From approximate factorization to root isolation

• Authors: Kurt Mehlhorn;Michael Sagraloff;Pengming Wang
• Venue: Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
• Year: 2013

We present an algorithm for isolating all roots of an arbitrary complex polynomial p which also works in the presence of multiple roots provided that arbitrary good approximations of the coefficients of p and the number of distinct roots are given. Its output consists of pairwise disjoint disks each containing one of the distinct roots of p, and its multiplicity. The algorithm uses approximate factorization as a subroutine. For the case, where Pan's algorithm [16] is used for the factorization, we derive complexity bounds for the problems of isolating and refining all roots which are stated in terms of the geometric locations of the roots only. Specializing the latter bounds to a polynomial of degree d and with integer coefficients of bitsize less than τ, we show that Õ(d3+d2τ+dκ) bit operations are sufficient to compute isolating disks of size less than 2-κ for all roots of p, where κ is an arbitrary positive integer. Our new algorithm has an interesting consequence on the complexity of computing the topology of a real algebraic curve specified as the zero set of a bivariate integer polynomial and for isolating the real solutions of a bivariate system. For input polynomials of degree n and bitsize τ, the currently best running time improves from Õ(n9τ+n8τ2) (deterministic) to Õ(n6+n5τ) (randomized) for topology computation and from Õ(n^{8}+n7τ) (deterministic) to Õ(n6+n5τ) (randomized) for solving bivariate systems.