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First, we compute the irreducible components of \(Y_\infty\), and compare them to the components of \(X_\infty\).
Macaulay2, version 1.20
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, Isomorphism,
LLLBases, MinimalPrimes, OnlineLookup, PrimaryDecomposition, ReesAlgebra,
Saturation, TangentCone
i1 : K=QQ;
i2 : t_0=1;
i3 : t_1=0;
i4 : R=K[x_0,x_12,x_13,x_14,x_15,x_23,x_24,x_25,x_34,x_35,x_45,x_1234,x_1235,x_1245,x_1345,x_2345];
i5 : Pinfty = ideal {x_2345,x_1245,x_1234,t_0*x_25-t_1*x_35-t_0*x_45,x_23-x_25,x_15+x_25-x_35-x_45,x_14,x_12-x_13,x_0+t_1*x_25-t_1*x_35-t_1*x_45};
o5 : Ideal of R
i6 : OG510 = ideal({x_0*x_2345-x_23*x_45+x_24*x_35-x_25*x_34,
x_12*x_1345-x_13*x_1245+x_14*x_1235-x_15*x_1234,
x_0*x_1345-x_13*x_45+x_14*x_35-x_15*x_34,
x_12*x_2345-x_23*x_1245+x_24*x_1235-x_25*x_1234,
x_0*x_1245-x_12*x_45+x_14*x_25-x_15*x_24,
x_13*x_2345-x_23*x_1345+x_34*x_1235-x_35*x_1234,
x_0*x_1235-x_12*x_35+x_13*x_25-x_15*x_23,
x_14*x_2345-x_24*x_1345+x_34*x_1245-x_45*x_1234,
x_0*x_1234-x_12*x_34+x_13*x_24-x_14*x_23,
x_15*x_2345-x_25*x_1345+x_35*x_1245-x_45*x_1235});
o6 : Ideal of R
i7 : Yinfty = Pinfty+OG510;
o7 : Ideal of R
i8 : Yinfty == radical Yinfty
o8 = true
i9 : Yinfty = primaryDecomposition(Yinfty)
o9 = {ideal (x , x , x , x , x , x , x , x , x - x , x , x , x , x ), ideal
2345 1245 1234 45 34 25 24 23 15 35 14 13 12 0
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(x , x , x , x , x , x , x , x , x - x , x , x , x , x - x ,
2345 1345 1245 1235 1234 45 35 25 24 34 23 15 14 12 13
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x ), ideal (x , x , x , x , x , x , x , x , x , x , x , x , x ,
0 2345 1345 1245 1235 1234 45 35 25 23 15 14 13 12
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x ), ideal (x , x , x , x , x , x , x + x , x - x , x - x , x ,
0 2345 1345 1245 1235 1234 35 34 45 25 45 23 45 15
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x , x , x , x ), ideal (x , x , x , x , x , x - x , x - x , x -
14 13 12 0 2345 1345 1245 1235 1234 25 45 24 34 23
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2
x , x - x , x , x + x + x , x + x + x , x , x x - x x - x ), ideal
45 15 35 14 13 34 45 12 34 45 0 34 35 34 45 45
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(x , x , x , x , x , x , x , x , x , x , x , x , x - x , x ), ideal
2345 1345 1245 1234 45 35 34 25 24 23 15 14 12 13 0
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(x , x , x , x , x , x , x , x , x , x , x , x , x , x ), ideal
2345 1245 1235 1234 45 35 25 24 23 15 14 13 12 0
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(x , x , x + x , x , x , x + x , x - x , x , x - x , x , x ,
