On polynomials arising from zonotopal algebra

38 mins 24 secs, 531.17 MB, MPEG-4 Video 480x360, 25.0 fps, 44100 Hz, 1.84 Mbits/sec

About this item

Description:	Holtz, OV (Berkeley) Wednesday 23 January 2008, 15:30-16:00 Zeros of Graph Polynomials

Created:

2008-02-06 16:15

Collection:

Combinatorics and Statistical Mechanics

Publisher:

Isaac Newton Institute

Holtz, OV

Language:

eng (English)

Credits:

Author:

Holtz, OV


Abstract:	Given a graph G, we consider its associated (linear) matroid X, and associate X with three kinds of algebraic structures called external, central, and internal. Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in n variables that are dual to each other. Algebraically, one encodes properties of the (generic) hyperplane arrangement H(X) associated to X, while the other encodes by duality the properties of the zonotope Z(X) built from the matrix X. These algebraic structures are then applied to obtain various statistics for the graph G. In particular, the Hilbert series of each of the three ideals turn out to be ultimately related to the Tutte polynomial of G, and the grading of the ideals turns out to be related to specific counting functions on subforests or on spanning trees of G. This leads to other combinatorial connections and several open problems as well. Related Links * http://arxiv.org/abs/0708.2632 - arXiv version of our paper "Zonotopal algebra"

Abstract:

Given a graph G, we consider its associated (linear) matroid X, and associate X with three kinds of algebraic structures called external, central, and internal. Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in n variables that are dual to each other. Algebraically, one encodes properties of the (generic) hyperplane arrangement H(X) associated to X, while the other encodes by duality the properties of the zonotope Z(X) built from the matrix X. These algebraic structures are then applied to obtain various statistics for the graph G. In particular, the Hilbert series of each of the three ideals turn out to be ultimately related to the Tutte polynomial of G, and the grading of the ideals turns out to be related to specific counting functions on subforests or on spanning trees of G. This leads to other combinatorial connections and several open problems as well.

Related Links
* http://arxiv.org/abs/0708.2632 - arXiv version of our paper "Zonotopal algebra"

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