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

TL;DR

This paper explores polynomial factorizations over the symmetrized tropical semiring, establishing a Descartes' rule of signs in this context and over ordered valued fields, extending classical results to new algebraic structures.

Contribution

It introduces a Descartes' rule of signs for polynomials over symmetrized tropical semirings and real closed fields with valuations, extending classical sign rules to these settings.

Findings

Established a Descartes' rule for signs and valuations.

Proved the inequality of Descartes' rule is tight for non-trivial value groups.

Extended Gunn's characterization to arbitrary value groups.

Abstract

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.