Consider the $n$-dimensional Torus $T^n = \prod_{i=1}^n S^1$ as a subset of $\mathbb R^{2n} = \prod_{i=1}^n \mathbb R^2$ in the standard way.

*Is it true that a generic $n$-multilinear functional on $\mathbb R^2$, i.e. a tensor in $T_0^n(\mathbb R^2)$, defines a Morse function on $T^n$? If so, does a generic element have the minimal number of critical points $2^n$ (among Morse functions)?*

Generic here refers to all but a Haar null set of exceptions (dense would also be fine). The statement is reasonable in my opinion since these multilinear functionals seem to be good "height functions" in this context and the statement is indeed true for $n=1,2$. For $n=1$ it is obviuously true for any nontrivial linear function on $\mathbb R^2$. For $n=2$ we can write any $\omega \in T_0^n(\mathbb R^2)$ as $$ \omega = a\, e_1^* \otimes e_1^* + b\, e_1^* \otimes e_2^* + c\, e_2^* \otimes e_1^* + d\, e_2^* \otimes e_2^* $$ with respect to the standard basis of $\mathbb R^2$. Define the function $f : \mathbb R \times \mathbb R \to \mathbb R$ by $f(s,t) = \omega(e^{is},e^{it})$ and solve for critical points. For generic $a,b,c,d$ it follows rather quickly (modulo calculation errors) using the addition formulas for sine and cosine \begin{align*} s & = \frac{1}{2}\arctan \frac{c+b}{a-d} - \frac{1}{2}\arctan \frac{b-c}{a+d} + m\pi\ , \\ t & = \frac{1}{2}\arctan \frac{c+b}{a-d} + \frac{1}{2}\arctan \frac{b-c}{a+d} + n\pi \ , \end{align*} for $m,n \in \mathbb Z$. This has exactly four solutions $(s,t) \in (-\pi,\pi]^2$. Similarly one can check that for generic $a,b,c,d$, the Hessian matrix of $f$ at critical points $(s,t)$ is nondegenerate.

It is interesting to note that the statement can't hold for all $\omega \neq 0$ because if the coordinates of $\omega$ are for example given by $b = c = 0$ and $a = d = 1$, then the associated function $f(s,t) = \cos(s-t)$ has a continuum of critical points.

Preferably there is an argument that avoids calculations as above for $n > 2$ as this seems to be quite a laborious task, unless I miss something (which is likely). Maybe this question has been considered before or follows directly from more general principles available in the literature, but I wasn't able to find something. Any references in this direction are welcome.

integercoefficients give well-defined maps from $T^n$ to $\mathbb R$. Are you referring to those? Then what does "generic" mean in this context? $\endgroup$ – Sebastian Goette Feb 26 '20 at 21:48