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

*
u _{i}*(

If *a _{i,j}*

*
H*=max_{σ}_{Î}_{Sn}_{
}Σ_{i} *a _{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 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 (a_{i,j}) be a square matrix n x n, find a permutation σ
such that Σ a_{i,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”.