457 research outputs found
Towards a splitter theorem for internally 4-connected binary matroids
This is the post-print version of the Article - Copyright @ 2012 ElsevierWe prove that if M is a 4-connected binary matroid and N is an internally 4-connected proper minor of M with at least 7 elements, then, unless M is a certain 16-element matroid, there is an element e of E(M) such that either M\e or M/e is internally 4-connected having an N-minor. This strengthens a result of Zhou and is a first step towards obtaining a splitter theorem for internally 4-connected binary matroids.This study is partially funded by Marsden Fund of New Zealand and the National Security Agency
A chain theorem for internally 4-connected binary matroids
This is the post-print version of the Article - Copyright @ 2011 ElsevierLet M be a matroid. When M is 3-connected, Tutte’s Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with |E(M) − E(N)| = 1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M) − E(N)| at most 3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder.This study was partially supported by the National Security Agency
Matroids on Eight Elements with the Half-plane Property and Related Concepts
We classify all matroids with at most 8 elements that have the half-plane
property, and we provide a list of some matroids on 9 elements that have, and
that do not have the half-plane property. Furthermore, we prove that several
classes of matroids and polynomials that are motivated by the theory of
semidefinite programming are closed under taking minors and under passing to
faces of the Newton polytope.Comment: Test results on the half-plane property of matroids on 9 elements are
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Combinatorial and Computational Methods for the Properties of Homogeneous Polynomials
In this manuscript, we provide foundations of properties of homogeneous polynomials such as the half-plane property, determinantal representability, being weakly determinantal, and having a spectrahedral hyperbolicity cone. One of the motivations for studying those properties comes from the ``generalized Lax conjecture'' stating that every hyperbolicity cone is spectrahedral. The conjecture has particular importance in convex optimization and has curious connections to other areas.
We take a combinatorial approach, contemplating the properties on matroids with a particular focus on operations that preserve these properties. We show that the spectrahedral representability of hyperbolicity cones and being weakly determinantal are minor-closed properties. In addition, they are preserved under passing to the faces of the Newton polytopes of homogeneous polynomials. We present a proved-to-be computationally feasible algorithm to test the half-plane property of matroids and another one for testing being weakly determinantal. Using the computer algebra system Macaulay2 and Julia, we implement these algorithms and conduct tests. We classify matroids on at most 8 elements with respect to the half-plane property and provide our test results on matroids with 9 elements. We provide 14 matroids on 8 elements of rank 4, including the Vámos matroid, that are potential candidates for the search of a counterexample for the conjecture.:1 Background 1
1.1 Some Properties of Homogeneous Polynomials . . . . . . . . . . 1
Hyperbolic Polynomials . . . . . . . . . . . . . . . . . . . . . . 1
The Half-Plane Property and Stability . . . . . . . . . . . . . . 8
Determinantal Representability . . . . . . . . . . . . . . . . . . 15
Spectrahedral Representability . . . . . . . . . . . . . . . . . . 19
1.2 Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Some Operations on Matroids . . . . . . . . . . . . . . . . . . . 29
The Half-Plane Property of Matroids . . . . . . . . . . . . . . . 36
2 Some Operations 43
2.1 Determinantal Representability of Matroids . . . . . . . . . . . 43
A Criterion for Determinantal Representability . . . . . . . . . 46
2.2 Spectrahedral Representability of Matroids . . . . . . . . . . . 50
2.3 Matroid Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . 54
Newton Polytopes of Stable Polynomials . . . . . . . . . . . . . 59
3 Testing the Properties: an Algorithm 61
The Half-Plane Property . . . . . . . . . . . . . . . . . . . . . . 61
Being SOS-Rayleigh and Weak Determinantal Representability 65
4 Test Results on Matroids on 8 and 9 Elements 71
4.1 Matroids on 8 Elements . . . . . . . . . . . . . . . . . . . . . . 71
SOS-Rayleigh and Weakly Determinantal Matroids . . . . . . . 76
4.2 Matroids on 9 Elements . . . . . . . . . . . . . . . . . . . . . . 80
5 Conclusion and Future Perspectives 85
5.1 Spectrahedral Matroids . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Non-negative Non-SOS Polynomials . . . . . . . . . . . . . . . 88
5.3 Completing the Classification of Matroids on 9 Elements and More 89
Bibliography 9
Cellular spanning trees and Laplacians of cubical complexes
We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell
complex in terms of the eigenvalues of its cellular Laplacian operators,
generalizing a previous result for simplicial complexes. As an application, we
obtain explicit formulas for spanning tree enumerators and Laplacian
eigenvalues of cubes; the latter are integers. We prove a weighted version of
the eigenvalue formula, providing evidence for a conjecture on weighted
enumeration of cubical spanning trees. We introduce a cubical analogue of
shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of
shifted cubical complexes, in particular, these eigenvalues are also integers.
Finally, we recover Adin's enumeration of spanning trees of a complete colorful
simplicial complex from the cellular Matrix-Tree Theorem together with a result
of Kook, Reiner and Stanton.Comment: 24 pages, revised version, to appear in Advances in Applied
Mathematic
Dimension, matroids, and dense pairs of first-order structures
A structure M is pregeometric if the algebraic closure is a pregeometry in
all M' elementarily equivalent to M. We define a generalisation: structures
with an existential matroid. The main examples are superstable groups of U-rank
a power of omega and d-minimal expansion of fields. Ultraproducts of
pregeometric structures expanding a field, while not pregeometric in general,
do have an unique existential matroid.
Generalising previous results by van den Dries, we define dense elementary
pairs of structures expanding a field and with an existential matroid, and we
show that the corresponding theories have natural completions, whose models
also have a unique existential matroid. We extend the above result to dense
tuples of structures.Comment: Version 2.8. 61 page
The moduli space of matroids
In the first part of the paper, we clarify the connections between several
algebraic objects appearing in matroid theory: both partial fields and
hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are
compatible with the respective matroid theories. Moreover, fuzzy rings are
ordered blueprints and lie in the intersection of tracts with ordered
blueprints; we call the objects of this intersection pastures.
In the second part, we construct moduli spaces for matroids over pastures. We
show that, for any non-empty finite set , the functor taking a pasture
to the set of isomorphism classes of rank- -matroids on is
representable by an ordered blue scheme , the moduli space of
rank- matroids on .
In the third part, we draw conclusions on matroid theory. A classical
rank- matroid on corresponds to a -valued point of
where is the Krasner hyperfield. Such a point defines a
residue pasture , which we call the universal pasture of . We show that
for every pasture , morphisms are canonically in bijection with
-matroid structures on .
An analogous weak universal pasture classifies weak -matroid
structures on . The unit group of can be canonically identified with
the Tutte group of . We call the sub-pasture of generated by
``cross-ratios' the foundation of ,. It parametrizes rescaling classes of
weak -matroid structures on , and its unit group is coincides with the
inner Tutte group of . We show that a matroid is regular if and only if
its foundation is the regular partial field, and a non-regular matroid is
binary if and only if its foundation is the field with two elements. This
yields a new proof of the fact that a matroid is regular if and only if it is
both binary and orientable.Comment: 83 page
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