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 characteristic polynomial of F. Also, we revisit the known sufficient condition in terms of Hankel matrices.

Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form ${\sigma }_G$. We define pseudo-real principal $G$-bundles on $X$. These are generalizations of real algebraic principal $G$-bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal $G$-bundles. Their relationships with the usual stable, semistable...