higher order analogues of sylvester's law of inertia?

Question

Sylvester's law of inertia (here I quote wikipedia)

If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number of negative entries in the diagonal of D is always the same, for all such S; and the same goes for the number of positive elements.

Is there an analgoue of Sylvester's theorem for say cubic forms? My gut feeling is that this should be false for higher order tensors. So maybe the question here is what is the right analogue for higher degree forms? Have such questions been investigated in the literature. Apologies if this is too elementary, but I couldn't find anything related when I searched on the web.

Answer 1

A generalization of Sylvester's classification of canonical quadratic forms (which is the "law of inertia") to cubic forms has been presented in Canonical forms for symmetric tensors (1984). The matrices $A,D$ are now $2\times 2\times 2$ tensors, and the congruence relation is $D_{ijk}=\sum_{l,m,n}S_{li}S_{mj}S_{nk}A_{lmm}$. The canonical cubic forms are given on page 276 of the cited paper.