  The computation of triangular decompositions are based on two fundamental
operations: polynomial GCDs modulo regular chains and regularity test modulo
saturated ideals. We propose new algorithms for these core operations relying
on modular methods and fast polynomial arithmetic. Our strategies take also
advantage of the context in which these operations are performed. We report on
extensive experimentation, comparing our code to pre-existing Maple
implementations, as well as more optimized Magma functions. In most cases, our
new code outperforms the other packages by several orders of magnitude.
