arXiv subjects

David E Speyer

Publications and source records attributed to David E Speyer.

42 records · Page 3Linked to original sources

arXiv2005

Cyclically Orientable Graphs

Barot, Geiss and Zelevinsky define a notion of a ``cyclically orientable graph'' and use it to devise a test for whether a cluster algebra is of finite type. Barot, Geiss and Zelivinsky's work leaves open the question of giving an efficient characterization of cyclically orientable graphs. In this paper, we give a simple recursive description of cyclically orientable graphs, and use this to give an O(n) algorithm to test whether a graph on $n$ vertices is cyclically orientable. Shortly after writing this paper, I learned that most of its results had been obtained independently by Gurvich; I am placing this paper on the arXiv to spread knowledge of these results.

math.CO↗

arXiv2004

Tropical Linear Spaces

We define tropical analogues of the notions of linear space and Plucker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated dualization and transverse intersection to be constructible. Our main result that all constructible tropical linear spaces have the same f-vector and are ``series-parallel''. We conjecture that this f-vector is maximal for all tropical linear spaces with equality precisely for the series-parallel tropical linear spaces. We present many partial results towards this conjecture. In addition we relate tropical linear spaces to linear spaces defined over power series fields and give many examples and counter-examples illustrating aspects of this relationship. We describe a family of particularly nice series-parallel linear spaces, which we term tree spaces, that realize the conjectured maximal f-vector and are constructed in a manner similar to the cyclic polytopes.

math.CO↗

arXiv2004

The Cube Recurrence

We construct a combinatorial model that is described by the cube recurrence, a nonlinear recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in $\mathbb{Z}^3$. In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs.

math.CO↗

arXiv2004

Perfect Matchings and the Octahedron Recurrence

We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the Laurent polynomial and the perfect matchings of certain graphs, generalizing the theory of Aztec diamonds. In particular, this shows that the coefficients of this polynomial, and polynomials obtained by specializing its variables, are positive, a conjecture of Fomin and Zelevinsky.

math.CO↗

arXiv2003

Horn's Problem, Vinnikov Curves and the Hive Cone

A Vinnikov curve is a projective plane curve which can be written in the form det(xX+yY+zZ)=0 for X, Y and Z positive definite Hermitian n by n matrices. Given three n-tuples of positive real numbers, alpha, beta and gamma, there exist A, B and C in GL_n \CC with singular values alpha, beta and gamma and ABC=1 if and only if there is a Vinnikov curve passing through the 3n points (-1: alpha_i^2:0), (0:-1:beta_i^2) and (gamma_i^2:0:-1). Knutson and Tao proved that another equivalent condition for such A, B and C to exist is that there is a hive (defined within) whose boundary is (log alpha, log beta, log gamma). The logarithms of the coefficients of F approximately form such a hive; this leads to a new proof of Knutson and Tao's result. This paper uses no representation theory and essentially no symplectic geometry. In their place, it uses Viro's patchworking method and a topological description of Vinnikov curves.

math.AG↗

arXiv2003

Reconstructing Trees from Subtree Weights

The tree-metric theorem provides a necessary and sufficient condition for a dissimilarity matrix to be a tree metric, and has served as the foundation for numerous distance-based reconstruction methods in phylogenetics. Our main result is an extension of the tree-metric theorem to more general dissimilarity maps. In particular, we show that a tree with n leaves is reconstructible from the weights of the m-leaf subtrees provided that n \geq 2m-1.

math.CO↗