i1 : isHomogeneous(ZZ)
o1 = true
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i2 : isHomogeneous(ZZ[x])
o2 = true
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i3 : isHomogeneous(ZZ[x]/(x^3-x-3))
o3 = false
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Quotients of multigraded rings are homogeneous, if the ideal is also multigraded.
i4 : R = QQ[a,b,c,Degrees=>{{1,1},{1,0},{0,1}}];
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i5 : I = ideal(a-b*c);
o5 : Ideal of R
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i6 : isHomogeneous I
o6 = true
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i7 : isHomogeneous(R/I)
o7 = true
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i8 : isHomogeneous(R/(a-b))
o8 = false
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Polyomial rings over polynomial rings are multigraded.
i9 : A = QQ[a]
o9 = A
o9 : PolynomialRing
|
i10 : B = A[x]
o10 = B
o10 : PolynomialRing
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i11 : degree x
o11 = {1, 0}
o11 : List
|
i12 : degree a_B
o12 = {0, 1}
o12 : List
|
i13 : isHomogeneous B
o13 = true
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A matrix is homogeneous if each entry is homogeneous of such a degree so that the matrix has a well-defined degree.
i14 : S = QQ[a,b];
|
i15 : F = S^{-1,2}
2
o15 = S
o15 : S-module, free, degrees {1, -2}
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i16 : isHomogeneous F
o16 = true
|
i17 : G = S^{1,2}
2
o17 = S
o17 : S-module, free, degrees {-1, -2}
|
i18 : phi = random(G,F)
o18 = {-1} | 8a2+ab+3b2 0 |
{-2} | 7a3+8a2b+3ab2+3b3 7 |
2 2
o18 : Matrix S <--- S
|
i19 : isHomogeneous phi
o19 = true
|
i20 : degree phi
o20 = {0}
o20 : List
|
Modules are homogeneous if their generator and relation matrices are homogeneous.
i21 : M = coker phi
o21 = cokernel {-1} | 8a2+ab+3b2 0 |
{-2} | 7a3+8a2b+3ab2+3b3 7 |
2
o21 : S-module, quotient of S
|
i22 : isHomogeneous(a*M)
o22 = true
|
i23 : isHomogeneous((a+1)*M)
o23 = false
|
Note that no implicit simplification is done.
i24 : R = QQ[x]
o24 = R
o24 : PolynomialRing
|
i25 : isHomogeneous ideal(x+x^2, x^2)
o25 = false
|