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quasidegreesAsVariables -- represents the quasidegree set in variables

Synopsis

Description

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.

Ways to use quasidegreesAsVariables :

For the programmer

The object quasidegreesAsVariables is a method function.