If Q is a real homogeneous quartic on $R^N$,

$Q(x) = \sum_{1 <= i,j,k,l <= N} Q_{ijkl} x_i x_j x_k x_l$

what is the condition on the (totally symmetric) coefficients $Q_{ijkl}$ for Q being bounded from below? I'm looking for the simplest expression in terms of $Q_{ijkl}$. Clearly, if $Q_{ijkl}$, as considered a map from the space of real symmetric matrices to the space of real symmetric matrices is positive semi-definite, is enough. But this is a too strong condition because $x_i x_j$ is a rank-1 real symmetric matrix, so in Q(x) Q is only evaluated on rank-1 matrices, not on every real symmetric matrix.