3,190 research outputs found
Improved algorithm for computing separating linear forms for bivariate systems
We address the problem of computing a linear separating form of a system of
two bivariate polynomials with integer coefficients, that is a linear
combination of the variables that takes different values when evaluated at the
distinct solutions of the system. The computation of such linear forms is at
the core of most algorithms that solve algebraic systems by computing rational
parameterizations of the solutions and this is the bottleneck of these
algorithms in terms of worst-case bit complexity. We present for this problem a
new algorithm of worst-case bit complexity \sOB(d^7+d^6\tau) where and
denote respectively the maximum degree and bitsize of the input (and
where \sO refers to the complexity where polylogarithmic factors are omitted
and refers to the bit complexity). This algorithm simplifies and
decreases by a factor the worst-case bit complexity presented for this
problem by Bouzidi et al. \cite{bouzidiJSC2014a}. This algorithm also yields,
for this problem, a probabilistic Las-Vegas algorithm of expected bit
complexity \sOB(d^5+d^4\tau).Comment: ISSAC - 39th International Symposium on Symbolic and Algebraic
Computation (2014
Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables
Analytic combinatorics studies the asymptotic behaviour of sequences through
the analytic properties of their generating functions. This article provides
effective algorithms required for the study of analytic combinatorics in
several variables, together with their complexity analyses. Given a
multivariate rational function we show how to compute its smooth isolated
critical points, with respect to a polynomial map encoding asymptotic
behaviour, in complexity singly exponential in the degree of its denominator.
We introduce a numerical Kronecker representation for solutions of polynomial
systems with rational coefficients and show that it can be used to decide
several properties (0 coordinate, equal coordinates, sign conditions for real
solutions, and vanishing of a polynomial) in good bit complexity. Among the
critical points, those that are minimal---a property governed by inequalities
on the moduli of the coordinates---typically determine the dominant asymptotics
of the diagonal coefficient sequence. When the Taylor expansion at the origin
has all non-negative coefficients (known as the `combinatorial case') and under
regularity conditions, we utilize this Kronecker representation to determine
probabilistically the minimal critical points in complexity singly exponential
in the degree of the denominator, with good control over the exponent in the
bit complexity estimate. Generically in the combinatorial case, this allows one
to automatically and rigorously determine asymptotics for the diagonal
coefficient sequence. Examples obtained with a preliminary implementation show
the wide applicability of this approach.Comment: As accepted to proceedings of ISSAC 201
On The Applications of Lifting Techniques
Lifting techniques are some of the main tools in solving a variety of different computational problems related to the field of computer algebra. In this thesis, we will consider two fundamental problems in the fields of computational algebraic geometry and number theory, trying to find more efficient algorithms to solve such problems.
The first problem, solving systems of polynomial equations, is one of the most fundamental problems in the field of computational algebraic geometry. In this thesis, We discuss how to solve bivariate polynomial systems over either k(T ) or Q using a combination of lifting and modular composition techniques. We will show that one can find an equiprojectable decomposition of a bivariate polynomial system in a better time complexity than the best known algorithms in the field, both in theory and practice.
The second problem, polynomial factorization over number fields, is one of the oldest problems in number theory. It has lots of applications in many other related problems and there have been lots of attempts to solve the problem efficiently, at least, in practice. Finding p-adic factors of a univariate polynomial over a number field uses lifting techniques. Improving this step can reduce the total running time of the factorization in practice. We first introduce a multivariate version of the Belabas factorization algorithm over number fields. Then we will compare the running time complexity of the factorization problem using two different representations of a number field, univariate vs multivariate, and at the end as an application, we will show the improvement gained in computing the splitting fields of a univariate polynomial over rational field
On the complexity of computing with zero-dimensional triangular sets
We study the complexity of some fundamental operations for triangular sets in
dimension zero. Using Las-Vegas algorithms, we prove that one can perform such
operations as change of order, equiprojectable decomposition, or quasi-inverse
computation with a cost that is essentially that of modular composition. Over
an abstract field, this leads to a subquadratic cost (with respect to the
degree of the underlying algebraic set). Over a finite field, in a boolean RAM
model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm
for modular composition. Conversely, we also show how to reduce the problem of
modular composition to change of order for triangular sets, so that all these
problems are essentially equivalent. Our algorithms are implemented in Maple;
we present some experimental results
An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks
We present an exact and complete algorithm to isolate the real solutions of a
zero-dimensional bivariate polynomial system. The proposed algorithm
constitutes an elimination method which improves upon existing approaches in a
number of points. First, the amount of purely symbolic operations is
significantly reduced, that is, only resultant computation and square-free
factorization is still needed. Second, our algorithm neither assumes generic
position of the input system nor demands for any change of the coordinate
system. The latter is due to a novel inclusion predicate to certify that a
certain region is isolating for a solution. Our implementation exploits
graphics hardware to expedite the resultant computation. Furthermore, we
integrate a number of filtering techniques to improve the overall performance.
Efficiency of the proposed method is proven by a comparison of our
implementation with two state-of-the-art implementations, that is, LPG and
Maple's isolate. For a series of challenging benchmark instances, experiments
show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201
Racah Polynomials and Recoupling Schemes of
The connection between the recoupling scheme of four copies of
, the generic superintegrable system on the 3 sphere, and
bivariate Racah polynomials is identified. The Racah polynomials are presented
as connection coefficients between eigenfunctions separated in different
spherical coordinate systems and equivalently as different irreducible
decompositions of the tensor product representations. As a consequence of the
model, an extension of the quadratic algebra is given. It is
shown that this algebra closes only with the inclusion of an additional shift
operator, beyond the eigenvalue operators for the bivariate Racah polynomials,
whose polynomial eigenfunctions are determined. The duality between the
variables and the degrees, and hence the bispectrality of the polynomials, is
interpreted in terms of expansion coefficients of the separated solutions
Computing the topology of a planar or space hyperelliptic curve
We present algorithms to compute the topology of 2D and 3D hyperelliptic
curves. The algorithms are based on the fact that 2D and 3D hyperelliptic
curves can be seen as the image of a planar curve (the Weierstrass form of the
curve), whose topology is easy to compute, under a birational mapping of the
plane or the space. We report on a {\tt Maple} implementation of these
algorithms, and present several examples. Complexity and certification issues
are also discussed.Comment: 34 pages, lot of figure
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