We describe a subdivision algorithm for isolating the complex roots of a polynomial F ∈ C [x]. Given an oracle that provides approximations of each of the coefficients of F to any absolute error bound and given an arbitrary square B in the complex plane containing only simple roots of F , our algorithm returns disjoint isolating disks for the roots of F in B.Our complexity analysis bounds the absolute error to which the coefficients of F have to be provided, the total number of iterations, and the overall bit complexity. It further shows that the complexity of our algorithm is controlled by the geometry of the roots in a near neighborhood of the input square B, namely, the number of roots, their absolute values and pairwise distances. The number of subdivision steps is near-optimal. For the benchmark problem, namely, to isolate all the roots of a polynomial of degree n with integer coefficients of bit size less than τ , our algorithm needsÕ(n 3 + n 2 τ ) bit operations, which is comparable to the record bound of Pan (2002). It is the first time that such a bound has been achieved using subdivision methods, and independent of divide-and-conquer techniques such as Schönhage's splitting circle technique.Our algorithm uses the quadtree construction of Weyl (1924) with two key ingredients: using Pellet's Theorem (1881) combined with Graeffe iteration, we derive a "soft-test" to count the number of roots in a disk. Using Schröder's modified Newton operator combined with bisection, in a form inspired by the quadratic interval method from Abbot (2006), we achieve quadratic convergence towards root clusters. Relative to the divide-conquer algorithms, our algorithm is quite simple with the potential of being practical. This paper is self-contained: we provide pseudo-code for all subroutines used by our algorithm.

Let F (z) be an arbitrary complex polynomial. We introduce the local root clustering problem, to compute a set of natural ε-clusters of roots of F (z) in some box region B0 in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is twofold: we provide an efficient certified subdivision algorithm for this problem, and we provide a bit-complexity analysis based on the local geometry of the root clusters. Our computational model assumes that arbitrarily good approximations of the coefficients of F are provided by means of an oracle at the cost of reading the coefficients. Our algorithmic techniques come from a companion paper [3] and are based on the Pellet test, Graeffe and Newton iterations, and are independent of Schönhage's splitting circle method. Our algorithm is relatively simple and promises to be efficient in practice.

We present an algorithm for isolating all roots of an arbitrary complex polynomial p that also works in the presence of multiple roots provided that (1) the number of distinct roots is given as part of the input and (2) the algorithm can ask for arbitrarily good approximations of the coefficients of p. The algorithm outputs pairwise disjoint disks each containing one of the distinct roots of p and the multiplicity of the root contained in the disk. The algorithm uses approximate factorization as a subroutine. For the case where Pan's algorithm [34] 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 thatÕ(d 3 + d 2 τ + 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.In addition, we apply our 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 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. (Pengming Wang). 1 An alternative restriction is to be content with the computation of well-separated clusters of roots, i.e., the computation of disks ∆ i and multiplicities m i such that D i contains exactly m i roots counted with multiplicity, ∑ i m i is equal to the degree of the polynomial, and substantially enlarged disks are disjoint. Our algorithm also applies to this version of the problem. We come back to it in Section 2.4. 2 The additional requirement for the leading coefficient p n yields a simpler presentation. Notice that, for general values p n , we first have to multiply the polynomial p by some 2 t , with t ∈ Z, such that 2 t · |p n | is contained in [1/4,1].

Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schönhage in 1982. The main drawbacks of Pan's method are that it is quite involved 1 and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity.In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted ANewDsc, which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. ANewDsc can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task.

We consider the problem of approximating all real roots of a squarefree polynomial f . Given isolating intervals, our algorithm refines each of them to a width at most 2 −L , that is, each of the roots is approximated to L bits after the binary point. Our method provides a certified answer for arbitrary real polynomials, only requiring finite approximations of the polynomial coefficient and choosing a suitable working precision adaptively. In this way, we get a correct algorithm that is simple to implement and practically efficient. Our algorithm uses the quadratic interval refinement method; we adapt that method to be able to cope with inaccuracies when evaluating f , without sacrificing its quadratic convergence behavior. We prove a bound on the bit complexity of our algorithm in terms of degree, coefficient size and discriminant. Our bound improves previous work on integer polynomials by a factor of deg f and essentially matches best known theoretical bounds on root approximation which are obtained by very sophisticated algorithms.

In this paper, we give improved bounds for the computational complexity of computing with planar algebraic curves. More specifically, for arbitrary coprime polynomials f , g ∈ Z[x, y] and an arbitrary polynomial h ∈ Z[x, y], each of total degree less than n and with integer coefficients of absolute value less than 2 τ , we show that each of the following problems can be solved in a deterministic way with a number of bit operations bounded byÕ(n 6 + n 5 τ), where we ignore polylogarithmic factors in n and τ:• The computation of isolating regions in C 2 for all complex solutions of the system f = g = 0,• the computation of a separating form for the solutions of f = g = 0,• the computation of the sign of h at all real valued solutions of f = g = 0, and• the computation of the topology of the planar algebraic curve C defined as the real valued vanishing set of the polynomial f .Our bound improves upon the best currently known bounds for the first three problems by a factor of n 2 or more and closes the gap to the state-of-the-art randomized complexity for the last problem.

Very recent work introduces an asymptotically fast subdivision algorithm, denoted ANewDsc, for isolating the real roots of a univariate real polynomial. The method combines Descartes' Rule of Signs to test intervals for the existence of roots, Newton iteration to speed up convergence against clusters of roots, and approximate computation to decrease the required precision. It achieves record bounds on the worst-case complexity for the considered problem, matching the complexity of Pan's method for computing all complex roots and improving upon the complexity of other subdivision methods by several magnitudes.In the article at hand, we report on an implementation of ANewDsc on top of the RS root isolator. RS is a highly efficient realization of the classical Descartes method and currently serves as the default real root solver in Maple. We describe crucial design changes within ANewDsc and RS that led to a highperformance implementation without harming the theoretical complexity of the underlying algorithm.With an excerpt of our extensive collection of benchmarks, available online at http://anewdsc.mpi-inf.mpg.de/, we illustrate that the theoretical gain in performance of ANewDsc over other subdivision methods also transfers into practice. These experiments also show that our new implementation outperforms both RS and mature competitors by magnitudes for notoriously hard instances with clustered roots. For all other instances, we avoid almost any overhead by integrating additional optimizations and heuristics.

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-ofthe-art implementations, that is, Lgp and Maple's Isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.