Improved complexity bounds for real root isolation using Continued Fractions

Abstract

We consider the problem of isolating the real roots of a square-free polynomial with integer coefficients using (variants of) the continued fraction algorithm (CF). We introduce a novel way to compute a lower bound on the positive real roots of univariate polynomials. This allows us to derive a worst case bound of $\sOB(d^6 + d^4\tau^2 + d^3\tau^2)$ for isolating the real roots of a polynomial with integer coefficients using the classic variant \cite{Akritas:implementation} of CF, where $d$ is the degree of the polynomial and $\tau$ the maximum bitsize of its coefficients. This improves the previous bound of Sharma \cite{sharma-tcs-2008} by a factor of $d^3$ and matches the bound derived by Mehlhorn and Ray \cite{mr-jsc-2009} for another variant of CF; it also matches the worst case bound of the subdivision-based solvers.