For each fake exponent of an A-hypergeometric system, the Hilbert series of an Artinian Stanley-Reisner quotient equals the graded dimension series of the orthogonal complement of the local fake indicial ideal, and under the Okuyama-Saito condition, the actual logarithmic solution space.
Nagamine, A-hypergeometric Series with Parameters in the Core, Math
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Hilbert Series and Logarithmic Degrees of $A$-Hypergeometric Series
For each fake exponent of an A-hypergeometric system, the Hilbert series of an Artinian Stanley-Reisner quotient equals the graded dimension series of the orthogonal complement of the local fake indicial ideal, and under the Okuyama-Saito condition, the actual logarithmic solution space.