 
Summary: Vincent's Theorem of 1836:
Overview and Future Research1
Alkiviadis G. Akritas
Department of Computer and Communication Engineering
University of Thessaly
Greece
akritas@uth.gr
Abstract We first present the two different versions of Vincent's theorem of 1836 and
discuss the various real root isolation methods derived from them: one using continued
fractions and two bisection methods  the former being the fastest real root isolation
method. We then concentrate on the Continued Fractions method and present: (a)
a recently developed quadratic complexity bound on the values of the positive roots
of polynomials, which helped improve its performance by an average of 40%, over its
initial implementation, and (b) directions for future research.
Key Words: Vincent's theorem, isolation of the real roots, real root isolation meth
ods, bisection methods, continued fractions method, positive root bounds.
1 Introduction
Isolation of the real roots of a polynomial is the process of finding real disjoint
intervals such that each contains one real root and every real root is contained
in some interval.
