Root Isolation of Zero-dimensional Polynomial Systems with Linear Univariate Representation

TL;DR

This paper introduces a linear univariate representation for roots of zero-dimensional polynomial systems, enabling precise control over root accuracy and simplifying the derivation of root isolation algorithms.

Contribution

It presents a novel linear univariate representation that allows precise control of root accuracy and straightforward root isolation for zero-dimensional polynomial systems.

Findings

Roots can be represented as linear combinations of univariate roots.

Exact precision requirements for roots are determined from the representation.

Root isolation algorithms are easily derived from the representation.

Abstract

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.