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 fromÕ(n 9 τ + n 8 τ 2 ) (deterministic) toÕ(n 6 + n 5 τ) (randomized) for topology computation and fromÕ(n 8 + n 7 τ) (deterministic) toÕ(n 6 + n 5 τ) (randomized) for solving bivariate systems.
Pan's factorization algorithm is also a key subroutine in a recent algorithm REF for isolating all roots of a complex polynomial in the time bound of our main theorem.
From Approximate Factorization to Root Isolation with Application to Cylindrical Algebraic Decomposition
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Mathematics Computer Science