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Báo cáo toán học: "Dissimilarity vectors of trees are contained in the tropical Grassmannian"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Dissimilarity vectors of trees are contained in the tropical Grassmannian. | Dissimilarity vectors of trees are contained in the tropical Grassmannian Benjamin Iriarte Giraldo Department of Mathematics San Francisco State University San Francisco CA USA biriarte@sfsu.edu Submitted Sep 1 2009 Accepted Jan 1 2010 Published Jan 14 2010 Mathematics Subject Classification 05C05 14T05 Abstract In this short writing we prove that the set of m-dissimilarity vectors of phylogenetic n-trees is contained in the tropical Grassmannian Qm.n answering a question of Pachter and Speyer. We do this by proving an equivalent conjecture proposed by Cools. 1 Introduction. This article essentially deals with the connection between phylogenetic trees and tropical geometry. That these two subjects are mathematically related can be traced back to Pachter and Speyer 7 Speyer and Sturmfels 9 and Ardila and Klivans 1 . The precise nature of this connection has been the matter of some recent papers by Bocci and Cools 2 and Cools 4 . In particular a relation between m-dissimilarity vectors of phylogenetic n-trees with the tropical Grassmannians Qm n has been noted. Theorem 1.1 Pachter and Sturmfels 8 . The set of 2-dissimilarity vectors is equal to the tropical Grassmannian Q2.n. This naturally raises the following question. Question 1.2 Pachter and Speyer 7 Problem 3 . Does the space of m-dissimilarity vectors lie in Qm n for m 3 The result in this article is of relevance in this direction and it is based on two papers of Cools 4 and Bocci and Cools 2 where the cases m 3 m 4 and m 5 are handled. We answer Question 1.2 affirmatively for all m Theorem 1.3. The set of m-dissimilarity vectors of phylogenetic n-trees is contained in the tropical Grassmannian Qm n. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N6 1 As we said we prove Theorem 1.3 by proving an equivalent conjecture Proposition 3.1 of this paper or see Conjecture 4.4 of 4 . 2 Definitions. 2.1 The Tropical Grassmannian. Let K C t be the field of Puiseux series. Recall that this is the algebraically closed .