11,267 research outputs found
Characterizing Triviality of the Exponent Lattice of A Polynomial through Galois and Galois-Like Groups
The problem of computing \emph{the exponent lattice} which consists of all
the multiplicative relations between the roots of a univariate polynomial has
drawn much attention in the field of computer algebra. As is known, almost all
irreducible polynomials with integer coefficients have only trivial exponent
lattices. However, the algorithms in the literature have difficulty in proving
such triviality for a generic polynomial. In this paper, the relations between
the Galois group (respectively, \emph{the Galois-like groups}) and the
triviality of the exponent lattice of a polynomial are investigated. The
\bbbq\emph{-trivial} pairs, which are at the heart of the relations between
the Galois group and the triviality of the exponent lattice of a polynomial,
are characterized. An effective algorithm is developed to recognize these
pairs. Based on this, a new algorithm is designed to prove the triviality of
the exponent lattice of a generic irreducible polynomial, which considerably
improves a state-of-the-art algorithm of the same type when the polynomial
degree becomes larger. In addition, the concept of the Galois-like groups of a
polynomial is introduced. Some properties of the Galois-like groups are proved
and, more importantly, a sufficient and necessary condition is given for a
polynomial (which is not necessarily irreducible) to have trivial exponent
lattice.Comment: 19 pages,2 figure
Algorithms on Ideal over Complex Multiplication order
We show in this paper that the Gentry-Szydlo algorithm for cyclotomic orders,
previously revisited by Lenstra-Silverberg, can be extended to
complex-multiplication (CM) orders, and even to a more general structure. This
algorithm allows to test equality over the polarized ideal class group, and
finds a generator of the polarized ideal in polynomial time. Also, the
algorithm allows to solve the norm equation over CM orders and the recent
reduction of principal ideals to the real suborder can also be performed in
polynomial time. Furthermore, we can also compute in polynomial time a unit of
an order of any number field given a (not very precise) approximation of it.
Our description of the Gentry-Szydlo algorithm is different from the original
and Lenstra- Silverberg's variant and we hope the simplifications made will
allow a deeper understanding. Finally, we show that the well-known speed-up for
enumeration and sieve algorithms for ideal lattices over power of two
cyclotomics can be generalized to any number field with many roots of unity.Comment: Full version of a paper submitted to ANT
Finding a closest point in a lattice of Voronoi's first kind
We show that for those lattices of Voronoi's first kind with known obtuse
superbasis, a closest lattice point can be computed in operations
where is the dimension of the lattice. To achieve this a series of relevant
lattice vectors that converges to a closest lattice point is found. We show
that the series converges after at most terms. Each vector in the series
can be efficiently computed in operations using an algorithm to
compute a minimum cut in an undirected flow network
QCD simulations with staggered fermions on GPUs
We report on our implementation of the RHMC algorithm for the simulation of
lattice QCD with two staggered flavors on Graphics Processing Units, using the
NVIDIA CUDA programming language. The main feature of our code is that the GPU
is not used just as an accelerator, but instead the whole Molecular Dynamics
trajectory is performed on it. After pointing out the main bottlenecks and how
to circumvent them, we discuss the obtained performances. We present some
preliminary results regarding OpenCL and multiGPU extensions of our code and
discuss future perspectives.Comment: 22 pages, 14 eps figures, final version to be published in Computer
Physics Communication
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