One way of obtaining strong combinatorial min-max theorems is to study if a system of linear inequalities Ax≤b is totally dual integral (TDI). The MaxFlow-MinCut theorem of Ford and Fulkerson is one fine example. In that specific case, the matrix A involved is totally unimodular (TU), meaning all its nonzero subdeterminants are ±1, and that implies that the system is TDI. In fact, since the matrix is TU, the system satisfies an even stronger property: it is box-TDI, that is, it remains TDI when we impose any bounds to the variables. Totally equimodular matrices appear in this context, as a generalization of totally unimodular matrices.

A full row rank matrix is equimodular when all its nonzero subdeterminants of maximal size have the same absolute value, this value is called the equideterminant of the matrix.

A matrix is totally equimodular (TE) if for all its row induced matrices are equimodular when they are of full row rank. When a matrix is TU, it is TE, and all the involved equideterminants equal to 1.

More generally, systems of linear inequalities defined by TE matrices are box-TDI.

We identify a matrix with the set of vectors composed of its rows.A set of vectors is TE if the associated matrix is TE.

After giving several examples of TE matrices, I will give a first result: a decomposition theorem for TE matrices with full row rank.

Incidentally, we obtain that systems defined by TE matrices with 0,±1 entries are totally dual dyadic.

We will then leverage this decomposition theorem to analyze the cones generated by linearly independent TE sets of vectors, such cones are called te-cones.

We first provide their Hilbert basis, which is the minimal set of integer vectors of the cone that generate all its integer points with nonnegative combinations.

Finally, we will find a unimodular regular Hilbert triangulation for te-cones in (almost) all cases of the decomposition.

This is a joint work with Patrick Chervet and Roland Grappe (arXiv:2504:05930).