i1 : R = ZZ/101[a..c];
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i2 : truncate(2,R^1)
o2 = image | a2 ab ac b2 bc c2 |
1
o2 : R-module, submodule of R
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i3 : truncate(2,R^1 ++ R^{-3})
o3 = image {0} | a2 ab ac b2 bc c2 0 |
{3} | 0 0 0 0 0 0 1 |
2
o3 : R-module, submodule of R
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i4 : truncate(2, ideal(a,b,c^3)/ideal(a^2,b^2,c^4))
o4 = subquotient (| ab ac bc c3 |, | a2 b2 c4 |)
1
o4 : R-module, subquotient of R
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i5 : truncate(2,ideal(a,b*c,c^7))
2 7
o5 = ideal (a , a*b, a*c, b*c, c )
o5 : Ideal of R
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The base may be ZZ, or another polynomial ring. In this case, the generators may not be minimal, but they do generate.
i6 : A = ZZ[x,y,z];
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i7 : truncate(2,ideal(3*x,5*y,15))
2 2 2
o7 = ideal (3x , 3x*y, 3x*z, 5x*y, 5y , 5y*z, 15z )
o7 : Ideal of A
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i8 : trim oo
2 2 2
o8 = ideal (15z , 5y*z, 3x*z, 5y , x*y, 3x )
o8 : Ideal of A
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i9 : truncate(2,comodule ideal(3*x,5*y,15))
o9 = subquotient (| x2 xz y2 yz z2 |, | 3x 5y 15 |)
1
o9 : A-module, subquotient of A
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If i is a multi-degree, then the result is the submodule generated by all elements of degree exactly i, together with all generators of M whose first degree is higher than the first degree of i. The following includes the generator of degree 8,20.
i10 : S = ZZ/101[x,y,z,Degrees=>{{1,3},{1,4},{1,-1}}];
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i11 : truncate({7,24}, S^1 ++ S^{{-8,-20}})
o11 = image {0, 0} | x4y3 0 |
{8, 20} | 0 1 |
2
o11 : S-module, submodule of S
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The coefficient ring may also be a polynomial ring. In this example, the coefficient variables also have degree one. The given generators will generate the truncation over the coefficient ring.
i12 : B = R[x,y,z, Join=>false]
o12 = B
o12 : PolynomialRing
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i13 : degree x
o13 = {1}
o13 : List
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i14 : degree B_3
o14 = {1}
o14 : List
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i15 : truncate(2, B^1)
o15 = image | x2 xy xz y2 yz z2 |
1
o15 : B-module, submodule of B
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i16 : truncate(4, ideal(b^2*y,x^3))
2 2 2 2 4 3 3
o16 = ideal (b x*y, b y , b y*z, x , x y, x z)
o16 : Ideal of B
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If the coefficient variables have degree 0:
i17 : A1 = ZZ/101[a,b,c,Degrees=>{3:{}}]
o17 = A1
o17 : PolynomialRing
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i18 : degree a
o18 = {}
o18 : List
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i19 : B1 = A1[x,y]
o19 = B1
o19 : PolynomialRing
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i20 : truncate(2,B1^1)
o20 = image | x2 xy y2 |
1
o20 : B1-module, submodule of B1
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i21 : truncate(2, ideal(a^3*x, b*y^2))
3 2 3 2
o21 = ideal (a x , a x*y, b*y )
o21 : Ideal of B1
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