English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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

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