31,507 research outputs found
Polymake and Lattice Polytopes
The polymake software system deals with convex polytopes and related objects
from geometric combinatorics. This note reports on a new implementation of a
subclass for lattice polytopes. The features displayed are enabled by recent
changes to the polymake core, which will be discussed briefly.Comment: 12 pages, 1 figur
Enumeration of points, lines, planes, etc
One of the earliest results in enumerative combinatorial geometry is the
following theorem of de Bruijn and Erd\H{o}s: Every set of points in a
projective plane determines at least lines, unless all the points are
contained in a line. Motzkin and others extended the result to higher
dimensions, who showed that every set of points in a projective space
determines at least hyperplanes, unless all the points are contained in a
hyperplane. Let be a spanning subset of a -dimensional vector space. We
show that, in the partially ordered set of subspaces spanned by subsets of ,
there are at least as many -dimensional subspaces as there are
-dimensional subspaces, for every at most . This confirms the
"top-heavy" conjecture of Dowling and Wilson for all matroids realizable over
some field. The proof relies on the decomposition theorem package for
-adic intersection complexes.Comment: 18 pages, major revisio
Geometric Reasoning with polymake
The mathematical software system polymake provides a wide range of functions
for convex polytopes, simplicial complexes, and other objects. A large part of
this paper is dedicated to a tutorial which exemplifies the usage. Later
sections include a survey of research results obtained with the help of
polymake so far and a short description of the technical background
A formally verified proof of the prime number theorem
The prime number theorem, established by Hadamard and de la Vall'ee Poussin
independently in 1896, asserts that the density of primes in the positive
integers is asymptotic to 1 / ln x. Whereas their proofs made serious use of
the methods of complex analysis, elementary proofs were provided by Selberg and
Erd"os in 1948. We describe a formally verified version of Selberg's proof,
obtained using the Isabelle proof assistant.Comment: 23 page
Permutation actions on equivariant cohomology
This survey paper describes two geometric representations of the permutation
group using the tools of toric topology. These actions are extremely useful for
computational problems in Schubert calculus. The (torus) equivariant cohomology
of the flag variety is constructed using the combinatorial description of
Goresky-Kottwitz-MacPherson, discussed in detail. Two permutation
representations on equivariant and ordinary cohomology are identified in terms
of irreducible representations of the permutation group. We show how to use the
permutation actions to construct divided difference operators and to give
formulas for some localizations of certain equivariant classes.
This paper includes several new results, in particular a new proof of the
Chevalley-Monk formula and a proof that one of the natural permutation
representations on the equivariant cohomology of the flag variety is the
regular representation. Many examples, exercises, and open questions are
provided.Comment: 24 page
Topological obstructions for vertex numbers of Minkowski sums
We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i
\ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the
maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of
Fukuda & Weibel (2006), who show that this is possible for up to d-1 summands.
The result is obtained by combining methods from discrete geometry (Gale
transforms) and topological combinatorics (van Kampen--type obstructions) as
developed in R\"{o}rig, Sanyal, and Ziegler (2007).Comment: 13 pages, 2 figures; Improved exposition and less typos.
Construction/example and remarks adde
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