Is there a standard notion of non-degeneracy for multilinear forms?


My motivation is simple curiosity, by the way!



Non-degenerate multilinear forms
I'm not sure if this notion is "standard", but there is one such notion, used for example in Nigel Hitchin's paper [*Stable forms and special metrics*](http://www.ams.org/mathscinet-getitem?mr=1871001) ([arXiv:math/0107101](http://arxiv.org/abs/math/0107101)) for alternating multilinear forms. The idea is that symplectic structures on a vector space $V$ can be characterised by the fact that they lie in an open orbit of $\mathrm{GL}(V)$ on $\Lambda^2V^\*$. Hitchin calls these *stable forms* and shows that apart from the the case of symplectic forms, there are stable $3$-forms in dimensions 6,7 and 8; the $G\_2$-invariant $3$-form in a seven-dimensional vector space (i.e., the imaginary component of the multiplication of imaginary octonions) being one such example.


Hitchin's notion is very fruitful, as it provides a variational approach to 7-dimensional riemannian metrics of weak $G\_2$ holonomy, for example.


