Univariate real root isolation in an extension field and applications

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 ÕB(N ) 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 ÕB(N ) and for the latter a bound of ÕB(N ). Using the results of Pan and Schönhage we can further reduce complexity to ÕB(N ). 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 ÕB(N ) or ÕB(N ). These bounds allows us to derive improved complexity results for the problems of isolating the real roots of a bivariate polynomial system and computing the topology of a real plane algebraic curve, that improve the previously known ones by a factor of N. 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.