Positive polynomials and hyperdeterminants

Let $F$ be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that $F$ is positive definite or positive semi-definite. In this note we produce a necessary condition for positivity, and a sufficient condition for non-negativity, in terms of positivity or semi-positivity of a one-variable \textit{characteristic polynomial} of $F$. Also, we revisit the known sufficient condition in terms of Hankel matrices.