Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields

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 signed 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 Descartes' rule of signs for real polynomials. More recently, Gunn extended these multiplicity definitions and characterization to the setting of “whole idylls”. We investigate here the factorizations of univariate polynomial functions over symmetrized tropical semirings, and relate them to the multiplicities of roots over these semirings. We deduce 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 Descartes' rule is tight when the value group is non-trivial. This extends a characterization of Gunn from the rank one case to arbitrary value groups, also answering the tightness question. Our results are obtained using the framework of semiring systems introduced by Rowen, together with model theory of valued fields.