 
Summary: Subdominant matroid ultrametrics
Federico Ardila
Abstract
Given a matroid M on the ground set E, the Bergman fan B(M), or
space of Multrametrics, is a polyhedral complex in RE
which arises in
several different areas, such as tropical algebraic geometry, dynamical
systems, and phylogenetics. Motivated by the phylogenetic situation,
we study the following problem: Given a point in RE
, we wish to
find an Multrametric which is closest to it in the metric.
The solution to this problem follows easily from the existence of the
subdominant Multrametric: a componentwise maximum Multrametric
which is componentwise smaller than . A procedure for computing
it is given, which brings together the points of view of matroid theory
and tropical geometry.
When the matroid in question is the graphical matroid of the com
plete graph Kn, the Bergman fan B(Kn) parameterizes the equidistant
phylogenetic trees with n leaves. In this case, our results provide a con
ceptual explanation for Chepoi and Fichet's method for computing the
