The Bernstein Basis and Real Root Isolation

Summary

In this mostly expository paper we explain how the Bernstein basis, widely used in computer-aided geometric design, provides an efficient method for real root isolation, using de Casteljau's algorithm. We discuss the link between this approach and more classical methods for real root isolation. We also present a new improved method for isolating real roots in the Bernstein basis inspired by Roullier and Zimmerman.

Introduction

Real root isolation is an important subroutine in many algorithms of real algebraic geometry [Basu et al. 2003] as well as in exact geometric computations, and is also interesting in its own right.

Our approach to real root isolation is based on properties of the Bernstein basis. We first recall Descartes’ Law of Signs and give a useful partial reciprocal to it. Section 2 contains the definition and main properties of the Bernstein basis. In the third section, several variants of real root isolation based on the Bernstein basis are given. In the fourth section, the link with more classical real root isolation methods [Uspensky 1948] is established. We end the paper with a few remarks on the computational efficiency of the algorithms described.