Sylvester's inertia law deals with the signatures of quadratic forms. I was thinking that it may be possible to extend this to multilinear forms; here is a first attempt.

Let $M$ be a $k$-linear form over the reals. For $1 \leq i \leq k$, define $\lambda_i(M)$ as the maximum dimension of a vector space $F$ such that

(*) $M(u_1,...,u_i,v_{i+1},...,v_k) \neq 0$ for every non-null $u_1,...,u_i \in F$, $v_{i+1},...,v_k \in F^{\bot}$.

If we define $\mu_i(M) = \lambda_i(M) - \lambda_{i-1}(M)$, is it true that these values form a partition of $n$? If so, can we formulate some "extended inertia law"? There might be some connection with Young tableaux but we probably don't require that for our needs.

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    $\begingroup$ It is an interesting question but I am not sure why you asked it in CSTheory. I think you are more likely to get an answer in Mathoverflow. $\endgroup$ – Sasho Nikolov Mar 22 '14 at 15:31
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    $\begingroup$ OTOH, there are many cstheory questions in MO and no one recommends they post them here :) $\endgroup$ – Suresh Venkat Mar 22 '14 at 20:45
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    $\begingroup$ The problem description is not at all clear. How is $n$ defined for example. And is all the coordinate of $M$ belongs to the same space so that $F$ can be well defined. A more precise formulation of the problem would be adding the condition that $M:R^n \times \dots \times R^n \to R$. $\endgroup$ – caozhu Mar 23 '14 at 7:32

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