The following code prints the combinatorial unprojection structure of a codimension 4 cyclic polytope.
Δ(0,4) -> Δ(2,6) -> Δ(4,8) -> Δ(6,10)
i1 : R=QQ[x_1..x_10]; |
i2 : cc=cycRes(6,R,verbose=>1);
delta(2,{z, x_2, x_3, x_4, x_5}) + delta(0,{z, x_2, x_3, x_4}) -> delta(2,{z, x_2, x_3, x_4, x_5, x_6})
delta(4,{z, x_2, x_3, x_4, x_5, x_6, x_7}) + delta(2,{z, x_2, x_3, x_4, x_5, x_6}) -> delta(4,{z, x_2, x_3, x_4, x_5, x_6, x_7, x_8})
delta(6,{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9}) + delta(4,{z, x_2, x_3, x_4, x_5, x_6, x_7, x_8}) -> delta(6,{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_10})
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i3 : betti cc
0 1 2 3 4
o3 = total: 1 25 48 25 1
0: 1 . . . .
1: . . . . .
2: . . . . .
3: . 25 48 25 .
4: . . . . .
5: . . . . .
6: . . . . 1
o3 : BettiTally
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We compute the differentials:
i4 : R=QQ[x_1..x_4]; |
i5 : cc=cycRes(0,R); |
i6 : print cc.dd_1 | x_1 x_3 x_4 x_2 | |
i7 : print cc.dd_2
{1} | x_3 x_4 x_2 0 0 0 |
{1} | -x_1 0 0 0 x_2 -x_4 |
{1} | 0 -x_1 0 -x_2 0 x_3 |
{1} | 0 0 -x_1 x_4 -x_3 0 |
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i8 : R=QQ[x_1..x_6]; |
i9 : cc=cycRes(2,R); |
i10 : print cc.dd_1 | x_2x_4 x_3x_5 x_1x_4 x_2x_5 x_1x_3 -x_1x_5 x_3x_6 x_4x_6 x_2x_6 | |
i11 : print cc.dd_2
{2} | 0 x_1 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 0 0 |
{2} | -x_1 0 x_2 0 0 -x_1 0 0 0 0 0 0 x_6 0 0 0 |
{2} | 0 -x_2 0 x_3 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 |
{2} | 0 0 -x_3 0 x_4 0 0 -x_1 0 0 0 0 0 0 x_6 0 |
{2} | x_5 0 0 -x_4 0 0 0 0 0 0 0 0 0 0 0 x_6 |
{2} | 0 0 0 0 0 -x_3 -x_4 -x_2 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 -x_2 x_4 0 -x_5 0 0 -x_1 |
{2} | 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 -x_1 0 0 |
{2} | 0 0 0 0 0 0 0 0 -x_4 x_3 0 -x_4 0 0 -x_5 0 |
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i12 : R=QQ[x_1..x_8]; |
i13 : cc=cycRes(4,R); |
i14 : print cc.dd_1 | x_2x_4x_6 x_3x_5x_7 x_1x_4x_6 x_2x_5x_7 x_1x_3x_6 x_2x_4x_7 x_1x_3x_5 x_2x_4x_8 x_3x_5x_8 -x_1x_4x_7 x_2x_5x_8 -x_1x_3x_7 x_1x_5x_7 x_3x_6x_8 x_4x_6x_8 x_2x_6x_8 | |
i15 : print cc.dd_2
{3} | 0 x_1 0 0 0 0 -x_7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 0 |
{3} | -x_1 0 x_2 0 0 0 0 -x_1 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 |
{3} | 0 -x_2 0 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_8 0 0 0 0 |
{3} | 0 0 -x_3 0 x_4 0 0 0 0 0 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 |
{3} | 0 0 0 -x_4 0 x_5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_8 0 0 |
{3} | 0 0 0 0 -x_5 0 x_6 0 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 |
{3} | x_7 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 x_5 0 0 0 0 0 0 -x_6 0 0 0 0 -x_6 0 0 0 0 -x_7 0 |
{3} | 0 0 0 0 0 0 0 0 0 -x_2 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 -x_7 0 0 0 0 -x_1 |
{3} | 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 x_5 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 x_3 0 -x_4 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 -x_7 0 0 0 |
{3} | 0 0 0 0 0 0 0 -x_5 0 0 x_4 0 0 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 x_3 x_4 x_2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 -x_4 0 x_5 0 0 0 0 0 0 0 -x_1 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_2 0 x_3 0 0 0 0 0 0 0 -x_1 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_4 -x_3 0 x_4 0 0 x_5 0 0 0 0 0 0 0 0 |
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