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 lextriangular1 algorithm [Laz92] and the TRIADE2 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.