2,987 research outputs found
Local Generic Position for Root Isolation of Zero-dimensional Triangular Polynomial Systems
International audienceWe present an algorithm based on local generic position (LGP) to isolate the complex or real roots and their multiplicities of a zero-dimensional triangular polynomial system. The Boolean complexity of the algorithm for computing the real roots is single exponential: , where , and , is the degree and the maximum coefficient bitsize of the polynomials, respectively, and is the number of variables
A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems
We improve the local generic position method for isolating the real roots of
a zero-dimensional bivariate polynomial system with two polynomials and extend
the method to general zero-dimensional polynomial systems. The method mainly
involves resultant computation and real root isolation of univariate polynomial
equations. The roots of the system have a linear univariate representation. The
complexity of the method is for the bivariate case, where
, resp., is an upper bound on the degree, resp., the
maximal coefficient bitsize of the input polynomials. The algorithm is
certified with probability 1 in the multivariate case. The implementation shows
that the method is efficient, especially for bivariate polynomial systems.Comment: 24 pages, 5 figure
Root Isolation of Zero-dimensional Polynomial Systems with Linear Univariate Representation
In this paper, a linear univariate representation for the roots of a
zero-dimensional polynomial equation system is presented, where the roots of
the equation system are represented as linear combinations of roots of several
univariate polynomial equations. The main advantage of this representation is
that the precision of the roots can be easily controlled. In fact, based on the
linear univariate representation, we can give the exact precisions needed for
roots of the univariate equations in order to obtain the roots of the equation
system to a given precision. As a consequence, a root isolation algorithm for a
zero-dimensional polynomial equation system can be easily derived from its
linear univariate representation.Comment: 19 pages,2 figures; MM-Preprint of KLMM, Vol. 29, 92-111, Aug. 201
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
On deflation and multiplicity structure
This paper presents two new constructions related to singular solutions of
polynomial systems. The first is a new deflation method for an isolated
singular root. This construction uses a single linear differential form defined
from the Jacobian matrix of the input, and defines the deflated system by
applying this differential form to the original system. The advantages of this
new deflation is that it does not introduce new variables and the increase in
the number of equations is linear in each iteration instead of the quadratic
increase of previous methods. The second construction gives the coefficients of
the so-called inverse system or dual basis, which defines the multiplicity
structure at the singular root. We present a system of equations in the
original variables plus a relatively small number of new variables that
completely deflates the root in one step. We show that the isolated simple
solutions of this new system correspond to roots of the original system with
given multiplicity structure up to a given order. Both constructions are
"exact" in that they permit one to treat all conjugate roots simultaneously and
can be used in certification procedures for singular roots and their
multiplicity structure with respect to an exact rational polynomial system.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0508
Certifying isolated singular points and their multiplicity structure
This paper presents two new constructions related to singular solutions of
polynomial systems. The first is a new deflation method for an isolated
singular root. This construc-tion uses a single linear differential form
defined from the Jacobian matrix of the input, and defines the deflated system
by applying this differential form to the original system. The advantages of
this new deflation is that it does not introduce new variables and the increase
in the number of equations is linear instead of the quadratic increase of
previous methods. The second construction gives the coefficients of the
so-called inverse system or dual basis, which defines the multiplicity
structure at the singular root. We present a system of equations in the
original variables plus a relatively small number of new vari-ables. We show
that the roots of this new system include the original singular root but now
with multiplicity one, and the new variables uniquely determine the
multiplicity structure. Both constructions are "exact", meaning that they
permit one to treat all conjugate roots simultaneously and can be used in
certification procedures for singular roots and their multiplicity structure
with respect to an exact rational polynomial system
Bivariate systems and topology of plane curves: algebraic and numerical methods
The work presented in this thesis belongs to the domain of non-linear computational geometry in lowdimension. More precisely it focuses on solving bivariate systems and computing the topology of curvesin the plane. When the input is given by polynomials, the natural tools come from computer algebra.Our contributions are algorithms proven efficient in a deterministic or a Las Vegas settings together witha practical efficient software for topology certified drawing of a plane algebraic curve. When the input isnot restricted to be polynomials but given by interval functions, we design algorithms based on certifiednumerical approches using subdivision and interval arithmetic. The input is then required to fulfill somegeneric assumptions and our algorithms are certified in the sense that they terminate if and only if theassumptions are satisfied.Le travail présenté dans cette thèse appartient au domaine de la géométrie computationnelle non linéaireen petite dimension. Plus précisément, il se concentre sur la résolution de systèmes bivariés et le calcul dela topologie des courbes dans le plan. Lorsque l’entrée est donnée par des polynômes, les outils naturelsproviennent du calcul formel. Nos contributions sont des algorithmes dont l’efficacité a été prouvée dansun cadre déterministe ou Las Vegas, ainsi qu’un logiciel efficace pour le dessin certifié de la topologied’une courbe algébrique plane. Lorsque les données d’entrée ne sont pas limitées aux polynômes maissont données par des fonctions d’intervalles, nous concevons des algorithmes basés sur des approchesnumériques certifiées utilisant la subdivision et l’arithmétique d’intervalles. L’entrée doit alors satisfairecertaines hypothèses génériques et nos algorithmes sont certifiés dans le sens où ils se terminent si etseulement si les hypothèses sont satisfaites
Exact Symbolic-Numeric Computation of Planar Algebraic Curves
We present a novel certified and complete algorithm to compute arrangements
of real planar algebraic curves. It provides a geometric-topological analysis
of the decomposition of the plane induced by a finite number of algebraic
curves in terms of a cylindrical algebraic decomposition. From a high-level
perspective, the overall method splits into two main subroutines, namely an
algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional
bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic
curve.
Compared to existing approaches based on elimination techniques, we
considerably improve the corresponding lifting steps in both subroutines. As a
result, generic position of the input system is never assumed, and thus our
algorithm never demands for any change of coordinates. In addition, we
significantly limit the types of involved exact operations, that is, we only
use resultant and gcd computations as purely symbolic operations. The latter
results are achieved by combining techniques from different fields such as
(modular) symbolic computation, numerical analysis and algebraic geometry.
We have implemented our algorithms as prototypical contributions to the
C++-project CGAL. They exploit graphics hardware to expedite the symbolic
computations. We have also compared our implementation with the current
reference implementations, that is, LGP and Maple's Isolate for polynomial
system solving, and CGAL's bivariate algebraic kernel for analyses and
arrangement computations of algebraic curves. For various series of challenging
instances, our exhaustive experiments show that the new implementations
outperform the existing ones.Comment: 46 pages, 4 figures, submitted to Special Issue of TCS on SNC 2011.
arXiv admin note: substantial text overlap with arXiv:1010.1386 and
arXiv:1103.469
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
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