Improved complexity bounds for real root isolation using Continued Fractions

TL;DR

This paper presents improved worst-case complexity bounds for real root isolation of square-free polynomials using continued fraction algorithms, introducing a new method to compute lower bounds on positive roots.

Contribution

It introduces a novel approach to compute lower bounds on positive roots, leading to tighter complexity bounds for continued fraction-based root isolation methods.

Findings

Derived a worst-case bound of $ ilde{O}(d^6 + d^4 au^2 + d^3 au^2)$ for CF-based root isolation.

Improved previous complexity bounds by a factor of $d^3$.

Matched bounds of subdivision-based solvers for root isolation.

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.