The concept of comprehensive triangular decomposition (CTD) was first
introduced by Chen et al. in their CASC'2007 paper and could be viewed as an
analogue of comprehensive Grobner systems for parametric polynomial systems.
The first complete algorithm for computing CTD was also proposed in that paper
and implemented in the RegularChains library in Maple. Following our previous
work on generic regular decomposition for parametric polynomial systems, we
introduce in this paper a so-called hierarchical strategy for computing CTDs.
Roughly speaking, for a given parametric system, the parametric space is
divided into several sub-spaces of different dimensions and we compute CTDs
over those sub-spaces one by one. So, it is possible that, for some benchmarks,
it is difficult to compute CTDs in reasonable time while this strategy can
obtain some "partial" solutions over some parametric sub-spaces. The program
based on this strategy has been tested on a number of benchmarks from the
literature. Experimental results on these benchmarks with comparison to
RegularChains are reported and may be valuable for developing more efficient
triangularization tools
