bounded homogeneous quartics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:04:36Z http://mathoverflow.net/feeds/question/17712 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17712/bounded-homogeneous-quartics bounded homogeneous quartics Daniel 2010-03-10T12:12:36Z 2010-03-10T12:18:28Z <p>If Q is a real homogeneous quartic on $R^N$,</p> <p>$Q(x) = \sum_{1 &lt;= i,j,k,l &lt;= N} Q_{ijkl} x_i x_j x_k x_l$</p> <p>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.</p>