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

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 τ , we improve the currently best running time from Õ(n9τ + n8τ2) (deterministic) to Õ(n6 + n5τ) (randomized) for topology computation and from Õ(n8 +n7τ) (deterministic) to Õ(n6 +n5τ) (randomized) for solving bivariate systems.