Univariate real root isolation in an extension field and applications

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(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 O˜B(N8) for isolating the real roots of Bα, where N is an upper bound on all the quantities (degree and bitsize) of the input polynomials. The bound becomes O˜B(N7) if we use Pan's algorithm for isolating the real roots. 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. For the former we prove a Boolean complexity bound of O˜B(N12) and for the latter a bound of O˜B(N5). We present aggregate separation bounds and complexity results for isolating the real roots of all polynomials Bαk, when αk runs over all the real conjugates of α. We show that we can isolate the real roots of all polynomials in O˜B(N5). Finally, we implemented the algorithms in C as part of the core library of MATHEMATICA and we illustrate their efficiency over various data sets.