# quasidegreesAsVariables -- represents the quasidegree set in variables

## Synopsis

• Usage:
quasidegreesAsVariables(I)
quasidegreesAsVariables(M)
• Inputs:
• I, an ideal,
• M, , 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 .