9,696 research outputs found
More nonexistence results for symmetric pair coverings
A -covering is a pair , where is a
-set of points and is a collection of -subsets of
(called blocks), such that every unordered pair of points in is contained
in at least blocks in . The excess of such a covering is
the multigraph on vertex set in which the edge between vertices and
has multiplicity , where is the number of blocks which
contain the pair . A covering is symmetric if it has the same number
of blocks as points. Bryant et al.(2011) adapted the determinant related
arguments used in the proof of the Bruck-Ryser-Chowla theorem to establish the
nonexistence of certain symmetric coverings with -regular excesses. Here, we
adapt the arguments related to rational congruence of matrices and show that
they imply the nonexistence of some cyclic symmetric coverings and of various
symmetric coverings with specified excesses.Comment: Submitted on May 22, 2015 to the Journal of Linear Algebra and its
Application
Algebraically constructible functions
An algebraic version of Kashiwara and Schapira's calculus of constructible
functions is used to describe local topological properties of real algebraic
sets, including Akbulut and King's numerical conditions for a stratified set of
dimension three to be algebraic. These properties, which include
generalizations of the invariants modulo 4, 8, and 16 of Coste and Kurdyka, are
defined using the link operator on the ring of constructible functions.Comment: AMS-TeX v2.1, 25 page
On the complexity of computing Gr\"obner bases for weighted homogeneous systems
Solving polynomial systems arising from applications is frequently made
easier by the structure of the systems. Weighted homogeneity (or
quasi-homogeneity) is one example of such a structure: given a system of
weights , -homogeneous polynomials are polynomials
which are homogeneous w.r.t the weighted degree
. Gr\"obner bases for weighted homogeneous systems can be
computed by adapting existing algorithms for homogeneous systems to the
weighted homogeneous case. We show that in this case, the complexity estimate
for Algorithm~\F5 \left(\binom{n+\dmax-1}{\dmax}^{\omega}\right) can be
divided by a factor . For zero-dimensional
systems, the complexity of Algorithm~\FGLM (where is the
number of solutions of the system) can be divided by the same factor
. Under genericity assumptions, for
zero-dimensional weighted homogeneous systems of -degree
, these complexity estimates are polynomial in the
weighted B\'ezout bound .
Furthermore, the maximum degree reached in a run of Algorithm \F5 is bounded by
the weighted Macaulay bound , and this bound is
sharp if we can order the weights so that . For overdetermined
semi-regular systems, estimates from the homogeneous case can be adapted to the
weighted case. We provide some experimental results based on systems arising
from a cryptography problem and from polynomial inversion problems. They show
that taking advantage of the weighted homogeneous structure yields substantial
speed-ups, and allows us to solve systems which were otherwise out of reach
Torsion-free, divisible, and Mittag-Leffler modules
We study (relative) K-Mittag-Leffler modules, with emphasis on the class K of
absolutely pure modules. A final goal is to describe the K-Mittag-Leffler
abelian groups as those that are, modulo their torsion part, aleph_1-free,
Cor.6.12. Several more general results of independent interest are derived on
the way. In particular, every flat K-Mittag-Leffler module (for K as before) is
Mittag-Leffler, Thm.3.9. A question about the definable subcategories generated
by the divisible modules and the torsion-free modules, resp., has been left
open, Quest.4.6
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