The moduli space of $n$ tropically collinear points in $\mathbb {R}^d$
Mike Develin · RACO (Revistes Catalanes amb Accés Obert) (Consorci de Serveis Universitaris de Catalunya) · 2005
The tropical semiring (R, min, +) has enjoyed a recent renaissance, owing to its connections to mathematical biology as well as optimization and algebraic geometry.In this paper, we investigate the space of labeled n-point configurations lying on a tropical line in d-space, which is interpretable as the space of n-species phylogenetic trees.This is equivalent to the space of n × d matrices of tropical rank two, a simplicial complex.We prove that this simplicial complex is shellable for dimension d = 3 and compute its homology in this case, conjecturing that this complex is shellable in general.We also investigate the space of d × n matrices of Barvinok rank two, a subcomplex directly related to optimization, giving a complete description of this subcomplex in the case d = 3.