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# quasidegreesAsVariables -- represents the quasidegree set in variables

- Usage:
    `quasidegreesAsVariables(I)`
    `quasidegreesAsVariables(M)`
- Inputs:
  - I, an ideal,
  - M, a module, a finitely generated module
- Outputs:
  - a list, a list that indexes the quasidegrees of M as variables

## 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:

- `quasidegreesAsVariables(Ideal)`
- `quasidegreesAsVariables(Module)`

## For the programmer

The object quasidegreesAsVariables is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Quasidegrees.m2:613:0.