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 x d matrices of tropical rank two, a simplicial complex. We prove that this simplicial complex is shellable for dimension d = 3 and compute is homology in this case, conjecturing that this complex is shellable in general. We also investigate the space of d x n matrices of Barvinok rank two, a subcomplex directly related to optimization, giving a complete description of this subcomplex in the case d = 3.
@article{urn:eudml:doc:41817,
title = {The moduli space of n tropically collinear points in Rd.},
journal = {Collectanea Mathematica},
volume = {56},
year = {2005},
pages = {1-19},
zbl = {1069.52014},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41817}
}
Develin, Mike. The moduli space of n tropically collinear points in Rd.. Collectanea Mathematica, Tome 56 (2005) pp. 1-19. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41817/