Introduction to numerical algebraic geometry (original) (raw)

Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components

SIAM Journal on Numerical Analysis, 2001

In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of di erent dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by nding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on its multiplicity. The bound is sharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroes of a nite number of polynomials.

Regeneration homotopies for solving systems of polynomials

Mathematics of Computation, 2010

We present a new technique, based on polynomial continuation, for solving systems of n polynomials in N complex variables. The method allows equations to be introduced one-by-one or in groups, obtaining at each stage a representation of the solution set that can be extended to the next stage until finally obtaining the solution set for the entire system. At any stage where positive dimensional solution components must be found, they are sliced down to isolated points by the introduction of hyperplanes. By moving these hyperplanes, one may build up the solution set to an intermediate system in which a union of hyperplanes "regenerates" the intersection of the component with the variety of the polynomial (or system of polynomials) brought in at the next stage. The theory underlying the approach guarantees that homotopy paths lead to all isolated solutions, and this capability can be used to generate witness supersets for solution components at any dimension, the first step in computing an irreducible decomposition of the solution set of a system of polynomial equations. The method is illustrated on several challenging problems, where it proves advantageous over both the polyhedral homotopy method and the diagonal equation-by-equation method, formerly the two leading approaches to solving sparse polynomial systems by numerical continuation.

PHoM ? a Polyhedral Homotopy Continuation Method for Polynomial Systems

Computing, 2004

PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedral-linear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations f (x) = 0. The second module CMPSc traces the solution curves of the homotopy equations to compute all isolated solutions of f (x) = 0. The third module Verify checks whether all isolated solutions of f (x) = 0 have been approximated correctly. We describe numerical methods used in each module and the usage of the package. Numerical results to demonstrate the performance of PHoM include some large polynomial systems that have not been solved previously.

Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets

Journal of Complexity, 2000

Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to various polynomial systems, such as the cyclic n-roots problem.

The complexity and geometry of numerically solving polynomial systems

2012

These pages contain a short overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Stephen Smale and Michael Shub, which set the foundations of this approach to polynomial system--solving, culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo, Peter Buergisser and Felipe Cucker.

Fast Linear Homotopy to Find Approximate Zeros of Polynomial Systems

Foundations of Computational Mathematics, 2011

We prove a new complexity bound, polynomial on the average, for the problem of finding an approximate zero of systems of polynomial equations. The average number of Newton steps required by this method is almost linear in the size of the input. We show that the method can also be used to approximate several or all the solutions of non-degenerate systems, and prove that this last task can be done in running time which is linear in the Bézout number of the system, on the average.

AN EXPLORATION OF HOMOTOPY SOLVING IN MAPLE

Computer Mathematics - Proceedings of the Sixth Asian Symposium (ASCM 2003), 2003

Homotopy continuation methods find approximate solutions of a given system by a continuous deformation of the solutions of a related exactly solvable system. There has been much recent progress in the theory and implementation of such path following methods for polynomial systems. In particular, exactly solvable related systems can be given which enable the computation of all isolated roots of a given polynomial system. Extension of such methods to determine manifolds of solutions has also been recently achieved. This progress, and our own research on extending continuation methods to identifying missing constraints for systems of differential equations, motivated us to implement higher order continuation methods in the computer algebra language Maple. By higher order, we refer to the iterative scheme used to solve for the roots of the homotopy equation at each step.

Software for Numerical Algebraic Geometry: A Paradigm and Progress Towards its Implementation

The IMA Volumes in Mathematics and its Applications, 2008

Though numerical methods to find all the isolated solutions of nonlinear systems of multivariate polynomials go back 30 years, it is only over the last decade that numerical methods have been devised for the computation and manipulation of algebraic sets coming from polynomial systems over the complex numbers. Collectively, these algorithms and the underlying theory have come to be known as numerical algebraic geometry. Several software packages are capable of carrying out some of the operations of numerical algebraic geometry, although no one package provides all such capabilities. This paper contains an enumeration of the operations that an ideal software package in this field would allow. The current and upcoming capabilities of Bertini, the most recently released package in this field, are also described.

Computation of all solutions to a system of polynomial equations

Mathematical Programming, 1983

This paper proposes a homotopy continuation method for approximating all solutions to a system of polynomial equations in several complex variables. The method is based on piecewise linear approximation and complementarity theory. It utilizes a skilful artificial map and two copies of the triangulation J3 with continuous refinement of grid size to increase the computational efficiency and to avoid the necessity of determining the grid size a priori. Some computational results are also reported.

A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations

SIAM Journal on Numerical Analysis, 2009

The solution set V of a polynomial system, i.e., the set of common zeroes of a set of multivariate polynomials with complex coefficients, may contain several components, e.g., points, curves, surfaces, etc. Each component has attached to it a number of quantities, one of which is its dimension. Given a numerical approximation to a point p on the set V , this article presents an efficient algorithm to compute the maximum dimension of the irreducible components of V which pass through p, i.e., a local dimension test. Such a test is a crucial element in the homotopy-based numerical irreducible decomposition algorithms of Sommese, Verschelde, and Wampler.