Journal
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
Volume 118, Issue 12, Pages -Publisher
NATL ACAD SCIENCES
DOI: 10.1073/pnas.2021244118
Keywords
phylogenetic tree; dissimilarity vector; Grassmannian; tropical geometry; rational normal curve
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Funding
- NSF [DMS-1802263]
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Researchers successfully demonstrate that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian by replacing the definition of the dissimilarity map, providing a geometric interpretation in terms of configurations of points on rational normal curves.
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.
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