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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
Supertropical linear algebra
The objective of this paper is to lay out the algebraic theory of
supertropical vector spaces and linear algebra, utilizing the key antisymmetric
relation of ``ghost surpasses.''Special attention is paid to the various
notions of ``base,'' which include d-base and s-base, and these are compared to
other treatments in the tropical theory. Whereas the number of elements in a
d-base may vary according to the d-base, it is shown that when an s-base
exists, it is unique up to permutation and multiplication by scalars, and can
be identified with a set of ``critical'' elements. Linear functionals and the
dual space are also studied, leading to supertropical bilinear forms and a
supertropical version of the Gram matrix, including its connection to linear
dependence, as well as a supertropical version of a theorem of Artin.Comment: 28 page
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