On the Complexity of Real Root Isolation

TL;DR

This paper presents a new, efficient, and practical method for isolating real roots of square-free polynomials, improving the bit complexity bounds over previous algorithms by leveraging approximate Descartes method techniques.

Contribution

It introduces an exact, complete, and deterministic root isolation method based on approximate Descartes, with significantly improved complexity bounds and a novel analysis of root geometry influence.

Findings

Achieves a bit complexity of n^3 u^2 for integer polynomials, improving previous bounds.

Shows that root isolation difficulty depends on root geometry, not coefficient size.

Provides an upper bound on the necessary precision for root isolation using approximate methods.

Abstract

We introduce a new approach to isolate the real roots of a square-free polynomial $F=\sum_{i=0}^n A_i x^i$ with real coefficients. It is assumed that each coefficient of $F$ can be approximated to any specified error bound. The presented method is exact, complete and deterministic. Due to its similarities to the Descartes method, we also consider it practical and easy to implement. Compared to previous approaches, our new method achieves a significantly better bit complexity. It is further shown that the hardness of isolating the real roots of $F$ is exclusively determined by the geometry of the roots and not by the complexity or the size of the coefficients. For the special case where $F$ has integer coefficients of maximal bitsize $\tau$, our bound on the bit complexity writes as $\tilde{O}(n^3\tau^2)$ which improves the best bounds known for existing practical algorithms by a factor of $n=deg F$. The crucial idea underlying the new approach is to run an approximate version of the Descartes method, where, in each subdivision step, we only consider approximations of the intermediate results to a certain precision. We give an upper bound on the maximal precision that is needed for isolating the roots of $F$. For integer polynomials, this bound is by a factor $n$ lower than that of the precision needed when using exact arithmetic explaining the improved bound on the bit complexity.