Design #1255: Ideals with trivial GBasis - CoCoALib - CoCoA Project

Design #1255

Ideals with trivial GBasis

Added by Anna Maria Bigatti about 5 years ago. Updated about 1 year ago.

Status:

New

Priority:

Normal

Assignee:

Category:

Improving

Target version:

CoCoALib-0.99880

Start date:

11 Mar 2019

Due date:

% Done:

Estimated time:

Spent time:

0.10 h

Description

If an ideal has a single (non-zero) generator then that generator is automatically a Gbasis.
CoCoALib does not currently recognize this. (#1249)
There are other cases like this: should they be detected?

Related issues

History

#1 Updated by Anna Maria Bigatti about 5 years ago

[by John Abbott: moved here from #1249]
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Also if the gens happen to have coprime LTs wrt to current ordering then they are a GBasis. Might be useful to have a function which checks if the gens are "obviously" a GBasis (without computing anything)?

Also it could be worth computing a GBasis with a low timeout...
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Here are some more minor points (after speaking to Anna on the phone):

Anna was concerned about potential cost if there are many gens;
if there are more (non-zero) gens than indets then the LTs cannot be coprime;
when scanning through the list of LTs, if the number of unseen indets is less than the number of remaining (non-zero) gens then they cannot all be pairwise coprime.

#2 Updated by Anna Maria Bigatti about 5 years ago

Related to Feature #1249: principal ideal has a Gbasis added

#3 Updated by John Abbott about 5 years ago

Generators which are monomials are also a special case.

Then one could imagine a more complicated "mixed" situation: monomial gens in the Xs, and gens with coprime LTs in Ys...

It seems that the simplest general solution would be to try to compute a GBasis with a low timeout. If we have a proper Buchberger "mill" then every time we call "ComputeGBasis" with a timeout should advance the computation by at least one step. This could lead to a strange situation where repeatedly calling IHaveGBasis(J) produces initially a string of false before suddenly changing to true. Is this desirable? (JAA: probably)