240 research outputs found
Secret-Sharing Matroids need not be Algebraic
We combine some known results and techniques with new ones to show that there
exists a non-algebraic, multi-linear matroid. This answers an open question by
Matus (Discrete Mathematics 1999), and an open question by Pendavingh and van
Zwam (Advances in Applied Mathematics 2013). The proof is constructive and the
matroid is explicitly given
New Representations of Matroids and Generalizations
We extend the notion of matroid representations by matrices over fields and
consider new representations of matroids by matrices over finite semirings,
more precisely over the boolean and the superboolean semirings. This idea of
representations is generalized naturally to include also hereditary
collections. We show that a matroid that can be directly decomposed as
matroids, each of which is representable over a field, has a boolean
representation, and more generally that any arbitrary hereditary collection is
superboolean-representable.Comment: 27 page
Searching for a Connection Between Matroid Theory and String Theory
We make a number of observations about matter-ghost string phase, which may
eventually lead to a formal connection between matroid theory and string
theory. In particular, in order to take advantage of the already established
connection between matroid theory and Chern-Simons theory, we propose a
generalization of string theory in terms of some kind of Kahler metric. We show
that this generalization is closely related to the Kahler-Chern-Simons action
due to Nair and Schiff. In addition, we discuss matroid/string connection via
matroid bundles and a Schild type action, and we add new information about the
relationship between matroid theory, D=11 supergravity and Chern-Simons
formalism.Comment: 28 pages, LaTex, section 6 and references adde
On the half-plane property and the Tutte group of a matroid
A multivariate polynomial is stable if it is non-vanishing whenever all
variables have positive imaginary parts. A matroid has the weak half-plane
property (WHPP) if there exists a stable polynomial with support equal to the
set of bases of the matroid. If the polynomial can be chosen with all of its
nonzero coefficients equal to one then the matroid has the half-plane property
(HPP). We describe a systematic method that allows us to reduce the WHPP to the
HPP for large families of matroids. This method makes use of the Tutte group of
a matroid. We prove that no projective geometry has the WHPP and that a binary
matroid has the WHPP if and only if it is regular. We also prove that T_8 and
R_9 fail to have the WHPP.Comment: 8 pages. To appear in J. Combin. Theory Ser.
Polynomials with the half-plane property and matroid theory
A polynomial f is said to have the half-plane property if there is an open
half-plane H, whose boundary contains the origin, such that f is non-zero
whenever all the variables are in H. This paper answers several open questions
regarding multivariate polynomials with the half-plane property and matroid
theory.
* We prove that the support of a multivariate polynomial with the half-plane
property is a jump system. This answers an open question posed by Choe, Oxley,
Sokal and Wagner and generalizes their recent result claiming that the same is
true whenever the polynomial is also homogeneous.
* We characterize multivariate multi-affine polynomial with real coefficients
that have the half-plane property (with respect to the upper half-plane) in
terms of inequalities. This is used to answer two open questions posed by Choe
and Wagner regarding strongly Rayleigh matroids.
* We prove that the Fano matroid is not the support of a polynomial with the
half-plane property. This is the first instance of a matroid which does not
appear as the support of a polynomial with the half-plane property and answers
a question posed by Choe et al.
We also discuss further directions and open problems.Comment: 17 pages. To appear in Adv. Mat
On the unique representability of spikes over prime fields
For an integer , a rank- matroid is called an -spike if it
consists of three-point lines through a common point such that, for all
, the union of every set of of these lines has
rank . Spikes are very special and important in matroid theory. In 2003 Wu
found the exact numbers of -spikes over fields with 2, 3, 4, 5, 7 elements,
and the asymptotic values for larger finite fields. In this paper, we prove
that, for each prime number , a ) representable -spike is only
representable on fields with characteristic provided that .
Moreover, is uniquely representable over .Comment: 8 page
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