  The symmetrized tropical semiring is an extension of the tropical semifield,
initially introduced to solve tropical linear systems using Cramer's rule. It
is equivalent to the real tropical hyperfield, which has been used in the study
of tropicalizations of semialgebraic sets. Polynomials over the symmetrized
tropical semiring, and their factorizations, were considered by Quadrat.
Recently, Baker and Lorscheid introduced a notion of multiplicity for the roots
of univariate polynomials over hyperfields. In the special case of the
hyperfield of signs, they related multiplicities with Descarte's rule of sign
for real polynomials. We investigate here the factorizations of univariate
polynomial functions over symmetrized tropical semirings, and relate them with
the multiplicities of roots over these semirings. We deduce a Descartes' rule
for "signs and valuations", which applies to polynomials over a real closed
field with a convex valuation and an arbitrary (divisible) value group. We show
in particular that the inequality of the Descartes' rule is tight when the
value group is non-trivial. This extends to arbitrary value groups a
characterization of Gunn in the rank one case, answering also to the tightness
question. Our results are obtained using the framework of semiring systems
introduced by Rowen, together with model theory of valued fields.
