On the boolean complexity of real root refinement

Abstract

We assume that a real square-free polynomial A has a degree d, a maximum coefficient bitsize τ and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then, we combine the Double Exponential Sieve algorithm (also called the Bisection of the Exponents), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of t=2^-L. The algorithm has Boolean complexity Õ_B(d² τ + d L ). Our algorithms support the same complexity bound for the refinement of r roots, for any r ≤ d.