2004/04/20 by Federico Ardila, Ardila, Federico · 1 citation
Computer Science · Engineering · Mathematics · #05B35 #51F99 #92B10 #Advanced Combinatorial Mathematics #Advanced Numerical Analysis Techniques #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Polynomial and algebraic computation #math.CO #math.MG #msc:05B35 #msc:51F99 #msc:92B10
paper · pdf · doi:10.48550/arxiv.math/0404370
13 pages, 4 figures
arxiv created 2004/04/20 · openalex publication_date 2004/04/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a matroid M on the ground set E, the Bergman fan B(M), or space of M-ultrametrics, 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 w in RE, we wish to find an M-ultrametric which is closest to it in the linfty metric. The solution to this problem follows easily from the existence of the subdominant M-ultrametric: a componentwise maximum M-ultrametric which is componentwise smaller than w. 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 complete graph Kn, the Bergman fan B(Kn) parameterizes the equidistant phylogenetic trees with n leaves. In this case, our results provide a conceptual explanation for Chepoi and Fichet's method for computing the tree that most closely matches measured data.