Univariate real root isolation in multiple extension fields

• Authors: Adam Strzeboński;Elias P. Tsigaridas
• Venue: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
• Year: 2012

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,..., αℓ) is an algebraic extension of the rational numbers. Our bounds are single exponential in ℓ and match the ones presented in [34] for the case ℓ = 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(N4ℓ+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 [equation], respectively. We implemented the algorithms in C as part of the core library of Mathematica and we illustrate their efficiency over various data sets.