  For several objects of interest in geometric complexity theory, namely for
the determinant, the permanent, the product of variables, the power sum, the
unit tensor, and the matrix multiplication tensor, we introduce and study a
fundamental SL-invariant function that relates the coordinate ring of the orbit
with the coordinate ring of its closure. For the power sums we can write down
this fundamental invariant explicitly in most cases. Our constructions
generalize the two Aronhold invariants on ternary cubics. For the other objects
we identify the invariant function conditional on intriguing combinatorial
problems much like the well-known Alon-Tarsi conjecture on Latin squares. We
provide computer calculations in small dimensions for these cases. As a main
tool for our analysis, we determine the stabilizers, and we establish the
polystability of all the mentioned forms and tensors (including the generic
ones).
