5,074 research outputs found
Algorithmic Algebraic Geometry and Flux Vacua
We develop a new and efficient method to systematically analyse four
dimensional effective supergravities which descend from flux compactifications.
The issue of finding vacua of such systems, both supersymmetric and
non-supersymmetric, is mapped into a problem in computational algebraic
geometry. Using recent developments in computer algebra, the problem can then
be rapidly dealt with in a completely algorithmic fashion. Two main results are
(1) a procedure for calculating constraints which the flux parameters must
satisfy in these models if any given type of vacuum is to exist; (2) a stepwise
process for finding all of the isolated vacua of such systems and their
physical properties. We illustrate our discussion with several concrete
examples, some of which have eluded conventional methods so far.Comment: 41 pages, 4 figure
The monodromy groups of lisse sheaves and overconvergent -isocrystals
It has been proven by Serre, Larsen-Pink and Chin, that over a smooth curve
over a finite field, the monodromy groups of compatible semi-simple pure lisse
sheaves have "the same" and neutral component. We generalize their
results to compatible systems of semi-simple lisse sheaves and overconvergent
-isocrystals over arbitrary smooth varieties. For this purpose, we extend
the theorem of Serre and Chin on Frobenius tori to overconvergent
-isocrystals. To put our results into perspective, we briefly survey recent
developments of the theory of lisse sheaves and overconvergent -isocrystals.
We use the Tannakian formalism to make explicit the similarities between the
two types of coefficient objects.Comment: 37 pages; to appear in Selecta Mathematic
Macaulay inverse systems revisited
Since its original publication in 1916 under the title "The Algebraic Theory
of Modular Systems", the book by F. S. Macaulay has attracted a lot of
scientists with a view towards pure mathematics (D. Eisenbud,...) or
applications to control theory (U. Oberst,...).However, a carefull examination
of the quotations clearly shows that people have hardly been looking at the
last chapter dealing with the so-called "inverse systems", unless in very
particular situations. The purpose of this paper is to provide for the first
time the full explanation of this chapter within the framework of the formal
theory of systems of partial differential equations (Spencer operator on
sections, involution,...) and its algebraic counterpart now called "algebraic
analysis" (commutative and homological algebra, differential modules,...). Many
explicit examples are fully treated and hints are given towards the way to work
out computer algebra packages.Comment: From a lecture at the International Conference : Application of
Computer Algebra (ACA 2008) july 2008, RISC, LINZ, AUSTRI
A geometric approach to alternating -linear forms
Given an -dimensional vector space over a field , let
. There is a natural correspondence between the alternating
-linear forms of and the linear functionals of
. Let be the Plucker embedding of the -Grassmannian
of . Then
is a
hyperplane of the point-line geometry . All hyperplanes of
can be obtained in this way. For a hyperplane of
, let be the subspace of formed by the -subspaces such that
contains all -subspaces that contain . In other words, if is
the (unique modulo a scalar) alternating -linear form defining , then the
elements of are the -subspaces of such that for all
. When is even it might be that . When
is odd, then , since every -subspace
of is contained in at least one member of . If every
-subspace of is contained in precisely one member of
we say that is spread-like. In this paper we obtain some
results on which answer some open questions from the literature
and suggest the conjecture that, if is even and at least , then
but for one exception with and , while if is odd and at least
then is never spread-like.Comment: 29 Page
Resolving zero-divisors using Hensel lifting
Algorithms which compute modulo triangular sets must respect the presence of
zero-divisors. We present Hensel lifting as a tool for dealing with them. We
give an application: a modular algorithm for computing GCDs of univariate
polynomials with coefficients modulo a radical triangular set over the
rationals. Our modular algorithm naturally generalizes previous work from
algebraic number theory. We have implemented our algorithm using Maple's RECDEN
package. We compare our implementation with the procedure RegularGcd in the
RegularChains package.Comment: Shorter version to appear in Proceedings of SYNASC 201
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