This method computes the Chow ring A=\ZZ[h_1,...,h_m]/(h_1^{n_1+1},...,h_m^{n_m+1}) of a product of projective spaces \PP^{n_1}\times \cdots \times\PP^{n_m}. It is needed for input into the methods Segre, Chern and CSM to ensure that these methods return results in the same ring. We give an example of the use of this method to work with elements of the Chow ring of \PP^3x\PP^4.
i1 : R=MultiProjCoordRing({3,4})
o1 = R
o1 : PolynomialRing
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i2 : A=ChowRing(R)
o2 = A
o2 : QuotientRing
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i3 : I=ideal(random({1,0},R));
o3 : Ideal of R
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i4 : K=ideal(random({1,1},R));
o4 : Ideal of R
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i5 : c=Chern(A,I)
3 4 3 3 2 4 3 2 2 3 4 3 2 2
o5 = 15h h + 30h h + 15h h + 30h h + 30h h + 5h h + 15h h + 30h h +
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
------------------------------------------------------------------------
3 3 2 2 2
10h h + 3h + 15h h + 10h h + 3h + 5h h + h
1 2 1 1 2 1 2 1 1 2 1
o5 : A
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i6 : s=Segre(A,K)
3 4 3 3 2 4 3 2 2 3 4 3 2 2
o6 = 35h h - 20h h - 15h h + 10h h + 10h h + 5h h - 4h h - 6h h -
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
------------------------------------------------------------------------
3 4 3 2 2 3 2 2
4h h - h + h + 3h h + 3h h + h - h - 2h h - h + h + h
1 2 2 1 1 2 1 2 2 1 1 2 2 1 2
o6 : A
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i7 : s-c
3 4 3 3 2 4 3 2 2 3 3 2 2 3
o7 = 20h h - 50h h - 30h h - 20h h - 20h h - 19h h - 36h h - 14h h -
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
------------------------------------------------------------------------
4 3 2 2 3 2 2
h - 2h - 12h h - 7h h + h - 4h - 7h h - h + h
2 1 1 2 1 2 2 1 1 2 2 2
o7 : A
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i8 : s*c
3 4 3 3 2 4 3 2 2 3 4 3 2 2
o8 = 12h h + 21h h + 12h h + 20h h + 19h h + 4h h + 10h h + 15h h +
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
------------------------------------------------------------------------
3 3 2 2 2
6h h + 2h + 6h h + 4h h + h + h h
1 2 1 1 2 1 2 1 1 2
o8 : A
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