Tropical mixtures of star tree metrics

TL;DR

This paper characterizes tree metrics that can be expressed as mixtures of two star tree metrics, revealing structural constraints and exploring their algebraic properties within tropical geometry.

Contribution

It provides a complete characterization of trees admitting such mixtures and analyzes the structure of the associated tropical secant varieties.

Findings

Only trees with at most one internal edge admit the mixture decomposition.

The weights of such trees satisfy specific linear inequalities.

The set of tree metric ranks on n taxa is unbounded.

Abstract

We study tree metrics that can be realized as a mixture of two star tree metrics. We prove that the only trees admitting such a decomposition are the ones coming from a tree with at most one internal edge, and whose weight satisfies certain linear inequalities. We also characterize the fibers of the corresponding mixture map. In addition, we discuss the general framework of tropical secant varieties and we interpret our results within this setting. Finally, we show that the set of tree metric ranks of metrics on $n$ taxa is unbounded.