  This paper studies the following question: given a piecewise-linear function,
find its minimal algebraic representation as a tropical rational signomial. We
put forward two different notions of minimality, one based on monomial length,
the other based on factorization length. We show that in dimension one, both
notions coincide, but this is not true in dimensions two or more. We prove
uniqueness of the minimal representation for dimension one and certain
subclasses of piecewise-linear functions in dimension two. As a proof step, we
obtain counting formulas and lower bounds for the number of regions in an
arrangement of tropical hypersurfaces, giving a small extension for a result by
Mont\'{u}far, Ren and Zhang. As an equivalent formulation, it gives a lower
bound on the number of vertices in a regular mixed subdivision of a Minkowski
sum, giving a small extension for Adiprasito's Lower Bound Theorem for
Minkowski sums.
