G-2012-101
The Injectivity Modules of a Tropical Map
BibTeX reference
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
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G-2012-101.pdf (300 KB)