57 research outputs found
A decomposition theorem for binary matroids with no prism minor
The prism graph is the dual of the complete graph on five vertices with an
edge deleted, . In this paper we determine the class of binary
matroids with no prism minor. The motivation for this problem is the 1963
result by Dirac where he identified the simple 3-connected graphs with no minor
isomorphic to the prism graph. We prove that besides Dirac's infinite families
of graphs and four infinite families of non-regular matroids determined by
Oxley, there are only three possibilities for a matroid in this class: it is
isomorphic to the dual of the generalized parallel connection of with
itself across a triangle with an element of the triangle deleted; it's rank is
bounded by 5; or it admits a non-minimal exact 3-separation induced by the
3-separation in . Since the prism graph has rank 5, the class has to
contain the binary projective geometries of rank 3 and 4, and ,
respectively. We show that there is just one rank 5 extremal matroid in the
class. It has 17 elements and is an extension of , the unique splitter
for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's
result identifying the binary internally 4-connected matroids with no prism
minor [5]
Galois groups of multivariate Tutte polynomials
The multivariate Tutte polynomial of a matroid is a
generalization of the standard two-variable version, obtained by assigning a
separate variable to each element of the ground set . It encodes
the full structure of . Let \bv = \{v_e\}_{e\in E}, let be an
arbitrary field, and suppose is connected. We show that is
irreducible over K(\bv), and give three self-contained proofs that the Galois
group of over K(\bv) is the symmetric group of degree , where
is the rank of . An immediate consequence of this result is that the
Galois group of the multivariate Tutte polynomial of any matroid is a direct
product of symmetric groups. Finally, we conjecture a similar result for the
standard Tutte polynomial of a connected matroid.Comment: 8 pages, final version, to appear in J. Alg. Comb. Substantial
revisions, including the addition of two alternative proofs of the main
resul
Relations between M\"obius and coboundary polynomial
It is known that, in general, the coboundary polynomial and the M\"obius
polynomial of a matroid do not determine each other. Less is known about more
specific cases. In this paper, we will try to answer if it is possible that the
M\"obius polynomial of a matroid, together with the M\"obius polynomial of the
dual matroid, define the coboundary polynomial of the matroid. In some cases,
the answer is affirmative, and we will give two constructions to determine the
coboundary polynomial in these cases.Comment: 12 page
Topological representations of matroid maps
The Topological Representation Theorem for (oriented) matroids states that
every (oriented) matroid can be realized as the intersection lattice of an
arrangement of codimension one homotopy spheres on a homotopy sphere. In this
paper, we use a construction of Engstr\"om to show that structure-preserving
maps between matroids induce topological mappings between their
representations; a result previously known only in the oriented case.
Specifically, we show that weak maps induce continuous maps and that the
process is a functor from the category of matroids with weak maps to the
homotopy category of topological spaces. We also give a new and conceptual
proof of a result regarding the Whitney numbers of the first kind of a matroid.Comment: Final version, 21 pages, 8 figures; Journal of Algebraic
Combinatorics, 201
Counting points on varieties over finite fields related to a conjecture of Kontsevich
We describe a characteristic-free algorithm for “reducing” an algebraic variety defined by the vanishing of a set of integer polynomials. In very special cases, the algorithm can be used to decide whether the number of points on a variety, as the ground field varies over finite fields, is a polynomial function of the size of the field. The algorithm is then used to investigate a conjecture of Kontsevich regarding the number of points on a variety associated with the set of spanning trees of any graph. We also prove several theorems describing properties of a (hypothetical) minimal counterexample to the conjecture, and produce counterexamples to some related conjectures.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43265/1/26_2005_Article_BF01608531.pd
Lines, Circles, Planes and Spheres
Let be a set of points in , no three collinear and not
all coplanar. If at most are coplanar and is sufficiently large, the
total number of planes determined is at least . For similar conditions and
sufficiently large , (inspired by the work of P. D. T. A. Elliott in
\cite{Ell67}) we also show that the number of spheres determined by points
is at least , and this bound is best
possible under its hypothesis. (By , we are denoting the
maximum number of three-point lines attainable by a configuration of
points, no four collinear, in the plane, i.e., the classic Orchard Problem.)
New lower bounds are also given for both lines and circles.Comment: 37 page
Ideal hierarchical secret sharing schemes
Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention from the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization deals with the properties of the hierarchically minimal sets of the access structure, which are the minimal qualified sets whose participants are in the lowest possible levels in the hierarchy. By using our characterization, it can be efficiently checked whether any given hierarchical access structure that is defined by its hierarchically minimal sets is ideal. We use the well known connection between ideal secret sharing and matroids and, in particular, the fact that every ideal access structure is a matroid port. In addition, we use recent results on ideal multipartite access structures and the connection between multipartite matroids and integer polymatroids. We prove that every ideal hierarchical access structure is the port of a representable matroid and, more specifically, we prove that every ideal structure in this family admits ideal linear secret sharing schemes over fields of all characteristics. In addition, methods to construct such ideal schemes can be derived from the results in this paper and the aforementioned ones on ideal multipartite secret sharing. Finally, we use our results to find a new proof for the characterization of the ideal weighted threshold access structures that is simpler than the existing one.Peer ReviewedPostprint (author's final draft
Extending Torelli map to toroidal compactifications of Siegel space
It has been known since the 1970s that the Torelli map ,
associating to a smooth curve its jacobian, extends to a regular map from the
Deligne-Mumford compactification to the 2nd Voronoi
compactification .
We prove that the extended Torelli map to the perfect cone (1st Voronoi)
compactification is also regular, and moreover
and share a common Zariski open
neighborhood of the image of . We also show that the map to the
Igusa monoidal transform (central cone compactification) is NOT regular for
; this disproves a 1973 conjecture of Namikawa.Comment: To appear in Inventiones Mathematica
Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial
We derive some new structural results for the transfer matrix of
square-lattice Potts models with free and cylindrical boundary conditions. In
particular, we obtain explicit closed-form expressions for the dominant (at
large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as
the solution of a special one-dimensional polymer model. We also obtain the
large-q expansion of the bulk and surface (resp. corner) free energies for the
zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47}
(resp. q^{-46}). Finally, we compute chromatic roots for strips of widths 9 <=
m <= 12 with free boundary conditions and locate roughly the limiting curves.Comment: 111 pages (LaTeX2e). Includes tex file, three sty files, and 19
Postscript figures. Also included are Mathematica files data_CYL.m and
data_FREE.m. Many changes from version 1: new material on series expansions
and their analysis, and several proofs of previously conjectured results.
Final version to be published in J. Stat. Phy
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