i1 : A = QQ[x,y];
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i2 : I = ideal(x^10+x^9*y^2, y^8-x^2*y^7);
o2 : Ideal of A
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i3 : f = x^2*y^7+y^14;
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i4 : f % I
12 8
o4 = - x*y + y
o4 : A
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So this f is not in the ideal I.
i5 : f = x*y^13+y^12;
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i6 : f % I
o6 = 0
o6 : A
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This f is in the ideal I.
Check inclusion and equality of ideals.
i7 : K = ideal(f,x^2*y^7+y^14);
o7 : Ideal of A
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i8 : (gens K) % I
o8 = | 0 -xy12+y8 |
1 2
o8 : Matrix A <-- A
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In Macaulay2, inclusion of ideals can be tested using
isSubset(Ideal,Ideal) and equality can be checked using
Ideal == Ideal. In both cases the necessary Groebner bases are computed, if they have not already been computed.
i9 : isSubset(K,I)
o9 = false
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i10 : K == I
o10 = false
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i11 : K = ideal(f,y^14+x*y^12);
o11 : Ideal of A
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i12 : (gens K) % I
o12 = 0
1 2
o12 : Matrix A <-- A
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i13 : isSubset(K,I)
o13 = true
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i14 : K == I
o14 = false
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