1,730 research outputs found
A recombination algorithm for the decomposition of multivariate rational functions
International audienceIn this paper we show how we can compute in a deterministic way the decomposition of a multivariate rational function with a recombination strategy. The key point of our recombination strategy is the used of Darboux polynomials. We study the complexity of this strategy and we show that this method improves the previous ones. In appendix, we explain how the strategy proposed recently by J. Berthomieu and G. Lecerf for the sparse factorization can be used in the decomposition setting. Then we deduce a decomposition algorithm in the sparse bivariate case and we give its complexit
A lifting and recombination algorithm for rational factorization of sparse polynomials
We propose a new lifting and recombination scheme for rational bivariate
polynomial factorization that takes advantage of the Newton polytope geometry.
We obtain a deterministic algorithm that can be seen as a sparse version of an
algorithm of Lecerf, with now a polynomial complexity in the volume of the
Newton polytope. We adopt a geometrical point of view, the main tool being
derived from some algebraic osculation criterions in toric varieties.Comment: 22 page
Nearly Optimal Algorithms for the Decomposition of Multivariate Rational Functions and the Extended L\"uroth's Theorem
The extended L\"uroth's Theorem says that if the transcendence degree of
\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK is 1 then there exists f \in
\KK(\underline{X}) such that \KK(\mathsf{f}_1,\dots,\mathsf{f}_m) is equal
to \KK(f). In this paper we show how to compute with a probabilistic
algorithm. We also describe a probabilistic and a deterministic algorithm for
the decomposition of multivariate rational functions. The probabilistic
algorithms proposed in this paper are softly optimal when is fixed and
tends to infinity. We also give an indecomposability test based on gcd
computations and Newton's polytope. In the last section, we show that we get a
polynomial time algorithm, with a minor modification in the exponential time
decomposition algorithm proposed by Gutierez-Rubio-Sevilla in 2001
Optimization as a design strategy. Considerations based on building simulation-assisted experiments about problem decomposition
In this article the most fundamental decomposition-based optimization method
- block coordinate search, based on the sequential decomposition of problems in
subproblems - and building performance simulation programs are used to reason
about a building design process at micro-urban scale and strategies are defined
to make the search more efficient. Cyclic overlapping block coordinate search
is here considered in its double nature of optimization method and surrogate
model (and metaphore) of a sequential design process. Heuristic indicators apt
to support the design of search structures suited to that method are developed
from building-simulation-assisted computational experiments, aimed to choose
the form and position of a small building in a plot. Those indicators link the
sharing of structure between subspaces ("commonality") to recursive
recombination, measured as freshness of the search wake and novelty of the
search moves. The aim of these indicators is to measure the relative
effectiveness of decomposition-based design moves and create efficient block
searches. Implications of a possible use of these indicators in genetic
algorithms are also highlighted.Comment: 48 pages. 12 figures, 3 table
Factoring bivariate polynomials using adjoints
One relates factorization of bivariate polynomials to singularities of
projective plane curves. One proves that adjoint polynomials permit to solve
the recombinations of the modular factors induced by the absolute and rational
factorizations, and so without using Hensel's lifting. One establishes in such
a way the relations between the algorithm of Duval-Ragot (locally constant
functions) and of Ch\`eze-Lecerf (lifting and recombinations), and one shows
that a fast computation of adjoint polynomials leads to a fast factorization.
The proof is based on cohomological sequences and residue theory.Comment: 22 pages, 2 figures. Extended version of arXiv.1201.578
Modular Las Vegas Algorithms for Polynomial Absolute Factorization
Let f(X,Y) \in \ZZ[X,Y] be an irreducible polynomial over \QQ. We give a
Las Vegas absolute irreducibility test based on a property of the Newton
polytope of , or more precisely, of modulo some prime integer . The
same idea of choosing a satisfying some prescribed properties together with
is used to provide a new strategy for absolute factorization of .
We present our approach in the bivariate case but the techniques extend to the
multivariate case. Maple computations show that it is efficient and promising
as we are able to factorize some polynomials of degree up to 400
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