1,373 research outputs found
Computing Algebraic Matroids
An affine variety induces the structure of an algebraic matroid on the set of
coordinates of the ambient space. The matroid has two natural decorations: a
circuit polynomial attached to each circuit, and the degree of the projection
map to each base, called the base degree. Decorated algebraic matroids can be
computed via symbolic computation using Groebner bases, or through linear
algebra in the space of differentials (with decorations calculated using
numerical algebraic geometry). Both algorithms are developed here. Failure of
the second algorithm occurs on a subvariety called the non-matroidal or NM-
locus. Decorated algebraic matroids have widespread relevance anywhere that
coordinates have combinatorial significance. Examples are computed from applied
algebra, in algebraic statistics and chemical reaction network theory, as well
as more theoretical examples from algebraic geometry and matroid theory.Comment: 15 pages; added link to references, note on page 1, and small
formatting fixe
Support Sets in Exponential Families and Oriented Matroid Theory
The closure of a discrete exponential family is described by a finite set of
equations corresponding to the circuits of an underlying oriented matroid.
These equations are similar to the equations used in algebraic statistics,
although they need not be polynomial in the general case. This description
allows for a combinatorial study of the possible support sets in the closure of
an exponential family. If two exponential families induce the same oriented
matroid, then their closures have the same support sets. Furthermore, the
positive cocircuits give a parameterization of the closure of the exponential
family.Comment: 27 pages, extended version published in IJA
Isotropical Linear Spaces and Valuated Delta-Matroids
The spinor variety is cut out by the quadratic Wick relations among the
principal Pfaffians of an n x n skew-symmetric matrix. Its points correspond to
n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this
paper we tropicalize this picture, and we develop a combinatorial theory of
tropical Wick vectors and tropical linear spaces that are tropically isotropic.
We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid
polytopes, and we examine to what extent the Wick relations form a tropical
basis. Our theory generalizes several results for tropical linear spaces and
valuated matroids to the class of Coxeter matroids of type D
Generic and special constructions of pure O-sequences
It is shown that the h-vectors of Stanley-Reisner rings of three classes of
matroids are pure O-sequences. The classes are (a) matroids that are
truncations of other matroids, or more generally of Cohen-Macaulay complexes,
(b) matroids whose dual is (rank + 2)-partite, and (c) matroids of
Cohen-Macaulay type at most five. Consequences for the computational search for
a counterexample to a conjecture of Stanley are discussed.Comment: 16 pages, v2: various small improvements, accepted by Bulletin of the
London Math. Societ
On perturbations of highly connected dyadic matroids
Geelen, Gerards, and Whittle [3] announced the following result: let be a prime power, and let be a proper minor-closed class of
-representable matroids, which does not contain
for sufficiently high . There exist integers
such that every vertically -connected matroid in is a
rank- perturbation of a frame matroid or the dual of a frame matroid
over . They further announced a characterization of the
perturbations through the introduction of subfield templates and frame
templates.
We show a family of dyadic matroids that form a counterexample to this
result. We offer several weaker conjectures to replace the ones in [3], discuss
consequences for some published papers, and discuss the impact of these new
conjectures on the structure of frame templates.Comment: Version 3 has a new title and a few other minor corrections; 38
pages, including a 6-page Jupyter notebook that contains SageMath code and
that is also available in the ancillary file
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