Is there a standard notion of nondegeneracy for multilinear forms?
My motivation is simple curiosity, by the way!
Is there a standard notion of nondegeneracy for multilinear forms? My motivation is simple curiosity, by the way! 


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 (arXiv: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 sevendimensional 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 7dimensional riemannian metrics of weak $G_2$ holonomy, for example. 


Basic idea: a bilinear form $B$ is degenerate if there are two nonzero vectors $v$ and $w$ so that not only is $$B(v,w)=0,$$ but also either vector is enough to kill $B$ without help from the other: $$B(v,) = 0 \mbox{ and } B(,w)=0.$$ Another way to say it: even if we perturb $v$ and $w$, $B(v \otimes w)=0$ to first order. From this point of view, we see that a bilinear form is degenerate iff it is an element of the variety dual to the Segre embedding of $\mathbb{P}V \times \mathbb{P}W$ in $\mathbb{P}(V \otimes W)$. This motivation generalizes gracefully to the following definition from Gelfand, Kapranov, and Zelevinsky's book Discriminants, resultants, and multidimensional determinants: A $p$linear form $T$ is said to be degenerate if either of the following equivalent conditions holds:
In certain favorable cases (when the dimensions of the vector spaces $V_i$ are not too different) the dual to the Segre is a hypersurface; in other words, there is a single polynomialthe hyperdeterminantwhich vanishes exactly at the degenerate multilinear forms. This polynomial possesses many magical properties and is much subtler than determinants of bilinear forms. I can attest that this definition is at least useful, if not standard, since it came up in a substantial way in an elementary question about coin flipping: http://arxiv.org/abs/1009.4188 . 

