484 research outputs found
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
Fast Computation of Special Resultants
We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series
Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time
International audienceIn this paper we study planar polynomial differential systems of this form: dX/dt=A(X, Y ), dY/dt= B(X, Y ), where A,B belongs to Z[X, Y ], degA ≤ d, degB ≤ d, and the height of A and B is smaller than H. A lot of properties of planar polynomial differential systems are related to irreducible Darboux polynomials of the corresponding derivation: D =A(X, Y )dX + B(X, Y )dY . Darboux polynomials are usually computed with the method of undetermined coefficients. With this method we have to solve a polynomial system. We show that this approach can give rise to the computation of an exponential number of reducible Darboux polynomials. Here we show that the Lagutinskii-Pereira's algorithm computes irreducible Darboux polynomials with degree smaller than N, with a polynomial number, relatively to d, log(H) and N, binary operations. We also give a polynomial-time method to compute, if it exists, a rational first integral with bounded degree
Algorithmic Contributions to the Theory of Regular Chains
Regular chains, introduced about twenty years ago, have emerged as one of the major
tools for solving polynomial systems symbolically. In this thesis, we focus on different
algorithmic aspects of the theory of regular chains, from theoretical questions to high-
performance implementation issues.
The inclusion test for saturated ideals is a fundamental problem in this theory.
By studying the primitivity of regular chains, we show that a regular chain generates
its saturated ideal if and only if it is primitive. As a result, a family of inclusion tests
can be detected very efficiently.
The algorithm to compute the regular GCDs of two polynomials modulo a regular
chain is one of the key routines in the various triangular decomposition algorithms. By
revisiting relations between subresultants and GCDs, we proposed a novel bottom-up
algorithm for this task, which improves the previous algorithm in a significant manner
and creates opportunities for parallel execution.
This thesis also discusses the accelerations towards fast Fourier transform (FFT)
over finite fields and FFT based subresultant chain constructions in the context of
massively parallel GPU architectures, which speedup our algorithms by several orders
of magnitude
Efficient Algorithms for Mixed Creative Telescoping
Creative telescoping is a powerful computer algebra paradigm -initiated by
Doron Zeilberger in the 90's- for dealing with definite integrals and sums with
parameters. We address the mixed continuous-discrete case, and focus on the
integration of bivariate hypergeometric-hyperexponential terms. We design a new
creative telescoping algorithm operating on this class of inputs, based on a
Hermite-like reduction procedure. The new algorithm has two nice features: it
is efficient and it delivers, for a suitable representation of the input, a
minimal-order telescoper. Its analysis reveals tight bounds on the sizes of the
telescoper it produces.Comment: To be published in the proceedings of ISSAC'1
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