i1 : V={{0,0},{0,1},{-1,-1},{1,0}};
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i2 : F={{0,1,2},{0,2,3},{0,1,3}};
|
i3 : phi=stanleyReisner(V,F)
QQ[e , e , e , t , t , t ]
0 1 2 0 1 2
o3 = map(---------------------------------------------,QQ[X , X , X , X ],{2e t - e t + e t - e t + 2e t + e t - e t - e t + e t , - e t + e t + e t , - e t - e t , e t - e t + e t })
2 2 2 0 1 2 3 0 0 0 1 0 2 1 0 1 1 1 2 2 0 2 1 2 2 0 0 0 1 2 1 0 0 1 1 1 0 1 1 2 0
(e e , e e , e e , e - e , e - e , e - e )
0 1 0 2 1 2 0 0 1 1 2 2
QQ[e , e , e , t , t , t ]
0 1 2 0 1 2
o3 : RingMap --------------------------------------------- <--- QQ[X , X , X , X ]
2 2 2 0 1 2 3
(e e , e e , e e , e - e , e - e , e - e )
0 1 0 2 1 2 0 0 1 1 2 2
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i4 : ker phi--decone of homogenized simplicial complex is three triangles meeting at a vertex
o4 = ideal(X X X )
1 2 3
o4 : Ideal of QQ[X , X , X , X ]
0 1 2 3
|
i5 : R=QQ[x,y];
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i6 : phi=stanleyReisner(V,F,BaseRing=>R,Homogenize=>false)
QQ[e , e , e , x, y]
0 1 2
o6 = map(---------------------------------------------,QQ[X , X , X ],{- e x + e y + e y, - e x - e y, e x - e y + e x})
2 2 2 0 1 2 0 0 2 0 1 1 1 2
(e e , e e , e e , e - e , e - e , e - e )
0 1 0 2 1 2 0 0 1 1 2 2
QQ[e , e , e , x, y]
0 1 2
o6 : RingMap --------------------------------------------- <--- QQ[X , X , X ]
2 2 2 0 1 2
(e e , e e , e e , e - e , e - e , e - e )
0 1 0 2 1 2 0 0 1 1 2 2
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i7 : ker phi--decone of simplicial complex is a three-cycle
o7 = ideal(X X X )
0 1 2
o7 : Ideal of QQ[X , X , X ]
0 1 2
|
i8 : V={{0,0},{0,1},{-1,-1},{1,0}};
|
i9 : F={{0,1,2},{0,2,3},{0,1,3}};
|
i10 : R=QQ[x,y];
|
i11 : CF = courantFunctions(V,F,Homogenize=>false,BaseRing=>R);
3 3
o11 : Matrix R <--- R
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i12 : phi=ringStructure(image CF,VariableGens=>false)
QQ[e , e , e , x, y]
0 1 2
o12 = map(---------------------------------------------,QQ[Y , Y , Y ],{- e x + e y + e y, - e x - e y, e x - e y + e x})
2 2 2 0 1 2 0 0 2 0 1 1 1 2
(e e , e e , e e , e - e , e - e , e - e )
0 1 0 2 1 2 0 0 1 1 2 2
QQ[e , e , e , x, y]
0 1 2
o12 : RingMap --------------------------------------------- <--- QQ[Y , Y , Y ]
2 2 2 0 1 2
(e e , e e , e e , e - e , e - e , e - e )
0 1 0 2 1 2 0 0 1 1 2 2
|
i13 : ker phi
o13 = ideal(Y Y Y )
0 1 2
o13 : Ideal of QQ[Y , Y , Y ]
0 1 2
|
i14 : V={{0,1},{-1,-1},{1,-1},{0,10},{-2,-2},{2,-2}};--symmetric triangular prism
|
i15 : V'={{0,1},{-1,-1},{1,-1},{1,10},{-2,-2},{2,-2}};--asymmetric triangular prism
|
i16 : F={{0,1,2},{0,1,3,4},{0,2,3,5},{1,2,4,5}};
|
i17 : S=QQ[x,y,z];
|
i18 : phi=stanleyReisner(V,F,BaseRing=>S) --four generators in degree one
QQ[e , e , e , e , x, y, z]
0 1 2 3
o18 = map(------------------------------------------------------------------------,QQ[X , X , X , X ],{e x + e x + e x + e x, e y + e y + e y + e y, e z + e z + e z + e z, e y + e z - 2e x + 2e y + 2e x + 2e y})
2 2 2 2 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 0 1 1 2 2
(e e , e e , e e , e e , e e , e e , e - e , e - e , e - e , e - e )
0 1 0 2 1 2 0 3 1 3 2 3 0 0 1 1 2 2 3 3
QQ[e , e , e , e , x, y, z]
0 1 2 3
o18 : RingMap ------------------------------------------------------------------------ <--- QQ[X , X , X , X ]
2 2 2 2 0 1 2 3
(e e , e e , e e , e e , e e , e e , e - e , e - e , e - e , e - e )
0 1 0 2 1 2 0 3 1 3 2 3 0 0 1 1 2 2 3 3
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i19 : phi'=stanleyReisner(V',F,BaseRing=>S) --six generators in degrees one and two
QQ[e , e , e , e , x, y, z]
0 1 2 3 2 2 2 2 2 2 2 2 2 2 2 2
