We introduce new ideas to improve the efficiency
and rationality of a triangulation decomposition algorithm.
On the one hand we identify and isolate the polynomial remainder sequences in
the triangulationdecomposition algorithm.
Subresultant polynomial remainder sequences are then used to compute
them and their specialisation properties are applied for the splittings.
The gain is two fold: control of expression swell
and reduction of the number of splittings.
On the other hand, we remove the role that initials had
in previous triangulationdecomposition algorithms.
They are not needed in theoretical results and it was expected
that they need not appear in the algorithms. This is the case
of the algorithm presented.
