  Using the interpretation of the ultradiscretization procedure as a
non-Archimedean valuation, we use results of tropical geometry to show how
roots and poles manifest themselves in piece-wise linear systems as points of
non-differentiability. This will allow us to demonstrate a correspondence
between singularity confinement for discrete integrable systems and
ultradiscrete singularity confinement for ultradiscrete integrable systems.
