Primary decomposition of large ideals












4












$begingroup$


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 too 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, $x_1cdots x_n$ is in the ideal iff $x_1cdots x_n$ 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 I may use.






algebraic-geometry commutative-algebra ideals macaulay2







share|cite|improve this question











$endgroup$












  • $begingroup$
    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.
    $endgroup$
    – mdp
    Aug 31 '12 at 11:07










  • $begingroup$
    @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?
    $endgroup$
    – Fredrik Meyer
    Aug 31 '12 at 12:52










  • $begingroup$
    When I did these kinds of calculations I used the Sage functions, but I don't know whether this is better or worse.
    $endgroup$
    – mdp
    Aug 31 '12 at 12:57
















4












$begingroup$


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 too 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, $x_1cdots x_n$ is in the ideal iff $x_1cdots x_n$ 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 I may use.






algebraic-geometry commutative-algebra ideals macaulay2







share|cite|improve this question











$endgroup$












  • $begingroup$
    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.
    $endgroup$
    – mdp
    Aug 31 '12 at 11:07










  • $begingroup$
    @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?
    $endgroup$
    – Fredrik Meyer
    Aug 31 '12 at 12:52










  • $begingroup$
    When I did these kinds of calculations I used the Sage functions, but I don't know whether this is better or worse.
    $endgroup$
    – mdp
    Aug 31 '12 at 12:57














4












4








4


1



$begingroup$


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 too 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, $x_1cdots x_n$ is in the ideal iff $x_1cdots x_n$ 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 I may use.






algebraic-geometry commutative-algebra ideals macaulay2







share|cite|improve this question











$endgroup$




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 too 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, $x_1cdots x_n$ is in the ideal iff $x_1cdots x_n$ 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 I may use.






algebraic-geometry commutative-algebra ideals macaulay2




algebraic-geometry commutative-algebra ideals macaulay2






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 9 '18 at 0:04









Rodrigo de Azevedo

12.9k41856




12.9k41856










asked Aug 31 '12 at 10:02









Fredrik MeyerFredrik Meyer

15.2k24065




15.2k24065












  • $begingroup$
    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.
    $endgroup$
    – mdp
    Aug 31 '12 at 11:07










  • $begingroup$
    @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?
    $endgroup$
    – Fredrik Meyer
    Aug 31 '12 at 12:52










  • $begingroup$
    When I did these kinds of calculations I used the Sage functions, but I don't know whether this is better or worse.
    $endgroup$
    – mdp
    Aug 31 '12 at 12:57


















  • $begingroup$
    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.
    $endgroup$
    – mdp
    Aug 31 '12 at 11:07










  • $begingroup$
    @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?
    $endgroup$
    – Fredrik Meyer
    Aug 31 '12 at 12:52










  • $begingroup$
    When I did these kinds of calculations I used the Sage functions, but I don't know whether this is better or worse.
    $endgroup$
    – mdp
    Aug 31 '12 at 12:57
















$begingroup$
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.
$endgroup$
– mdp
Aug 31 '12 at 11:07




$begingroup$
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.
$endgroup$
– mdp
Aug 31 '12 at 11:07












$begingroup$
@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?
$endgroup$
– Fredrik Meyer
Aug 31 '12 at 12:52




$begingroup$
@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?
$endgroup$
– Fredrik Meyer
Aug 31 '12 at 12:52












$begingroup$
When I did these kinds of calculations I used the Sage functions, but I don't know whether this is better or worse.
$endgroup$
– mdp
Aug 31 '12 at 12:57




$begingroup$
When I did these kinds of calculations I used the Sage functions, but I don't know whether this is better or worse.
$endgroup$
– mdp
Aug 31 '12 at 12:57










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Dijon

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Dijon.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} De haut en bas, de gauche à droite : Panorama du centre historique vu depuis la Cathédrale Saint-Bénigne, le tramway passant sous l'Auditorium , la place François Rude, les Tuiles vernissées de l'Hôtel Aubriot, le centre commercial Toison d'or et le jardin Darcy. Blason Logo Administration Pays   France Région Bourgogne-Franche-Comté (préfecture) Département Côte-d'Or (préfecture) Arrondissement Dijon (chef-lieu) Canton bureau centralisateur de 6 cantons : Dijon-1, 2, 3, 4, 5, 6 Intercommunalité Dijon Métropole Maire Mandat François Rebsamen 2015-2020 Code postal 21000 (comportait aussi le code 21100 quand la ville possédait deux bureaux distributeurs) Code commune 21231 Démographie Gentilé ...
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Determine an Integral..

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up vote 0 down vote favorite Let $ B:= {(x,y,z) in mathbb{R}^3 | 1 leq z leq 3, x^2+ y^2 leq 16 } $ I want to determine $ int_B frac{z}{cosh^2 sqrt{x^2+y^2}} d(x,y,z) $ I know it is a cylinder with radius 4. I tried transforming with polarcoordinates-that was just not helpful. Can you give me some pretty hints how I finally this integral :-)? thanks in advance! real-analysis integration share | cite | improve this question asked yesterday constant94 13 2 New contributor constant94 is a new...
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