Congruences and coordinate semirings of tropical varieties

Zur Izhakian*, Louis Rowen

*Corresponding author for this work

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence. Our main definition applies Zariski density to the algebraic structure of the coordinate semiring of an affine supertropical algebraic set, which we tie to tropical geometry, especially in connection with the dimension of an affine variety, obtaining the analogs of classical results from dimension theory including catenarity. The second approach, based on the layered structure, is given in the appendix.

Original languageEnglish
Pages (from-to)231-259
Number of pages29
JournalBulletin des Sciences Mathematiques
Volume140
Issue number3
Early online date29 Dec 2015
DOIs
Publication statusPublished - Apr 2016

Fingerprint

Semiring
Congruence
Tropical Geometry
Dimension Theory
Algebraic Set
Tie
Algebraic Structure
Correspondence
Analogue

Keywords

  • Admissible variety
  • Catenarity
  • Coordinate semiring
  • Dimension
  • Primary
  • Secondary
  • Supertropical algebra
  • Tropical geometry

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Congruences and coordinate semirings of tropical varieties. / Izhakian, Zur; Rowen, Louis.

In: Bulletin des Sciences Mathematiques, Vol. 140, No. 3, 04.2016, p. 231-259.

Research output: Contribution to journalArticle

@article{bb2fd19d9d524bf192a2e3f2ec85d3d5,
title = "Congruences and coordinate semirings of tropical varieties",
abstract = "In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence. Our main definition applies Zariski density to the algebraic structure of the coordinate semiring† of an affine supertropical algebraic set, which we tie to tropical geometry, especially in connection with the dimension of an affine variety, obtaining the analogs of classical results from dimension theory including catenarity. The second approach, based on the layered structure, is given in the appendix.",
keywords = "Admissible variety, Catenarity, Coordinate semiring, Dimension, Primary, Secondary, Supertropical algebra, Tropical geometry",
author = "Zur Izhakian and Louis Rowen",
year = "2016",
month = "4",
doi = "10.1016/j.bulsci.2015.12.001",
language = "English",
volume = "140",
pages = "231--259",
journal = "Bulletin des Sciences Mathematiques",
issn = "0007-4497",
publisher = "Elsevier",
number = "3",

}

TY - JOUR

T1 - Congruences and coordinate semirings of tropical varieties

AU - Izhakian, Zur

AU - Rowen, Louis

PY - 2016/4

Y1 - 2016/4

N2 - In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence. Our main definition applies Zariski density to the algebraic structure of the coordinate semiring† of an affine supertropical algebraic set, which we tie to tropical geometry, especially in connection with the dimension of an affine variety, obtaining the analogs of classical results from dimension theory including catenarity. The second approach, based on the layered structure, is given in the appendix.

AB - In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence. Our main definition applies Zariski density to the algebraic structure of the coordinate semiring† of an affine supertropical algebraic set, which we tie to tropical geometry, especially in connection with the dimension of an affine variety, obtaining the analogs of classical results from dimension theory including catenarity. The second approach, based on the layered structure, is given in the appendix.

KW - Admissible variety

KW - Catenarity

KW - Coordinate semiring

KW - Dimension

KW - Primary

KW - Secondary

KW - Supertropical algebra

KW - Tropical geometry

UR - http://www.scopus.com/inward/record.url?scp=84961782053&partnerID=8YFLogxK

U2 - 10.1016/j.bulsci.2015.12.001

DO - 10.1016/j.bulsci.2015.12.001

M3 - Article

VL - 140

SP - 231

EP - 259

JO - Bulletin des Sciences Mathematiques

JF - Bulletin des Sciences Mathematiques

SN - 0007-4497

IS - 3

ER -