The Injectivity Modules of a Tropical Map

In [Wagneur, E., Linear Algebra and its Application, 435, 1786-1795, 2011], we show that any \(m\)-dimensional tropical torsion module can be embedded in \(\underline{R}^d\), with \(d\leq m(m-1)\), and that \(m\)-dimensional tropical torsion modules are classified by a \(p\)-parameter family, with \(p\le (m-1)[m(m-1) - 1]\).

The aim of the paper is to revisit and extend some of these results by showing that - at least in the 3-dimensional case - the two upper bounds are tight. More precisely, we show that for \(m =3\), we can find tropical torsion modules which cannot be embedded in \(\underline{R}^d\) for \(d<6\), and that all the \(p= 2\cdot(2\cdot 3 -1)=10\) parameters required for the unambiguous specification of the 3 generators of \(m\) are necessary for the characterization of \(m\).