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 triangulation-decomposition 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 triangulation-decomposition 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.