207 research outputs found
Lifting matroid divisors on tropical curves
Tropical geometry gives a bound on the ranks of divisors on curves in terms
of the combinatorics of the dual graph of a degeneration. We show that for a
family of examples, curves realizing this bound might only exist over certain
characteristics or over certain fields of definition. Our examples also apply
to the theory of metrized complexes and weighted graphs. These examples arise
by relating the lifting problem to matroid realizability. We also give a proof
of Mn\"ev universality with explicit bounds on the size of the matroid, which
may be of independent interest.Comment: 27 pages, 7 figures, final submitted version: several proofs
clarified and various minor change
A blow-up construction and graph coloring
Given a graph G (or more generally a matroid embedded in a projective space),
we construct a sequence of varieties whose geometry encodes combinatorial
information about G. For example, the chromatic polynomial of G (giving at each
m>0 the number of colorings of G with m colors, such that no adjacent vertices
are assigned the same color) can be computed as an intersection product between
certain classes on these varieties, and other information such as Crapo's
invariant find a very natural geometric counterpart. The note presents this
construction, and gives `geometric' proofs of a number of standard
combinatorial results on the chromatic polynomial.Comment: 22 pages, amstex 2.
Matroid Regression
We propose an algebraic combinatorial method for solving large sparse linear
systems of equations locally - that is, a method which can compute single
evaluations of the signal without computing the whole signal. The method scales
only in the sparsity of the system and not in its size, and allows to provide
error estimates for any solution method. At the heart of our approach is the
so-called regression matroid, a combinatorial object associated to sparsity
patterns, which allows to replace inversion of the large matrix with the
inversion of a kernel matrix that is constant size. We show that our method
provides the best linear unbiased estimator (BLUE) for this setting and the
minimum variance unbiased estimator (MVUE) under Gaussian noise assumptions,
and furthermore we show that the size of the kernel matrix which is to be
inverted can be traded off with accuracy
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
Representing some non-representable matroids
We extend the notion of representation of a matroid to algebraic structures
that we call skew partial fields. Our definition of such representations
extends Tutte's definition, using chain groups. We show how such
representations behave under duality and minors, we extend Tutte's
representability criterion to this new class, and we study the generator
matrices of the chain groups. An example shows that the class of matroids
representable over a skew partial field properly contains the class of matroids
representable over a skew field.
Next, we show that every multilinear representation of a matroid can be seen
as a representation over a skew partial field.
Finally we study a class of matroids called quaternionic unimodular. We prove
a generalization of the Matrix Tree theorem for this class.Comment: 29 pages, 2 figure
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