Let K be a field of multivariate rational functions over the rational numbers. Let L be a direct product of fields extending K by a tower of algebraic extensions. We present a modular algorithm for computing polynomial gcds over L based on a Hensel lifting strategy.
The rational reconstruction is avoided by guessing the denominator of the gcd before the lifting step. The algorithm may not stop but does terminate with probability one. Our implementation shows a significant improvement with respect to other methods.