Given a list L, this function gives a $\ZZ^r$-graded polynomial ring (where $r$ is the length of L) containing L_i+1 variables of multidegree equal to the i-th basis vector of $\ZZ^r$, i.e. the coordinate ring of the product of projective spaces with dimensions the entries of L. Given an integer n it returns the coordinate ring of a product of n copies of $\PP^1$.
i1 : S = multigradedPolynomialRing({1,3,4})
o1 = S
o1 : PolynomialRing
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i2 : gens S
o2 = {x , x , x , x , x , x , x , x , x , x , x }
0,0 0,1 1,0 1,1 1,2 1,3 2,0 2,1 2,2 2,3 2,4
o2 : List
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i3 : degrees S
o3 = {{1, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0,
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0, 1}, {0, 0, 1}, {0, 0, 1}, {0, 0, 1}, {0, 0, 1}}
o3 : List
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i4 : gens multigradedPolynomialRing 4
o4 = {x , x , x , x , x , x , x , x }
0,0 0,1 1,0 1,1 2,0 2,1 3,0 3,1
o4 : List
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By default the output will be a ring over ZZ/32003 in variables of the form x_(i,j). The coefficients can be changed using the option CoefficientField and the variable name with Variables (which takes a string). Setting the option Standard to false will produce variables with no indices, starting at a.
i5 : multigradedPolynomialRing({1,2},CoefficientField => ZZ/5,Variables=>"y")
ZZ
o5 = --[y ..y , y ..y ]
5 0,0 0,1 1,0 1,2
o5 : PolynomialRing
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i6 : multigradedPolynomialRing(3,Standard=>false)
ZZ
o6 = -----[a..f]
32003
o6 : PolynomialRing
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The output of multigradedPolynomialRing is not compatible with some functions from the package TateOnProducts, such as cohomologyHashTable. Use productOfProjectiveSpaces instead.
The object multigradedPolynomialRing is a method function with options.