9 research outputs found
Solving Polynomial Systems via a Stabilized Representation of Quotient Algebras
We consider the problem of finding the isolated common roots of a set of
polynomial functions defining a zero-dimensional ideal I in a ring R of
polynomials over C. We propose a general algebraic framework to find the
solutions and to compute the structure of the quotient ring R/I from the null
space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous
and multi-homogeneous cases are treated. In the presented framework, the
concept of a border basis is generalized by relaxing the conditions on the set
of basis elements. This allows for algorithms to adapt the choice of basis in
order to enhance the numerical stability. We present such an algorithm and show
numerical results
A Robust Numerical Path Tracking Algorithm for Polynomial Homotopy Continuation
We propose a new algorithm for numerical path tracking in polynomial homotopy
continuation. The algorithm is `robust' in the sense that it is designed to
prevent path jumping and in many cases, it can be used in (only) double
precision arithmetic. It is based on an adaptive stepsize predictor that uses
Pad\'e techniques to detect local difficulties for function approximation and
danger for path jumping. We show the potential of the new path tracking
algorithm through several numerical examples and compare with existing
implementations.Comment: 30 pages, 12 figures, 7 table
Toric Eigenvalue Methods for Solving Sparse Polynomial Systems
We consider the problem of computing homogeneous coordinates of points in a
zero-dimensional subscheme of a compact toric variety . Our starting point
is a homogeneous ideal in the Cox ring of , which gives a global
description of this subscheme. It was recently shown that eigenvalue methods
for solving this problem lead to robust numerical algorithms for solving
(nearly) degenerate sparse polynomial systems. In this work, we give a first
description of this strategy for non-reduced, zero-dimensional subschemes of
. That is, we allow isolated points with arbitrary multiplicities.
Additionally, we investigate the regularity of to provide the first
universal complexity bounds for the approach, as well as sharper bounds for
weighted homogeneous, multihomogeneous and unmixed sparse systems, among
others. We disprove a recent conjecture regarding the regularity and prove an
alternative version. Our contributions are illustrated by several examples.Comment: 41 pages, 7 figure
Truncated Normal Forms for Solving Polynomial Systems: Generalized and Efficient Algorithms
International audienceWe consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem. The framework presented in Telen et al. (2018) uses truncated normal forms (TNFs) to compute the algebra structure of R/I and the solutions of I. This framework allows for the use of much more general bases than the standard monomials for R/I. This is exploited in this paper to introduce the use of two special (nonmonomial) types of basis functions with nice properties. This allows, for instance, to adapt the basis functions to the expected location of the roots of I. We also propose algorithms for efficient computation of TNFs and a generalization of the construction of TNFs in the case of non-generic zero-dimensional systems. The potential of the TNF method and usefulness of the new results are exposed by many experiments
A Stabilized Normal Form Algorithm for Generic Systems of Polynomial Equations
We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring associated to a zero-dimensional ideal generated by -polynomials in variables. We assume that the polynomials are generic in the sense that the number of solutions in equals the B\'ezout number. The main contribution of this paper is an automated choice of basis for , which is crucial for the feasibility of normal form methods in finite precision arithmetic. This choice is based on numerical linear algebra techniques and it depends on the given generators of .status: publishe