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.