On basic concepts of tropical geometry
type: Item
instance of: scholarly article
title: On basic concepts of tropical geometry
zbMATH Open document ID: 1237.14074
DOI: 10.1134/S0081543811040134
author: Oleg Viro
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2011-10-25T00:00:00Z
review text: When introducing tropical geometry to a newcomer, a common question is: ``Why is it impossible to define a tropical hypersurface by the equation \(f=0\), where \(f\) is a tropical polynomial and \(0\) is the tropical zero (i.e., \(-\infty\)), instead of taking the corner locus of \(f\), which is so different from conventional algebraic geometry?'' The easy answer is ``because \(f=0\) has no solutions'', but the question itself is not silly, and points out an important gap in the common way to establish a framework for tropical geometry (see, for example, [\textit{B. Sturmfels}, CBMS Regional Conference Series in Mathematics 97. Providence, RI: AMS (2002; Zbl 1101.13040)] or [\textit{G. Mikhalkin}, Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures. Zürich: EMS. 827--852 (2006; Zbl 1103.14034)] for commonly accepted introductions to the subject). The author suggests an unexpectedly elementary and fruitful way to fix this problem: instead of defining the tropical sum of \(a\) and \(b\) as \(\max(a,b)\), one should preserve this definition for \(a\neq b\), and define the tropical sum of \(a\) and \(a\) as \([ -\infty, a]\), in the same way as it happens for the degree of the sum of two polynomials \(F\) and \(G\): we have \(\deg(F+G) = \max (\deg F, \deg G)\) provided that \(\deg F \neq \deg G\), otherwise \(\deg(F+G)\) may be equal to any number less than or equal to \(\deg F\). With this convention, a tropical hypersurface can be defined as the set of solutions of a tropical algebraic equation. Besides this toy application, the idea of defining tropical operations as multivalued is used to introduce tropical operations on the set of complex numbers, resulting in the so-called complex tropical hyperfield. Complex tropical varieties, which appeared in many works in the role of crucial, but purely technical constructions (see, for example, complex tropical curves in [\textit{G. Mikhalkin}, J…
reviewed by: Alexander Esterov
Mathematics Subject Classification ID: 14T05
Mathematics Subject Classification ID: 12K10
Mathematics Subject Classification ID: 20N20
Mathematics Subject Classification ID: 54A99
zbMATH DE Number: 5963538
zbMATH Keywords: tropical geometry
zbMATH Keywords: tropical semifield
zbMATH Keywords: multivalued operation
zbMATH Keywords: hyperfield
zbMATH Keywords: complex tropical curve
zbMATH Keywords: Vietoris topology
zbMATH Keywords: Fell topology
zbMATH Keywords: Litvinov-Maslov dequantization
MaRDI profile type: Publication
cites work: The Logarithmic Limit-Set of an Algebraic Variety
cites work: Q3998951
cites work: Q3041316
cites work: Multivalued groups, their representations and Hopf algebras
cites work: Combinatorial aspects of relations
cites work: Non-archimedean amoebas and tropical varieties
cites work: A Hausdorff Topology for the Closed Subsets of a Locally Compact Non-Hausdorff Space
cites work: Tropical algebraic geometry
cites work: Q4293510
cites work: Patchworking algebraic curves disproves the Ragsdale conjecture
cites work: A tropical toolkit
cites work: Q4550194
cites work: Q4395236
cites work: Real reduced multirings and multifields
cites work: Decomposition into pairs-of-pants for complex algebraic hypersurfaces
cites work: Enumerative tropical algebraic geometry in ℝ²
cites work: Q5490930
cites work: Q3533399
cites work: Q1106472
cites work: Exploded Fibrations
cites work: Q5702484
cites work: Q4781203
cites work: Hypergroups
Recommended article: Q5702484
Recommended article: Un peu de géométrie tropicale
Recommended article: Tropical Geometry
Recommended article: Tropical algebraic geometry
Recommended article: Q3103290
Recommended article: Q5490930
Recommended article: Q3603140
description: scientific article; zbMATH DE number 5963538