Extremal decompositions of tropical varieties and relations with rigidity theory

Farhad Babaee; Sean Dewar; James W Maxwell

doi:10.48550/arXiv.2403.00655

Extremal decompositions of tropical varieties and relations with rigidity theory

Farhad Babaee, Sean Dewar, James W Maxwell

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

Abstract

Extremality and irreducibility constitute fundamental concepts in mathematics, particularly within tropical geometry. While extremal decomposition is typically computationally hard, this article presents a fast algorithm for identifying the extremal decomposition of tropical varieties with rational balanced weightings. Additionally, we explore connections and applications related to rigidity theory. In particular, we prove that a tropical hypersurface is extremal if and only if it has a unique reciprocal diagram up to homothety.

Original language	English
Article number	102461
Number of pages	30
Journal	Journal of Symbolic Computation
Early online date	19 May 2025
DOIs	https://doi.org/10.48550/arXiv.2403.00655 https://doi.org/10.1016/j.jsc.2025.102461
Publication status	E-pub ahead of print - 19 May 2025

Research Groups and Themes

Pure Mathematics

Keywords

tropical varieties
infinitesimal rigidity
reciprocal diagrams
parallel redrawings
extremal decomposition
Newton polytopes
duality

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10.48550/arXiv.2403.00655
10.1016/j.jsc.2025.102461Licence: CC BY

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title = "Extremal decompositions of tropical varieties and relations with rigidity theory",

abstract = "Extremality and irreducibility constitute fundamental concepts in mathematics, particularly within tropical geometry. While extremal decomposition is typically computationally hard, this article presents a fast algorithm for identifying the extremal decomposition of tropical varieties with rational balanced weightings. Additionally, we explore connections and applications related to rigidity theory. In particular, we prove that a tropical hypersurface is extremal if and only if it has a unique reciprocal diagram up to homothety.",

keywords = "tropical varieties, infinitesimal rigidity, reciprocal diagrams, parallel redrawings, extremal decomposition, Newton polytopes, duality",

author = "Farhad Babaee and Sean Dewar and Maxwell, \{James W\}",

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day = "19",

doi = "10.48550/arXiv.2403.00655",

language = "English",

journal = "Journal of Symbolic Computation",

issn = "0747-7171",

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TY - JOUR

T1 - Extremal decompositions of tropical varieties and relations with rigidity theory

AU - Babaee, Farhad

AU - Dewar, Sean

AU - Maxwell, James W

PY - 2025/5/19

Y1 - 2025/5/19

N2 - Extremality and irreducibility constitute fundamental concepts in mathematics, particularly within tropical geometry. While extremal decomposition is typically computationally hard, this article presents a fast algorithm for identifying the extremal decomposition of tropical varieties with rational balanced weightings. Additionally, we explore connections and applications related to rigidity theory. In particular, we prove that a tropical hypersurface is extremal if and only if it has a unique reciprocal diagram up to homothety.

AB - Extremality and irreducibility constitute fundamental concepts in mathematics, particularly within tropical geometry. While extremal decomposition is typically computationally hard, this article presents a fast algorithm for identifying the extremal decomposition of tropical varieties with rational balanced weightings. Additionally, we explore connections and applications related to rigidity theory. In particular, we prove that a tropical hypersurface is extremal if and only if it has a unique reciprocal diagram up to homothety.

KW - tropical varieties

KW - infinitesimal rigidity

KW - reciprocal diagrams

KW - parallel redrawings

KW - extremal decomposition

KW - Newton polytopes

KW - duality

U2 - 10.48550/arXiv.2403.00655

DO - 10.48550/arXiv.2403.00655

M3 - Article (Academic Journal)

SN - 0747-7171

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

M1 - 102461

ER -

Extremal decompositions of tropical varieties and relations with rigidity theory

Abstract

Research Groups and Themes

Keywords

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