2345 1245 1235 1345 1234 35 34 45 25 45 24 23 45 15 14
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x , x , x )}
13 12 0
o9 : List
i10 : Xinfty= new HashTable from {
{0, ideal {x_2345, x_1245, x_1235, x_1234, x_45, x_35, x_25, x_24, x_23, x_15, x_14, x_13, x_12, x_0}},
{1, ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_45, x_35, x_25, x_23, x_15, x_14, x_13, x_12, x_0}},
{2, ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_45, x_35, x_25, x_24-x_34, x_23, x_15, x_14, x_12-x_13, x_0}},
{34,ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_24-x_34, x_23-x_25, x_15+x_25-x_35-x_45, x_14, x_13+x_25+x_34, x_12+x_25+x_34, x_25*x_34-x_34*x_35+x_25*x_45, t_1*x_25-t_1*x_35-t_1*x_45+x_0, t_0*x_25-t_1*x_35-t_0*x_45, t_1*x_34*x_45+t_1*x_35*x_45+t_1*x_45^2-x_0*x_34-x_0*x_45, t_0*x_34*x_35-t_1*x_34*x_35-t_0*x_34*x_45-t_1*x_35*x_45-t_0*x_45^2, t_0*t_1*x_35-t_1^2*x_35-t_0*x_0}},
{5, ideal {x_2345, x_1345, x_1245, x_1234, x_45, x_34, x_24, x_23-x_25, x_15+x_25-x_35, x_14, x_12-x_13, t_1*x_1235-x_13-x_25, x_13*x_25+x_25^2-x_13*x_35-x_25*x_35+x_0*x_1235, t_1*x_25-t_1*x_35+x_0, t_0*x_25-t_1*x_35, t_0*x_13*x_35-t_1*x_13*x_35-t_0*x_0*x_1235+x_0*x_35, t_0*t_1*x_35-t_1^2*x_35-t_0*x_0}},
{6, ideal {x_2345, x_1245, x_1234, x_34+x_45, x_24, x_23-x_25, x_15+x_25-x_35-x_45, x_14, x_13, x_12, t_1*x_1345+x_45, x_45*x_1235+x_25*x_1345, t_1*x_1235-x_25, t_0*x_1235+t_0*x_1345-x_35, x_25*x_45-x_35*x_45-x_45^2-x_0*x_1345, x_25^2-x_25*x_35-x_35*x_45-x_45^2+x_0*x_1235-x_0*x_1345, t_1*x_25-t_1*x_35-t_1*x_45+x_0, t_0*x_25-t_1*x_35-t_0*x_45, t_0*x_35*x_45-t_1*x_35*x_45+t_0*x_0*x_1345, t_0*t_1*x_35-t_1^2*x_35-t_0*x_0}},
{7, ideal {x_2345, x_1245, x_1235+x_1345, x_1234, x_35, x_34+x_45, x_25-x_45, x_24, x_23-x_45, x_15, x_14, x_13, x_12, x_0}},
{8, ideal {x_2345, x_1245, x_1234, x_45, x_35, x_34, x_25, x_24, x_23, x_15, x_14, x_13, x_12, x_0}},
{9, ideal {x_2345, x_1245, x_1235, x_1234, x_25, x_24, x_23, x_15-x_35-x_45, x_14, x_13, x_12, t_1*x_1345-x_34, t_0*x_45-t_1*x_45+x_0, x_34*x_35+x_34*x_45-x_0*x_1345, t_1*x_35+t_1*x_45-x_0}},
{10, ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_35, x_34+x_45, x_25-x_45, x_23-x_45, x_15, x_14, x_13, x_12, x_0}},
{11, ideal {x_2345, x_1345, x_1245, x_1234, x_45, x_35, x_34, x_25, x_24, x_23, x_15, x_14, x_12-x_13, x_0}}
};
i11 : apply(#Yinfty, i -> select(keys Xinfty, k -> Yinfty#i==Xinfty#k))
o11 = {{}, {2}, {1}, {10}, {34}, {11}, {0}, {7}}
o11 : List
i12 : dim(Yinfty_0)
o12 = 3
i13 : linearSpan = (L) -> (
ideal(select(flatten entries gens intersect(L), f -> degree(f)=={1}))
);
i14 : L=linearSpan({Xinfty#5,Xinfty#6,Xinfty#8,Xinfty#9});
o14 : Ideal of R
i15 : Yinfty#0==L
o15 = true