Non-degenerate multilinear forms Is there a standard notion of non-degeneracy for multilinear forms?
My motivation is simple curiosity, by the way!
 A: 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:


*

*there exist nonzero vectors $\beta_i$ so that, for any $1 \leq j \leq p$,
$$ T \left( \beta_1, \ldots , \beta_{j-1} , x_j ,\beta_{j+1}, \ldots , \beta_{p} \right) = 0 \mbox{ for all $x_{j}$;} $$

*there exist nonzero vectors $\beta_{i}$ so that $T$ vanishes at $\otimes \beta_{i}$ along with every partial derivative with respect to an entry of some $\beta_{i}$:
$$ T \mbox{ and } \frac{\partial T}{\partial \beta^{(j)}_{i}} \mbox{ vanish at $\otimes \beta_{i}$.}$$
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 polynomial---the hyperdeterminant---which 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 .
A: 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 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.
