Section:
New Results
Univariate Real Root Isolation in Multiple Extension Fields
In [31]
we present algorithmic, complexity and
implementation results for the problem of isolating
the real roots of a univariate polynomial in , where is an algebraic extension of the
rational numbers. Our bounds are single exponential
in and match the ones presented for the case . 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
. The Boolean complexity of this approach is
, where 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
and
, respectively. We
implemented the algorithms in C as part of
the core library of Mathematica and we illustrate
their efficiency over various data sets.