64 research outputs found
Convex normality of rational polytopes with long edges
We introduce the property of convex normality of rational polytopes and give
a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing
the property. As an application, we show that if every edge of a lattice
d-polytope P has lattice length at least 4d(d+1) then P is normal. This answers
in the positive a question raised in 2007. If P is a lattice simplex whose
edges have lattice lengths at least d(d+1) then P is even covered by lattice
parallelepipeds. For the approach developed here, it is necessary to involve
rational polytopes even for applications to lattice polytopes.Comment: 16 pages, final version, to appear in Advances in Mathematic
The Steinberg group of a monoid ring, nilpotence, and algorithms
For a regular ring R and an affine monoid M the homotheties of M act
nilpotently on the Milnor unstable groups of R[M]. This strengthens the K_2
part of the main result of [G5] in two ways: the coefficient field of
characteristic 0 is extended to any regular ring and the stable K_2-group is
substituted by the unstable ones. The proof is based on a
polyhedral/combinatorial techniques, computations in Steinberg groups, and a
substantially corrected version of an old result on elementary matrices by
Mushkudiani [Mu]. A similar stronger nilpotence result for K_1 and algorithmic
consequences for factorization of high Frobenius powers of invertible matrices
are also derived.Comment: final version, to appear in Journal of Algebr
Polytopal linear algebra
We investigate similarities between the category of vector spaces and that of
polytopal algebras, containing the former as a full subcategory. In Section 2
we introduce the notion of a polytopal Picard group and show that it is trivial
for fields. The coincidence of this group with the ordinary Picard group for
general rings remains an open question. In Section 3 we survey some of the
previous results on the automorphism groups and retractions. These results
support a general conjecture proposed in Section 4 about the nature of
arbitrary homomorphisms of polytopal algebras. Thereafter a further
confirmation of this conjecture is presented by homomorphisms defined on
Veronese singularities.
This is a continuation of the project started in our papers "Polytopal linear
groups" (J. Algebra 218 (1999), 715--737), "Polytopal linear retractions"
preprint, math.AG/0001049) and "Polyhedral algebras, arrangements of toric
varieties, and their groups" (preprint,
http://www.mathematik.uni-osnabrueck.de/K-theory/0232/index.html). The higher
-theoretic aspects of polytopal linear objects will be treated in
"Polyhedral -theory" (in preparation).Comment: 21 pages, uses pstricks and P. Taylor's CD package. Beitr. Algebra
Geom., to appea
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