Given a finitely generated module over a $\ZZ^d$-graded polynomial ring $R$, quasidegreesAsVariables gives a representation of the quasidegree set of $M$ using the variables of $R$. This method captures the plane arrangement of the quasidegree set of the module.
If the input is an ideal $I$, then quasidegreesAsVariables executes for the module $R/I$ where $R$ is the ring of $I$.
A synonym for this function is qav.
i1 : R = QQ[x,y,Degrees=>{{1,0},{0,1}}]
o1 = R
o1 : PolynomialRing
|
i2 : I = ideal(x^2*y,x*y^2,y^3)
2 2 3
o2 = ideal (x y, x*y , y )
o2 : Ideal of R
|
i3 : M = R^1/I
o3 = cokernel | x2y xy2 y3 |
1
o3 : R-module, quotient of R
|
i4 : quasidegreesAsVariables M
2
o4 = {{1, {x}}, {y, {}}, {x*y, {}}, {y , {}}}
o4 : List
|
In the above example, the first element in the list \{1,\{x\}\} corresponds to a line in the $x$ direction with no shift. The element \{y,\{\}\} corresponds to a point shifted in the direction of the degree of $y$, the element \{x*y,\{\}\} corresponds to a point shifted in the direction of the degree $xy$, and the element \{y^2,\{\}\} corresponds to a point shifted in the direction of the degree of $y^2$.
The next example has a 2 dimensional quasidegree set.
i5 : R=QQ[x,y,z,Degrees=>{{1,0,0},{0,1,0},{0,0,1}}]
o5 = R
o5 : PolynomialRing
|
i6 : I=ideal(y) o6 = ideal y o6 : Ideal of R |
i7 : M=R^1/I
o7 = cokernel | y |
1
o7 : R-module, quotient of R
|
i8 : quasidegreesAsVariables M
o8 = {{1, {x, z}}}
o8 : List
|
The quasidegree set of $\QQ[x,y,z]/<y>$ with the standard $\ZZ^3$-grading is the (unshifted) $xz$-plane.
The object quasidegreesAsVariables is a method function.