6,469 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
Smoothness of stabilisers in generic characteristic
Let be a commutative unital ring. Given a finitely-presented affine
-group acting on a finitely-presented -scheme of finite type, we
show that there is a prime so that for any -algebra which is a
field of characteristic , the centralisers in of all subsets are smooth. We prove this using the Lefschetz principle
together with careful application of Gr\"{o}bner basis techniques.Comment: 15 page
On the Tropicalization of the Hilbert Scheme
In this article we study the tropicalization of the Hilbert scheme and its
suitability as a parameter space for tropical varieties. We prove that the
points of the tropicalization of the Hilbert scheme have a tropical variety
naturally associated to them. To prove this, we find a bound on the degree of
the elements of a tropical basis of an ideal in terms of its Hilbert
polynomial.
As corollary, we prove that the set of tropical varieties defined over an
algebraically closed valued field only depends on the characteristic pair of
the field and the image group of the valuation.
In conclusion, we examine some simple examples that suggest that the
definition of tropical variety should include more structure than what is
currently considered.Comment: 19 page
Parallelization of Modular Algorithms
In this paper we investigate the parallelization of two modular algorithms.
In fact, we consider the modular computation of Gr\"obner bases (resp. standard
bases) and the modular computation of the associated primes of a
zero-dimensional ideal and describe their parallel implementation in SINGULAR.
Our modular algorithms to solve problems over Q mainly consist of three parts,
solving the problem modulo p for several primes p, lifting the result to Q by
applying Chinese remainder resp. rational reconstruction, and a part of
verification. Arnold proved using the Hilbert function that the verification
part in the modular algorithm to compute Gr\"obner bases can be simplified for
homogeneous ideals (cf. \cite{A03}). The idea of the proof could easily be
adapted to the local case, i.e. for local orderings and not necessarily
homogeneous ideals, using the Hilbert-Samuel function (cf. \cite{Pf07}). In
this paper we prove the corresponding theorem for non-homogeneous ideals in
case of a global ordering.Comment: 16 page
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