78 research outputs found
Multivariate Subresultants in Roots
We give rational expressions for the subresultants of n+1 generic polynomials
f_1,..., f_{n+1} in n variables as a function of the coordinates of the common
roots of f_1,..., f_n and their evaluation in f_{n+1}. We present a simple
technique to prove our results, giving new proofs and generalizing the
classical Poisson product formula for the projective resultant, as well as the
expressions of Hong for univariate subresultants in roots.Comment: 22 pages, no figures, elsart style, revised version of the paper
presented in MEGA 2005, accepted for publication in Journal of Algebr
Thomas Decomposition of Algebraic and Differential Systems
In this paper we consider disjoint decomposition of algebraic and non-linear
partial differential systems of equations and inequations into so-called simple
subsystems. We exploit Thomas decomposition ideas and develop them into a new
algorithm. For algebraic systems simplicity means triangularity, squarefreeness
and non-vanishing initials. For differential systems the algorithm provides not
only algebraic simplicity but also involutivity. The algorithm has been
implemented in Maple
On the equations of the moving curve ideal of a rational algebraic plane curve
Given a parametrization of a rational plane algebraic curve C, some explicit
adjoint pencils on C are described in terms of determinants. Moreover, some
generators of the Rees algebra associated to this parametrization are
presented. The main ingredient developed in this paper is a detailed study of
the elimination ideal of two homogeneous polynomials in two homogeneous
variables that form a regular sequence.Comment: Journal of Algebra (2009
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
Numeric certified algorithm for the topology of resultant and discriminant curves
Let be a real plane algebraic curve defined by the resultant of
two polynomials (resp. by the discriminant of a polynomial). Geometrically such
a curve is the projection of the intersection of the surfaces
(resp. ), and generically its singularities are nodes (resp. nodes and
ordinary cusps). State-of-the-art numerical algorithms compute the topology of
smooth curves but usually fail to certify the topology of singular ones. The
main challenge is to find practical numerical criteria that guarantee the
existence and the uniqueness of a singularity inside a given box , while
ensuring that does not contain any closed loop of . We solve
this problem by first providing a square deflation system, based on
subresultants, that can be used to certify numerically whether contains a
unique singularity or not. Then we introduce a numeric adaptive separation
criterion based on interval arithmetic to ensure that the topology of in is homeomorphic to the local topology at . Our algorithms are
implemented and experiments show their efficiency compared to state-of-the-art
symbolic or homotopic methods
Cache-Friendly, Modular and Parallel Schemes For Computing Subresultant Chains
The RegularChains library in Maple offers a collection of commands for solving polynomial systems symbolically with taking advantage of the theory of regular chains. The primary goal of this thesis is algorithmic contributions, in particular, to high-performance computational schemes for subresultant chains and underlying routines to extend that of RegularChains in a C/C++ open-source library.
Subresultants are one of the most fundamental tools in computer algebra. They are at the core of numerous algorithms including, but not limited to, polynomial GCD computations, polynomial system solving, and symbolic integration. When the subresultant chain of two polynomials is involved in a client procedure, not all polynomials of the chain, or not all coefficients of a given subresultant, may be needed. Based on that observation, we design so-called speculative and caching strategies which yield great performance improvements within our polynomial system solver.
Our implementation of these techniques has been highly optimized. We have implemented optimized core arithmetic routines and multithreaded subresultant algorithms for univariate, bivariate and multivariate polynomials. We further examine memory access patterns and data locality for computing subresultants of multivariate polynomials, and study different optimization techniques for the fraction-free LU decomposition algorithm to compute subresultants based on determinant of Bezout matrices.
Our code is publicly available at www.bpaslib.org as part of the Basic Polynomial Algebra Subprograms (BPAS) library that is mainly written in C, with concurrency support and user interfaces written in C++
The resultant on compact Riemann surfaces
We introduce a notion of resultant of two meromorphic functions on a compact
Riemann surface and demonstrate its usefulness in several respects. For
example, we exhibit several integral formulas for the resultant, relate it to
potential theory and give explicit formulas for the algebraic dependence
between two meromorphic functions on a compact Riemann surface. As a particular
application, the exponential transform of a quadrature domain in the complex
plane is expressed in terms of the resultant of two meromorphic functions on
the Schottky double of the domain.Comment: 44 page
The Design and Implementation of a High-Performance Polynomial System Solver
This thesis examines the algorithmic and practical challenges of solving systems of polynomial equations. We discuss the design and implementation of triangular decomposition to solve polynomials systems exactly by means of symbolic computation.
Incremental triangular decomposition solves one equation from the input list of polynomials at a time. Each step may produce several different components (points, curves, surfaces, etc.) of the solution set. Independent components imply that the solving process may proceed on each component concurrently. This so-called component-level parallelism is a theoretical and practical challenge characterized by irregular parallelism. Parallelism is not an algorithmic property but rather a geometrical property of the particular input system’s solution set.
Despite these challenges, we have effectively applied parallel computing to triangular decomposition through the layering and cooperation of many parallel code regions. This parallel computing is supported by our generic object-oriented framework based on the dynamic multithreading paradigm. Meanwhile, the required polynomial algebra is sup- ported by an object-oriented framework for algebraic types which allows type safety and mathematical correctness to be determined at compile-time.
Our software is implemented in C/C++ and have extensively tested the implementation for correctness and performance on over 3000 polynomial systems that have arisen in practice.
The parallel framework has been re-used in the implementation of Hensel factorization as a parallel pipeline to compute roots of a polynomial with multivariate power series coefficients. Hensel factorization is one step toward computing the non-trivial limit points of quasi-components
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