1. For 0-dimensional ideals we often only care about the solutions. With the multiplicity we can deal later. Radical ideals could be used for that. The problem being, calculationg the radical of an ideal is twice exponential, and does not even use Gröbner bases. I do not know whether it is twice in the worst case, or in a random case.
2. From our physical knowledge of a system, we know that some small factors cannot be admissible solutions. Throwing them out greatly improves the chances of finishing calculations.
Now the questions. Suppose that after each iteration of Buchberger you do a factorization of the polynomial in hand (and a factorization in the very beginning). Is it possible for Singular to interactively ask the user whether to keep or throw a given factor? I assume that it then remembers that decision for the rest of its calculations. Also, cannot use Rabinowitz trick because cannot know beforehand what these factors are. Rabinowitz most of the time makes matters worse, even if you know the factors.
As I said, I only care about solutions, not multiplicity. It would be awesome if the algorithm automatically throws away multiple factors. Also, if the algorithm calculates the radical of the ideal <p, q> for any two p, q polynomials that it can, factors it, and puts it in the system. I am assuming that calculating the whole radical is computationally impossible.
Thank you very much for your attention.
Best regards.
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