The proposed projects aim to develop new techniques or enhance existing ones for computing triangular decompositions. Each of these projects involves algorithmic and experimental activities and should lead to very interesting results.

The *lextriangular*^{1} algorithm [Laz92] and
the TRIADE^{2}
algorithm [Mor99] implemented in AXIOM, ALDOR and MAPLE
will be the core of these projects.

One of the key ideas behind these two algorithms is the *D5 Principle*
[DDD85] presented during the lectures in February.
The *lextriangular* algorithm assumes that its input is a
lexicographical Gröbner Basis of a zero-dimensional ideal whereas
the TRIADE algorithm can take as input any finite set of polynomials.
In addition, when the same input can be used with both algorithms,
the output is essentially the same.
A more detailed presentation of these algorithms will be given during
the lectures in March.

Section 2 shows how to use the AXIOM
implementations of *lextriangular* and TRIADE.
Remember that AXIOM is freely accessible
on the web.
It is also available on the ORCCA machines.
The MAPLE version of TRIADE will be made available
at the beginning of March.

Section 3 specifies the guidelines
for the project and, in particular, for the *preliminary
mini-project*.

Section 4 describes the projects offered to the students.

2005-02-04