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(α) is a simple algebraic extension of the rational numbers. We revisit two approaches for the problem. In the first approach, using resultant computations, we perform a reduction to a polynomial with integer coefficients and we deduce a bound of OB(N 10 ) for isolating the real roots of Bα, 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 the roots and we prove that they are optimal, under mild assumptions. For isolating the real roots we consider a modified Sturm algorithm, and a modified version of descartes' algorithm introduced by Sagraloff. For the former we prove a complexity bound of OB(N 8 ) and for the latter a bound of OB(N 7 ). We implemented the algorithms in 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.
The latter claim has already been proven by means of a recent implementation from A. Strzebonski and E. Tsigaridas REF "in C as part of the core library of MATHEMATICA."
Univariate real root isolation in an extension field
ISSAC '11
Mathematics Computer Science