点构型、系统发育树和相异度向量
Point configurations, phylogenetic trees, and dissimilarity vectors
摘要：
2004年，Pachter和Speyer引入了系统发育树的高相异度图，并提出了两个关于它们与热带格拉斯曼的关系的重要问题。多个作者，使用独立的方法，肯定地回答了这些问题中的第一个，表明相异性向量位于热带格拉斯曼，但第二个问题，是否一组相异性向量形成一个热带亚种，仍然开放。我们通过证明热带平衡条件不成立来解决这个问题。然而，通过替换的定义的相异度图与加权的变体，我们表明，加权相异度向量形成一个热带的热带Grassmannian的方式，Pachter-Speyer设想的。此外，我们提供了一个几何解释的配置点的合理正态曲线，并构建了一个有限的热带基础，产生一个明确的特征加权相异性向量。
英文摘要：
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--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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