Univariate real root isolation in an extension field

TL;DR

This paper develops and compares algorithms for isolating real roots of univariate polynomials over algebraic extension fields, providing complexity bounds, optimal separation bounds, and practical implementation results.

Contribution

It introduces two approaches for root isolation in extension fields, with improved complexity bounds and optimal separation bounds, and includes a practical implementation in Mathematica.

Findings

Resultant-based approach isolates roots in O(N^{10}) complexity.

Direct approach with modified Sturm's algorithm achieves O(N^{8}) complexity.

Implementation demonstrates efficiency on various data sets.

Abstract

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in $B_{\alpha} \in L[y]$, where $L=\QQ(\alpha)$ is a simple algebraic extension of the rational numbers. We consider two approaches for tackling the problem. In the first approach using resultant computations we perform a reduction to a polynomial with integer coefficients. We compute separation bounds for the roots, and using them we deduce that we can isolate the real roots of $B_{\alpha}$ in $\sOB(N^{10})$, where $N$ is an upper bound on all the quantities (degree and bitsize) of the input polynomials. In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for real roots and we prove that they are optimal, under mild assumptions. For isolating the roots we consider a modified Sturm's algorithm, and a modified version of \func{descartes}' algorithm introduced by Sagraloff. For the former we prove a complexity bound of $\sOB(N^8)$ and for the latter a bound of $\sOB(N^{7})$. We implemented the algorithms in \func{C} as part of the core library of \mathematica and we illustrate their efficiency over various data sets. Finally, we present complexity results for the general case of the first approach, where the coefficients belong to multiple extensions.