Computing isolated roots of sparse polynomial systems in affine space

María Isabel Herrero,Gabriela Jeronimo,Juan Sabia

doi:10.1016/j.tcs.2010.07.015

María Isabel Herrero, Gabriela Jeronimo + Show 1 more

Open Access

https://doi.org/10.1016/j.tcs.2010.07.015

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Abstract

We present a symbolic probabilistic algorithm to compute the isolated roots in Cn of sparse polynomial equation systems. As some already known numerical algorithms solving this task, our procedure is based on polyhedral deformations and homotopies, but it amounts to solving a smaller number of square systems of equations and in fewer variables. The output of the algorithm is a geometric resolution of a finite set of points including the isolated roots of the system. The complexity is polynomial in the size of the combinatorial structure of the system supports up to a pre-processing yielding the mixed cells in a subdivision of the family of these supports.

Highlights

The known algorithmic methods to solve general polynomial equation systems require a large number of calculations, which results in a long computing time
A deformation method to compute the isolated roots of a polynomial system consists in considering the given system as a particular instance of a parametric family of generic zerodimensional systems; the isolated roots of the input system are obtained from the zeroes of a sufficiently generic instance which is easy to solve
We present a new symbolic algorithm to compute the isolated solutions in Cn of a sparse system with n equations

Summary

Introduction

The known algorithmic methods to solve general polynomial equation systems require a large number of calculations, which results in a long computing time. A deformation method to compute the isolated roots of a polynomial system consists in considering the given system as a particular instance of a parametric family of generic zerodimensional systems; the isolated roots of the input system are obtained from the zeroes of a sufficiently generic instance which is easy to solve These techniques, originally applied for numerical solving of equations (see, for instance, [20,27] and the references therein), have been used in symbolic procedures by means of the so-called Newton–Hensel lifting (see, for instance, [10,12,19,15]).

Basic definitions and notation

Sparse systems and subdivisions

Deformation of polynomial systems

Generic sparse systems

The algorithm

Examples