CoCoA » CoCoALib

What output do we want or need from the Buchberger-Moeller implementations?

1. generators of the ideal (list of polynomials)
2. separators (list of polynomials)
3. quotient basis (list of PPs, or perhaps list of polynomials)

The definition of separator in the case of projective points is not unique. In C4 we opted for polynomials which evaluate to 1 on the given representatives; should we continue to use this definition in CoCoA-5? Or perhaps there should no way to compute the separators in this case?

Opinions? Comments?

JAA proposes offering two functions IdealOfPoints (for the affine case) and IdealOfProjectivePoints (for the projective case) which produce ideals.

In CoCoALib (and perhaps also in CoCoA-5) we could offer further functions for computing a complete result (presumably represented as a RECORD). In C4 these had cumbersome names such as IdealAndSeparatorsOfProjectivePoints. JAA proposes simpler names such as BuchbergerMoeller and ProjectiveBuchbergerMoeller. In the result JAA prefers a list of generators rather than an ideal -- it is quite easy for a user to produce an ideal from a list of generators (or vice versa).

Opinions?

My old impls in CoCoA-4 returned a matrix as the result: the entries in the matrix are the coeffs of the basis elements and of the separators.

The result also contained a list of PPs so that the appropriate polys could be reconstructed easily.

JAA now thinks that this is probably the most sensible type of value to return (for the internal BM fns that actually do the work).

The conversion from matrix to list of polys is straightforward, and can be effected by some auxiliary fns.