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 $K\left[x\right]\in L\left[y\right]$, where $L=\mathbb{Q}(\alpha \_1,\cdots ,\alpha \_\ell )$ is an algebraic extension of the rational numbers. Our bounds are single exponential in $\ell$ and match the ones presented for the case $\ell =1$. 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 $B_{\alpha }$. The Boolean complexity of this approach is ${\tilde{O}}_B\left(N^{4\ell +4}\right)$, where $N$ 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 ${\tilde{O}}_B(min\{N^{4\ell +7},N^{2{\ell }^2+6}\})$ and ${\tilde{O}}_B\left(N^{2\ell +4}\right)$, respectively. We implemented the algorithms in C as part of the core library of Mathematica and we illustrate their efficiency over various data sets.