6,551 research outputs found
Few smooth d-polytopes with n lattice points
We prove that, for fixed n there exist only finitely many embeddings of
Q-factorial toric varieties X into P^n that are induced by a complete linear
system. The proof is based on a combinatorial result that for fixed nonnegative
integers d and n, there are only finitely many smooth d-polytopes with n
lattice points. We also enumerate all smooth 3-polytopes with at most 12
lattice points. In fact, it is sufficient to bound the singularities and the
number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result
Multiplicative structure of 2x2 tropical matrices
We study the algebraic structure of the semigroup of all
tropical matrices under multiplication. Using ideas from tropical geometry, we
give a complete description of Green's relations and the idempotents and
maximal subgroups of this semigroup.Comment: 21 pages, 5 figure
Good Random Matrices over Finite Fields
The random matrix uniformly distributed over the set of all m-by-n matrices
over a finite field plays an important role in many branches of information
theory. In this paper a generalization of this random matrix, called k-good
random matrices, is studied. It is shown that a k-good random m-by-n matrix
with a distribution of minimum support size is uniformly distributed over a
maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and
vice versa. Further examples of k-good random matrices are derived from
homogeneous weights on matrix modules. Several applications of k-good random
matrices are given, establishing links with some well-known combinatorial
problems. Finally, the related combinatorial concept of a k-dense set of m-by-n
matrices is studied, identifying such sets as blocking sets with respect to
(m-k)-dimensional flats in a certain m-by-n matrix geometry and determining
their minimum size in special cases.Comment: 25 pages, publishe
Classification of real Bott manifolds and acyclic digraphs
We completely characterize real Bott manifolds up to affine diffeomorphism in
terms of three simple matrix operations on square binary matrices obtained from
strictly upper triangular matrices by permuting rows and columns
simultaneously. We also prove that any graded ring isomorphism between the
cohomology rings of real Bott manifolds with coefficients is
induced by an affine diffeomorphism between the real Bott manifolds.
Our characterization can also be described in terms of graph operations on
directed acyclic graphs. Using this combinatorial interpretation, we prove that
the decomposition of a real Bott manifold into a product of indecomposable real
Bott manifolds is unique up to permutations of the indecomposable factors.
Finally, we produce some numerical invariants of real Bott manifolds from the
viewpoint of graph theory and discuss their topological meaning. As a
by-product, we prove that the toral rank conjecture holds for real Bott
manifolds.Comment: 27 pages, 5 figures. It is a combination of arXiv:0809.2178 and
arXiv:1002.4704, including some new result
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