I know that a necessary and sufficient condition for the positivity of a quartic polynomial of many variables is in general difficult. I have a somewhat special case, maybe here more can be said. Let $x\in {\mathbb R}^{n \times m}$ be a real $n\times m$ matrix, $n<m$, and $i=1,\ldots,n$ and $a=1,\ldots,m$. I'm interested in the quartic,

$$Q(x) = \sum_{i,j,a,b,c,d} \;\; x_{ia}\; x_{ib} \; x_{jc} \; x_{jd} \; W_{abcd}$$

where $W_{abcd}\;$ has a number of symmetry properties obviously.

Clearly, if $W_{abcd}\;$, viewed as a quadratic form on symmetric $m\times m$ matrices -- via $W(M) = M_{ab}\; M_{cd}\; W_{abcd}\;$ where $M_{ab}$ is symmetric -- is positive semi-definite, then $Q(x)$ is also non-negative. This is a sufficient condition then for the non-negativeness of $Q$. Were $n=m$ it'd be also necessary.

This requirement is however overly strict because $\sum_i x_{ia}\;x_{ib}$ is not a general $m \times m$ matrix but of rank-n, where we've said $n<m$.

So what would be a necessary and sufficient condition in this case?