  We propose new algorithms for computing triangular decompositions of
polynomial systems incrementally. With respect to previous works, our
improvements are based on a {\em weakened} notion of a polynomial GCD modulo a
regular chain, which permits to greatly simplify and optimize the
sub-algorithms. Extracting common work from similar expensive computations is
also a key feature of our algorithms. In our experimental results the
implementation of our new algorithms, realized with the {\RegularChains}
library in {\Maple}, outperforms solvers with similar specifications by several
orders of magnitude on sufficiently difficult problems.
