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GroebnerFanIdeals

all reduced Groebner bases of an ideal

Syntax
GroebnerFanIdeals(I: IDEAL): LIST of IDEAL

Description
Returns a LIST of ideals, one for each possible distinct reduced Groebner basis of I ; these ideals are generated by their GBasis, and each is in a different ring; their associated term orderings lead to all the different possible reduced Groebner bases.

See CallOnGroebnerFanIdeals for a way of computing with all the various ideal (but without storing the whole list).

Verbosity:

* with verbosity >=10 (recursive, CallOnRecursive)

. with verbosity >=20 (GetFlippableInequalities)

maxdeg with verbosity >=80 (GetFlippableInequalities)

timings with verbosity >=90 (GroebnerFanIdeals, CallOnGroebnerFanIdeals)

This function used to be called AllReducedGroebnerBases up to version 5.1.4, and used to return the ideals encoded with the same set of generators as I (now generated by GBasis).

Example
/**/ Use R ::= QQ[a,b,c];
/**/ I := ideal(b^3+c^2-1, b^2+a^2+c-1, a^2+b^3-1);
/**/ l := GroebnerFanIdeals(I);
/**/ [ len(GBasis(I)) | I in l];
[4, 4, 6, 6, 5, 6, 4, 4, 4, 3, 4, 3, 3, 3, 4, 3, 3]

-- The ideal in [Sturmfels, Example 3.9] has 360 marked reduced Groebner bases
/**/ Use R ::= QQ[a,b,c];
/**/ I := ideal(a^5+b^3+c^2-1, b^2+a^2+c-1, c^3+a^6+b^5-1);
/**/ l := GroebnerFanIdeals(I);
/**/ len(l);
360

See Also