Groupe d’études et de recherche en analyse des décisions

# The Injectivity Modules of a Tropical Map

## Edouard Wagneur

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 $$m$$.

, 10 pages