From Approximate Factorization to Root Isolation with Application to Cylindrical Algebraic Decomposition

Abstract

We present an algorithm for isolating the roots of an arbitrary complex polynomial $p$ that also works for polynomials with multiple roots provided that the number $k$ of distinct roots is given as part of the input. It outputs $k$ pairwise disjoint disks each containing one of the distinct roots of $p$, and its multiplicity. The algorithm uses approximate factorization as a subroutine. In addition, we apply the new root isolation algorithm to a recent algorithm for computing the topology of a real planar algebraic curve specified as the zero set of a bivariate integer polynomial and for isolating the real solutions of a bivariate polynomial system. For input polynomials of degree $n$ and bitsize $\tau$, we improve the currently best running time from $\tO(n^{9}\tau+n^{8}\tau^{2})$ (deterministic) to $\tO(n^{6}+n^{5}\tau)$ (randomized) for topology computation and from $\tO(n^{8}+n^{7}\tau)$ (deterministic) to $\tO(n^{6}+n^{5}\tau)$ (randomized) for solving bivariate systems.