# Primary decomposition of large ideals

Short version:

I'd like to do a primary decomposition of an ideal with 38 generators in a polynomial ring with 44 generators. However, the ideal seems far to large to naively decompose in, say, Macaulay2 (suffice to say I let my computer work all night with no result). Does there exist any "tricks" to do such a computation?

More context:

The ideal I'm trying to decompose is this:

I = ideal matrix {{t_2*t_22-t_7*t_27}, {t_4*t_22+t_2*t_23-t_10*t_27-t_7*t_29}, {t_5*t_22+t_2*t_24-t_12*t_27-t_7*t_31}, {t_4*t_23-t_10*t_29}, {t_5*t_24-t_12*t_31}, {t_5*t_72*t_81-t_12*t_67*t_76+t_24*t_68*t_80-t_31*t_66*t_71}, {t_63*t_73-t_70*t_79},
{t_67*t_71-t_72*t_80}, {t_41*t_69*t_81-t_34*t_76*t_78-t_54*t_66*t_5+t_61*t_64*t_68}, {t_64*t_69-t_75*t_78}, {t_66*t_76-t_68*t_81}, {t_4*t_64*t_73-t_10*t_75*t_79+t_23*t_63*t_69-t_29*t_70*t_78}, {t_39*t_73*t_80-t_33*t_71*t_79-t_52*t_67*t_70+t_59*t_63*t_72}, {t_33*t_52-t_39*t_59}, {t_35*t_52+t_33*t_55-t_42*t_59-t_39*t_62}, {t_34*t_54-t_41*t_61},
{t_35*t_54+t_34*t_55-t_42*t_61-t_41*t_62}, {t_35*t_55-t_42*t_62}, {t_2*t_65-t_7*t_74}, {t_4*t_65-t_10*t_74}, {t_5*t_65-t_12*t_74}, {t_27*t_65-t_22*t_74}, {t_29*t_65-t_23*t_74},
{t_31*t_65-t_24*t_74}, {t_33*t_77-t_39*t_82}, {t_34*t_77-t_41*t_82}, {t_35*t_77-t_42*t_82}, {t_59*t_77-t_52*t_82}, {t_61*t_77-t_54*t_82}, {t_62*t_77-t_55*t_82}, {t_64*t_74-t_75*t_77},
{t_63*t_65-t_70*t_77}, {t_69*t_77-t_74*t_78}, {t_73*t_77-t_65*t_79}, {t_65*t_67-t_72*t_82}, {t_65*t_80-t_71*t_82}, {t_66*t_74-t_68*t_82}, {t_74*t_81-t_76*t_82}}


(not all the $t_i$ variables are used, that's why I claim the ring has 44, not 82 generators). Note that most of the generators are binomials - and there exists effective (or at least, more effective) algorithms for decomposing binomial ideals. Is this possible to use somehow?

So how did this ideal come about? Well, it all started with a triangulated simplicial 3-sphere, which give rise to a monomial ideal. (Specifially, $x_1\cdots x_n$ is in the ideal iff $x_1\cdots x_n$ is not a face of the sphere) Then we try to perturb this ideal and look at the deformation space.

Now, there are results saying that smoothings (in the sense of deformation theory) of the sphere give rise to interesting varieties, for example Calabi-Yau varieties. In my case, we know there's a deformation to the Grassmannian $G(3,6)$. Anyways - I used a Macaulay2 package by Nathan Ilten to start computing the deformation space. The large ideal above consists of the initial terms of the generators of this space (so I guess this is in some sense the tangent space).

What I'm looking for is a yes/no-answer whether this is a feasible task, and if so, what strategies may I use.

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Naive suggestion: have you tried Sage online? I often found this much quicker than macaulay2, mostly because the code will run on a Sage server instead of your own computer. If that doesn't work then you'll need to be more cunning, but it's something I'd try if you haven't already. – Matthew Pressland Aug 31 '12 at 11:07
@MattPressland: Thanks, I hadn't thought of that. Trying it now. Do you have any thoughts on whether I should use Sage's polynomial ring functionality, or use the built-in Macaulay2-interface? – Fredrik Meyer Aug 31 '12 at 12:52
When I did these kinds of calculations I used the Sage functions, but I don't know whether this is better or worse. – Matthew Pressland Aug 31 '12 at 12:57