CoCoA » CoCoA-5

I have given this a higher priority because I think it will be quick and easy to resolve.

@Anna: can you do it? (before releasing 5.1.6?)

I tried GroebnerFanIdeals on the following input:

use P::=QQ[x,y,z];
I := ideal(x^5-3*y^3*z+x*y*z-2, y^3-2*z+5, z^2-12*x+7*y);
GF := GroebnerFanIdeals(I);
indent(GF);

NOTE comparing the printed forms of the ideals there are only 11 different results (perhaps ones that appear equal actually have different LPPs?)

John Abbott wrote:

I tried GroebnerFanIdeals on the following input:
[...]
The resulting list GF has 195 elements, but many of them seem to be repeats. Is this intended? (Or should there be another issue for this?)

NOTE comparing the printed forms of the ideals there are only 11 different results (perhaps ones that appear equal actually have different LPPs?)

The function GroebnerFanIdeals returns the IDEALS.
(Renzo's happy you had the same problem as his!)
Inside the ideals there are the GBases.
If you want them do

indent([ReducedGBasis(J) | J in GroebnerFanIdeals(I)]);

It would be nice to have result the ideals generated by the reduced GBases, but in that way (in CoCoA-5!) we would lose that those are the GBases! In CoCoALib we can declare that the generators are a GBasis, but in CoCoA-5 we cannot.
Should we allow that? (indeed it would be quite useful in a few places)

moved to #977

Does this code correctly compute the "universal denominator" for the ideal I? (...)

I think the confusion came from the way the documentation is worded.

I have rerun the example from comment 2, and can (fairly) safely say that no-one wants to see the reduced G-bases (the whole output was more than 2Mbytes).

What was not entirely clear to me was whether these ideals already had their reduced G-bases memorized inside, or whether the G-bases are computed "on demand". Maybe it does not matter; maybe "on demand" is actually a better option, as I imagine non-trivial examples quickly give gigantic results (i.e. the sum of the sizes of all the various G-bases).

I'm still amazed at how "ugly" some of the reduced G-bases are (i.e. the coeffs are very complicated).

John Abbott wrote:

I think the confusion came from the way the documentation is worded.

I have rerun the example from comment 2, and can (fairly) safely say that no-one wants to see the reduced G-bases (the whole output was more than 2Mbytes).

indeed.

What was not entirely clear to me was whether these ideals already had their reduced G-bases memorized inside, or whether the G-bases are computed "on demand". Maybe it does not matter; maybe "on demand" is actually a better option, as I imagine non-trivial examples quickly give gigantic results (i.e. the sum of the sizes of all the various G-bases).

This actually memorizes the GBases.
The clever function is "CallOnGroebnerFanIdeals".

But it is true that the user gets confused. (both you and Robbiano were tricked by lists which were identical, but did NOT LOOK so)
One possibility, for small examples (or for convincing the user it is not a good idea ;-) is writing the function "GroebnerFanReducedGBases", which is trivial to write, and exactly what the user expects (even though it is often not what he wants!)

Or maybe I should write some good examples in the manual?

Anna Maria Bigatti wrote:

One possibility, for small examples (or for convincing the user it is not a good idea ;-) is writing the function "GroebnerFanReducedGBases"

Or maybe I should write some good examples in the manual?

I will do both: indeed I'll write a good example in the manual of the new function with an explicatory example.

Out of curiosity.... Does GFan produce order matrices with smallest possible non-negative entries?
In the example from comment 2, there are two order matrices with entries larger than 1000...
If negative entries are allowed then these number can be greatly reduced (though internally they are probably made non-negative again because of the coding we use for order vectors).

• with verbosity >=10 (each call of recursive, CallOnRecursive)
  . with verbosity >=20 (each call of GetFlippableInequalities)
  maxdeg with verbosity >=80 (GetFlippableInequalities)
  timings with verbosity >=90 (GroebnerFanIdeals, CallOnGroebnerFanIdeals)

Now using IdealOfGBasis, so that the generators of the ideals are the different GBasis.

/**/ use R ::= QQ[a,b,c];
/**/ I := ideal(b^2-1, a^2+c-1, c^2-b);
/**/ indent(GroebnerFanIdeals(I));
[
  ideal(c^2 -b, b^2 -1, a^2 +c -1),
  ideal(b -c^2, a^2 +c -1, c^4 -1),
  ideal(c +a^2 -1, b -a^4 +2*a^2 -1, a^8 -4*a^6 +6*a^4 -4*a^2),
  ideal(c +a^2 -1, b^2 -1, a^4 -b -2*a^2 +1)
]