univariate real root isolation in an extension field
We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in B ? ? L [ y ], where L = Q ? 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 ( N 10 ) for isolating the real roots of B ? , 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 O B ( N 8 ) and for the latter a bound of O B ( 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.
real root isolation separation bounds sturm algebraic polynomial field extension descartes' rule of sign