Σi maxj ai,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 equation in order to produce an auxiliary system, generically allowing to compute an equation depending only of one chosen variable.

These two paper 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 (ai,j) be a square matrix n x n,  find a permutation σ such that  Σ ai,σ(i) be maximal. Such an algorithm was only rediscovered in 1955 par Kuhn, who called it “Hungarian method”, having obtained it from results Kőnig and Egerváry. It is a well-known problem in mathematical economy, under the name of  “assignment problem”.