Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909216785039360 |
|---|---|
| author | Akian, Marianne Gaubert, Stephane Tavakolipour, Hanieh |
| author_facet | Akian, Marianne Gaubert, Stephane Tavakolipour, Hanieh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_05483 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields Akian, Marianne Gaubert, Stephane Tavakolipour, Hanieh Rings and Algebras Primary 12J15, 12J25, 15A80, 12D05, 16Y60, Secondary 14T10, 06F20, 16Y20 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. |
| title | Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields |
| topic | Rings and Algebras Primary 12J15, 12J25, 15A80, 12D05, 16Y60, Secondary 14T10, 06F20, 16Y20 |
| url | https://arxiv.org/abs/2301.05483 |