CoCoA » CoCoALib

Part of this task has already been solved by #90 CoefficientsWRT.

JAA now needs a fn which converts a polynomial into a vector of coeffs.

Inputs are a polynomial and an ordered set of PPs;
it is assumed that the poly is a lin comb of the given PPs (o/w error).
output is a vector of coeffs giving the lin comb.

The simplest way to specify an ordered set of PPs is to represent them as a polynomial (whose coeffs can be ignored?) -- this does constrain the order of the PPs to be that of the PPordering.

A possibly more natural way to represent an ordered set of PPs is as a vector, but this leads to some complications (e.g. there may be duplicates, the PPs can appear in any order (which is tedious rather than truly problematic), we would have to check that each entry is indeed a PP in the correct poly ring) compared to representing the set as the support of a polynomial.

We need a good name for this fn. The name CoefficientsWRT would be fairly good, but could be confusing (or even lead to ambiguity). How about CoeffsWRTBasis? Other possibilities are CoeffVec or CoeffList.

JAA thinks that the most useful form of CoeffsWRTBasis is one where the basis is an ordered set of PPs. Mathematically the basis can be any collection of (lin indep) polynomials; but computationally this is likely to be much costlier as the basis must be suitably prepared (e.g. triangularized or diagonalized).

It is likely that we will want an analogous function for module elements.
Perhaps the name should be chosen so that it can apply equally well to module elements.

John Abbott wrote:

We need a good name for this fn. The name CoefficientsWRT would be fairly good, but could be confusing (or even lead to ambiguity). How about CoeffsWRTBasis? Other possibilities are CoeffVec or CoeffList.

my preferred is CoefficientsWRTBasis, the name is long, but it reflects a function whose meaning is quite specialized.

After talking to AMB, it seems best to "forget" the general function for the time being (until we have a genuine need for it).

The long name for the special fn is appropriate because it expects an unusual repr for the basis of PPs. The fn will also check that all coeffs in the "basis polynomial" are equal to 1 (o/w error).

Once this fn has been impl'ed and documented this task will finish.

I think that, by its current implementation, the order of the elements in the result of CoefficientsWRT is set (increasing on "PP" in the current ordering).

We should state in the documentation whether this is guaranteed or not guaranteed.