Question

The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{*}$ with weights (1,1,-1,-1) is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series

$\frac{1 - abcd}{(1-ac)(1-ad)(1-bc)(1-bd)}$

Is there a software package that can compute multigraded Hilbert series? Can it be computed using Macaulay2?

Alternatively, is there software that can compute the multigraded Hilbert series of a toric variety, specified by its fan?

For this example $v_1 = (0,0,1), v_2 = (1,0,1), v_3 = (1,1,1), v_4 = (0,1,1)$ specify the vertices of the toric fan. The multigraded Hiblert series is given by the index which counts points in the dual cone $S_{C^{*}}$ $\sum_{m \in S_{C^{*}}} q^m = \frac{(1 - q_1)}{ (1 - q_2)(1 - q_3)(1 - q_1 q_2^{-1}) (1 - q_1 q_3^{-1}) }$

References:

"Linear diophantine equations and local cohomology," R.P. Stanley, 1982.

"Combinatorial commutative algebra," E. Miller and B. Sturmfels, 2005.

"Sasaki-Einstein manifolds and volume minimisation," Martelli, Sparks, Yau, 2006.

Answer 1

Macaulay 2 can do multigraded Hilbert series. Let's first assume that you have a presentation of your multigraded ring. I'll mention how to calculate this below. So for your $S = k[ac,ad,bc,bd]$, we'll write it as $S = k[x,y,z,w] / (xz - yw)$.

Assuming that each of $a,b,c,d$ has its own degree direction (so the grading is by ${\bf Z}^4$), we input $k[x,y,z,w]$ as

S = QQ[x,y,z,w, Degrees=> {{1,0,1,0}, {1,0,0,1}, {0,1,1,0}, {0,1,0,1}}]

where here QQ means the rationals. Then we want the Hilbert series of the ideal $(xz - yw)$, so we put

i8 : hilbertSeries ideal(y*z-x*w)

and the answer is:

                   1 - T T T T
                        0 1 2 3
o8 = ----------------------------------------
     (1 - T T )(1 - T T )(1 - T T )(1 - T T )
            1 3        1 2        0 3        0 2

If you also need to get the presentation, we can do this as follows. First, download normaliz: http://www.mathematik.uni-osnabrueck.de/normaliz/

and the Macaulay 2 interface to normaliz (I think this is automatically there in Macaulay 2 1.3.1): http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.2/share/doc/Macaulay2/Normaliz/html/

After loading the package with

loadPackage "Normaliz" 

Set the path to normaliz in Macaulay 2 using the command

setNmzExecPath("path to the executables norm32 and norm64");

The torusInvariants command will give you generators for the subring of invariants. Finally, we can use the ringmap command to define a surjection from a polynomial ring onto the subring to get the desired ideal.

Answer 2

Piggyback question: What if the grading is by a f.g. abelian group?