o19 = map(------------------------------------------------------------------------,QQ[X , X , X , X , X , X ],{e x + e x + e x + e x, e y + e y + e y + e y, e z + e z + e z + e z, - 14e x*y - 7e y - 14e x*z + 7e z + 72e x - 80e x*y + 8e y + 8e x*z - 8e y*z, 2e x*y + 9e y + 2e x*z + 9e y*z - 20e x*y + 20e y + 2e x*z - 2e y*z + 16e x*y + 16e y + 2e x*z + 2e y*z, 14e x*y - 47e y + 14e x*z - 36e y*z + 11e z + 80e x*y - 80e y - 8e x*z + 8e y*z + 72e x - 72e y })
2 2 2 2 0 1 2 3 4 5 0 1 2 3 0 1 2 3 0 1 2 3 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 2 2 2 2 0 0 0 0 0 1 1 1 1 2 2
(e e , e e , e e , e e , e e , e e , e - e , e - e , e - e , e - e )
0 1 0 2 1 2 0 3 1 3 2 3 0 0 1 1 2 2 3 3
QQ[e , e , e , e , x, y, z]
0 1 2 3
o19 : RingMap ------------------------------------------------------------------------ <--- QQ[X , X , X , X , X , X ]
2 2 2 2 0 1 2 3 4 5
(e e , e e , e e , e e , e e , e e , e - e , e - e , e - e , e - e )
0 1 0 2 1 2 0 3 1 3 2 3 0 0 1 1 2 2 3 3
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i20 : ker phi --kernel generated by single polynomial of degree four
2 3 2 2 2 2 2 2 2 3 3 4
o20 = ideal(4X X X - 4X X + 4X X X - 4X X X - 4X X + 8X X + 4X X X - 5X X - X X + X )
0 1 3 1 3 0 2 3 1 2 3 0 3 1 3 1 2 3 1 3 2 3 3
o20 : Ideal of QQ[X , X , X , X ]
0 1 2 3
|
i21 : mingens ker phi' --kernel has six minimal generators of degree four
o21 = | 76X_0X_1X_4-428X_1^2X_4-22X_0X_2X_4-22X_1X_2X_4+11X_4^2-16X_0X_1X_5-72X_1^2X_5-2X_0X_2X_5-9X_1X_2X_5+X_4X_5
---------------------------------------------------------------------------------------------------------------------------
67032X_0^2X_4-1663704X_1^2X_4-99792X_0X_2X_4-99792X_1X_2X_4-2299X_3X_4+49896X_4^2-9576X_0^2X_5-110880X_0X_1X_5-278712X_1^2X
---------------------------------------------------------------------------------------------------------------------------
_5-13860X_0X_2X_5-36036X_1X_2X_5-209X_3X_5+5999X_4X_5+133X_5^2
---------------------------------------------------------------------------------------------------------------------------
14630X_1X_2X_3-1463X_2^2X_3+145152X_1^2X_4+61740X_0X_2X_4-43596X_1X_2X_4+1368X_3X_4-4536X_4^2+10080X_0X_1X_5+26208X_1^2X_5+
---------------------------------------------------------------------------------------------------------------------------
1260X_0X_2X_5-8428X_1X_2X_5-1463X_2^2X_5+342X_3X_5-630X_4X_5
---------------------------------------------------------------------------------------------------------------------------
146300X_1^2X_3-1463X_2^2X_3+1132992X_1^2X_4+123480X_0X_2X_4+18144X_1X_2X_4+1368X_3X_4-35406X_4^2+20160X_0X_1X_5+146048X_1^
---------------------------------------------------------------------------------------------------------------------------
2X_5+2520X_0X_2X_5+6552X_1X_2X_5-1463X_2^2X_5+342X_3X_5-1260X_4X_5
---------------------------------------------------------------------------------------------------------------------------
4180X_0X_1X_3-418X_0X_2X_3+116928X_1^2X_4+7308X_0X_2X_4+7308X_1X_2X_4+209X_3X_4-3654X_4^2+6448X_0X_1X_5+19440X_1^2X_5+806X_
---------------------------------------------------------------------------------------------------------------------------
0X_2X_5+2430X_1X_2X_5-403X_4X_5 2633400X_0^2X_3-26334X_2^2X_3-36575X_3^2+281667456X_1^2X_4+18552240X_0X_2X_4+16656192X_1X_
---------------------------------------------------------------------------------------------------------------------------
2X_4+640224X_3X_4-8802108X_4^2+2633400X_0^2X_5+18506880X_0X_1X_5+47169864X_1^2X_5+2313360X_0X_2X_5+6014736X_1X_2X_5-26334X_
---------------------------------------------------------------------------------------------------------------------------
2^2X_5+86906X_3X_5-1156680X_4X_5-36575X_5^2 |
1 6
o21 : Matrix (QQ[X , X , X , X , X , X ]) <--- (QQ[X , X , X , X , X , X ])
0 1 2 3 4 5 0 1 2 3 4 5
|