Triangular systems and factorized Gröbner bases

In a preceding paper (9) we reported on some experience with a new version of the well known Grobner algorithm with factorization and constraint inequalities. Here we discuss, how this approach may be refined to produce triangular systems in the sense of (12) and (13). Such a refinement guarantees, dierent to the usual Grobner factorizer, to produce a quasi prime decomposition, i.e. the resulting components are at least pure dimensional radical ideals. As in (9) our method weakens the usual restriction to lexicographic term orders. Triangular systems are a very helpful tool between factorization at a heuristical level and full decomposition into prime components. Our approach grew up from a consequent interpretation of the algorithmic ideas in (5) as a delayed quotient computation in favour of early use of (multivariate) factorization. It is implemented in version 2.2 of the REDUCE package CALI (8).