scholarly journals The degree of a tropical basis

2017 ◽  
Vol 146 (3) ◽  
pp. 961-970 ◽  
Author(s):  
Michael Joswig ◽  
Benjamin Schröter
Keyword(s):  
Tropical Basis

scholarly journals Point configurations, phylogenetic trees, and dissimilarity vectors

2021 ◽  
Vol 118 (12) ◽  
pp. e2021244118
Author(s):  
Alessio Caminata ◽  
Noah Giansiracusa ◽  
Han-Bom Moon ◽  
Luca Schaffler
Keyword(s):  
Phylogenetic Trees ◽  
Geometric Interpretation ◽  
Explicit Characterization ◽  
Point Configurations ◽  
Definition Of ◽  
Tropical Basis ◽  
The Way

In 2004, Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter and Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors.

scholarly journals The 4×4 minors of a 5×n matrix are a tropical basis

2011 ◽  
Vol 435 (7) ◽  
pp. 1598-1611 ◽  
Author(s):  
Melody Chan ◽  
Anders Jensen ◽  
Elena Rubei
Keyword(s):  
Tropical Basis

scholarly journals Isotropical Linear Spaces and Valuated Delta-Matroids

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Felipe Rincón
Keyword(s):  
Symmetric Matrix ◽  
Combinatorial Theory ◽  
Linear Spaces ◽  
Dimensional Vector Space ◽  
Type D ◽  
Matroid Polytopes ◽  
International Audience ◽  
Isotropic Subspaces ◽  
Tropical Basis ◽  
Espace Vectoriel

International audience The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an $n \times n$ skew-symmetric matrix. Its points correspond to $n$-dimensional isotropic subspaces of a $2n$-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type $D$. La variété spinorielle est decoupée par les relations quadratiques de Wick parmi les Pfaffiens principaux d'une matrice antisymétrique $n \times n$. Ses points correspondent aux sous-espaces isotropes à $n$ dimensions d'un espace vectoriel de dimension $2n$. Dans cet article nous tropicalisons cette description, et nous développons une théorie combinatoire de vecteurs tropicaux de Wick et d'espaces linéaires tropicaux qui sont tropicalement isotropes. Nous caractérisons des vecteurs tropicaux de Wick en termes de subdivisions des polytopes Delta-matroïde, et nous étudions dans quelle mesure les relations de Wick forment une base tropicale. Notre théorie généralise plusieurs résultats pour les espaces linéaires tropicaux et évaluait des matroïdes à la classe des matroïdes de Coxeter du type $D$.

scholarly journals The Tropical Symplectic Grassmannian

2021 ◽  
Author(s):  
George Balla ◽  
Jorge Alberto Olarte
Keyword(s):  
Symplectic Form ◽  
Linear Subspaces ◽  
Tropical Basis

Abstract We launch the study of the tropicalization of the symplectic Grassmannian, that is, the space of all linear subspaces isotropic with respect to a fixed symplectic form. We formulate tropical analogues of several equivalent characterizations of the symplectic Grassmannian and determine all implications between them. In the process, we show that the Plücker and symplectic relations form a tropical basis if and only if the rank is at most 2. We provide plenty of examples that show that several features of the symplectic Grassmannian do not hold after tropicalizing. We show exactly when do conormal fans of matroids satisfy these characterizations, as well as doing the same for a valuated generalization. Finally, we propose several directions to extend the study of the tropical symplectic Grassmannian.

scholarly journals Detecting tropical defects of polynomial equations

2019 ◽  
Author(s):  
Paul Görlach ◽  
Yue Ren ◽  
Jeff Sommars
Keyword(s):  
Del Pezzo Surfaces ◽  
Polynomial Equations ◽  
Zero Set ◽  
Open Problems ◽  
Affine Spaces ◽  
Del Pezzo ◽  
System Of Polynomial Equations ◽  
Tropical Basis

Abstract We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these techniques to solve open problems regarding del Pezzo surfaces of degree 3 and realizability of valuated gaussoids on 4 elements.

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