Univariate real root isolation in multiple extension fields

Abstract

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in Bα ∈ L[y], where L = ( Q(α1, . . . , α ) is an algebraic extension of the rational numbers. Our bounds are single exponential in and match the ones presented in [34] for the case = 1. We consider two approaches. The first, indirect approach, using multivariate resultants, computes a univariate polynomial with integer coefficients, among the real roots of which are the real roots of Bα. The Boolean complexity of this approach is OB(N 4 +4 ), where N is the maximum of the degrees and the coefficient bitsize of the involved polynomials. The second, direct approach, tries to solve the polynomial directly, without reducing the problem to a univariate one. We present an algorithm that generalizes Sturm algorithm from the univariate case, and modified versions of well known solvers that are either numerical or based on Descartes' rule of sign. We achieve a Boolean complexity of OB(min{N 4 +7 , N 2 2 +6 }) and OB(N 2 +4 ), respectively. We implemented the algorithms in C as part of the core library of mathematica and we illustrate their efficiency over various data sets.

On the computation of the topology of plane curves

“…Using the results of [20] (see also [8,12,24]) we have the following Proposition 9 [20] Let P ∈ Z[X, Y ] be a square-free polynomial of total degree d and integer coefficients of bitsize bounded by τ . Supposing that, for every real root α of D, deg(gcd(P (α, Y )), ∂Y P (α, Y )) is known, there is an algorithm with bit complexityÕ(d 5 τ + d 6 ) for − computing a set of special boxes …”

Computation in Real Closed Infinitesimal and Transcendental Extensions of the Rationals

“…If K depends on infinitesimals, to guarantee termination, we interrupt the binary search if the size of the interval in a branch is smaller than a user provided parameter, and switch to the approach based on sign determination. Note that if p does not depend on infinitesimal values, then more efficient root isolation methods can be used [23].…”

Univariate Real Root Isolation over a Single Logarithmic Extension of Real Algebraic Numbers

“…For a detailed treatment of numerical solvers we refer the reader to [25,Chapter 15]. Problem 1 is a generalization of the problem of solving polynomials with coefficients in an extension field, [22,36,35], see also [9,40,39,23] and references therein. We also refer to the recent work of Bates and Sottile [4] on Khovanskii-Rolle continuation algorithm that exploits logarithms of polynomial expressions.…”