Analogoue of a determinant test for $3$-tensors

Question

Suppose we have a matrix $H$ and I want to find out whether there exists a vector $r\ne 0$ such that $H^{ij}r_i r_j=0$. To this end we could simply compute if the determinant $\det(H)$ is zero, which is computationally inexpensive and explicit (we don't have to find such $r$ explicitly).

Is there an analogue of such a test to see if there exists a vector $r\ne0$ such that $G^{ijk}r_i r_j r_k=0$ for a $3$-tensor $G$?

I didn't specify the number of dimensions of the vector space, but if it's relevant $r\in \mathbb R^3$.

This question could be related to Determinant of a tensor.

Answer 1

There's always such a vector. To see this, let $$ f(r) = G^{ijk}r_i r_j r_k, \qquad r \neq 0 . $$ If $f$ is identically zero, any $r \neq 0$ will do. Otherwise, $f$ assumes both positive and negative values, since $f(-r)=-f(r)$. Then, by continuity of $f$ and connectedness of the space $\mathbb{R}^3\setminus \{ 0 \}$, we see that $f$ must also take the value zero for some $r \neq 0$.

Answer 2

Example: $$ H = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} $$ Then $\det H = -1 \ne 0$, but for $r = (0,1,0)^\top \ne 0$ we have $$ H^{ij} r_i r_j = r_1 r_1 + r_2 r_3 + r_3 r_2 = 0 $$ So your criterion is not good enough for you, it leaves out the cases $x^\top H x$ where $x \perp Hx$, covers only $Hx = 0$.