### G-2012-101

# 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 `\(m\)`

.

Published **December 2012**
,
10 pages