A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems

TL;DR

This paper presents an improved and extended generic position method for isolating real roots of zero-dimensional polynomial systems, utilizing resultants and univariate root isolation, with proven complexity and certified correctness.

Contribution

The authors extend the generic position method to general zero-dimensional systems and analyze its complexity, providing an efficient, certified algorithm for real root isolation.

Findings

Complexity is $ ilde{O}_B(N^{10})$ for bivariate systems.

Method is efficient and particularly effective for bivariate polynomial systems.

Algorithm is certified with probability 1 in multivariate cases.

Abstract

We improve the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials and extend the method to general zero-dimensional polynomial systems. The method mainly involves resultant computation and real root isolation of univariate polynomial equations. The roots of the system have a linear univariate representation. The complexity of the method is $\tilde{O}_B(N^{10})$ for the bivariate case, where $N=\max(d,\tau)$, $d$ resp., $\tau$ is an upper bound on the degree, resp., the maximal coefficient bitsize of the input polynomials. The algorithm is certified with probability 1 in the multivariate case. The implementation shows that the method is efficient, especially for bivariate polynomial systems.