1,417 research outputs found
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
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
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
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
Counting Solutions of a Polynomial System Locally and Exactly
We propose a symbolic-numeric algorithm to count the number of solutions of a
polynomial system within a local region. More specifically, given a
zero-dimensional system , with
, and a polydisc
, our method aims to certify the existence
of solutions (counted with multiplicity) within the polydisc.
In case of success, it yields the correct result under guarantee. Otherwise,
no information is given. However, we show that our algorithm always succeeds if
is sufficiently small and well-isolating for a -fold
solution of the system.
Our analysis of the algorithm further yields a bound on the size of the
polydisc for which our algorithm succeeds under guarantee. This bound depends
on local parameters such as the size and multiplicity of as well
as the distances between and all other solutions. Efficiency of
our method stems from the fact that we reduce the problem of counting the roots
in of the original system to the problem of solving a
truncated system of degree . In particular, if the multiplicity of
is small compared to the total degrees of the polynomials ,
our method considerably improves upon known complete and certified methods.
For the special case of a bivariate system, we report on an implementation of
our algorithm, and show experimentally that our algorithm leads to a
significant improvement, when integrated as inclusion predicate into an
elimination method
On the Complexity of Computing with Planar Algebraic Curves
In this paper, we give improved bounds for the computational complexity of
computing with planar algebraic curves. More specifically, for arbitrary
coprime polynomials , and an arbitrary polynomial , each of total degree less than and with integer
coefficients of absolute value less than , we show that each of the
following problems can be solved in a deterministic way with a number of bit
operations bounded by , where we ignore polylogarithmic
factors in and :
(1) The computation of isolating regions in for all complex
solutions of the system ,
(2) the computation of a separating form for the solutions of ,
(3) the computation of the sign of at all real valued solutions of , and
(4) the computation of the topology of the planar algebraic curve
defined as the real valued vanishing set of the polynomial .
Our bound improves upon the best currently known bounds for the first three
problems by a factor of or more and closes the gap to the
state-of-the-art randomized complexity for the last problem.Comment: 41 pages, 1 figur
On the computation of the topology of plane curves
International audienceLet P be a square free bivariate polynomial of degree at most d and with integer coefficients of bit size at most t. We give a deterministic algorithm for the computation of the topology of the real algebraic curve definit by P, i.e. a straight-line planar graph isotopic to the curve. Our main result is an algorithm for the computation of the local topology in a neighbourhood of each of the singular and critical points of the projection wrt the X axis in bit operations where means that we ignore logarithmic factors in and . Combined to state of the art sub-algorithms used for computing a Cylindrical Algebraic Decomposition, this result avoids a generic shear and gives a deterministic algorithm for the computation of the topology of the curve in bit operations
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
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