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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 $Q_{ijkl}$ for Q being bounded from below? I'm looking for the simplest expression in terms of Q_ijkl. $Q_{ijkl}$. Clearly, if Q_ijkl, $Q_{ijkl}$, 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 $x_i x_j x_j$ 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.

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# bounded homogeneous quartics

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.