48,427 research outputs found
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
Computing invariants of algebraic group actions in arbitrary characteristic
Let G be an affine algebraic group acting on an affine variety
X. We present an algorithm for computing generators of the invariant ring
K[X]^G in the case where G is reductive. Furthermore, we address the case where
G is connected and unipotent, so the invariant ring need not be finitely
generated. For this case, we develop an algorithm which computes K[X]^G in
terms of a so-called colon-operation. From this, generators of K[X]^G can be
obtained in finite time if it is finitely generated. Under the additional
hypothesis that K[X] is factorial, we present an algorithm that finds a
quasi-affine variety whose coordinate ring is K[X]^G. Along the way, we develop
some techniques for dealing with non-finitely generated algebras. In
particular, we introduce the finite generation locus ideal.Comment: 43 page
A Geometric Index Reduction Method for Implicit Systems of Differential Algebraic Equations
This paper deals with the index reduction problem for the class of
quasi-regular DAE systems. It is shown that any of these systems can be
transformed to a generically equivalent first order DAE system consisting of a
single purely algebraic (polynomial) equation plus an under-determined ODE
(that is, a semi-explicit DAE system of differentiation index 1) in as many
variables as the order of the input system. This can be done by means of a
Kronecker-type algorithm with bounded complexity
Canonical Characteristic Sets of Characterizable Differential Ideals
We study the concept of canonical characteristic set of a characterizable
differential ideal. We propose an efficient algorithm that transforms any
characteristic set into the canonical one. We prove the basic properties of
canonical characteristic sets. In particular, we show that in the ordinary case
for any ranking the order of each element of the canonical characteristic set
of a characterizable differential ideal is bounded by the order of the ideal.
Finally, we propose a factorization-free algorithm for computing the canonical
characteristic set of a characterizable differential ideal represented as a
radical ideal by a set of generators. The algorithm is not restricted to the
ordinary case and is applicable for an arbitrary ranking.Comment: 26 page
Convex Hulls of Algebraic Sets
This article describes a method to compute successive convex approximations
of the convex hull of a set of points in R^n that are the solutions to a system
of polynomial equations over the reals. The method relies on sums of squares of
polynomials and the dual theory of moment matrices. The main feature of the
technique is that all computations are done modulo the ideal generated by the
polynomials defining the set to the convexified. This work was motivated by
questions raised by Lov\'asz concerning extensions of the theta body of a graph
to arbitrary real algebraic varieties, and hence the relaxations described here
are called theta bodies. The convexification process can be seen as an
incarnation of Lasserre's hierarchy of convex relaxations of a semialgebraic
set in R^n. When the defining ideal is real radical the results become
especially nice. We provide several examples of the method and discuss
convergence issues. Finite convergence, especially after the first step of the
method, can be described explicitly for finite point sets.Comment: This article was written for the "Handbook of Semidefinite, Cone and
Polynomial Optimization: Theory, Algorithms, Software and Applications
Improved Complexity Bounds for Counting Points on Hyperelliptic Curves
We present a probabilistic Las Vegas algorithm for computing the local zeta
function of a hyperelliptic curve of genus defined over . It
is based on the approaches by Schoof and Pila combined with a modeling of the
-torsion by structured polynomial systems. Our main result improves on
previously known complexity bounds by showing that there exists a constant
such that, for any fixed , this algorithm has expected time and space
complexity as grows and the characteristic is large
enough.Comment: To appear in Foundations of Computational Mathematic
Solving multivariate polynomial systems and an invariant from commutative algebra
The complexity of computing the solutions of a system of multivariate
polynomial equations by means of Gr\"obner bases computations is upper bounded
by a function of the solving degree. In this paper, we discuss how to
rigorously estimate the solving degree of a system, focusing on systems arising
within public-key cryptography. In particular, we show that it is upper bounded
by, and often equal to, the Castelnuovo Mumford regularity of the ideal
generated by the homogenization of the equations of the system, or by the
equations themselves in case they are homogeneous. We discuss the underlying
commutative algebra and clarify under which assumptions the commonly used
results hold. In particular, we discuss the assumption of being in generic
coordinates (often required for bounds obtained following this type of
approach) and prove that systems that contain the field equations or their fake
Weil descent are in generic coordinates. We also compare the notion of solving
degree with that of degree of regularity, which is commonly used in the
literature. We complement the paper with some examples of bounds obtained
following the strategy that we describe
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