  We study chiral rings of 4d $\mathcal{N}=1$ supersymmetric gauge theories via
the notion of K-stability. We show that when using Hilbert series to perform
the computations of Futaki invariants, it is not enough to only include the
test symmetry information in the former's denominator. We discuss a way to
modify the numerator so that K-stability can be correctly determined, and a
rescaling method is also applied to simplify the calculations involving test
configurations. All of these are illustrated with a host of examples, by
considering vacuum moduli spaces of various theories. Using Gr\"obner basis and
plethystic techniques, many non-complete intersections can also be addressed,
thus expanding the list of known theories in the literature.
