2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69717Let 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 review another well-known sufficient condition.Several minor changes and new Prop. (1.15)Algebraic Geometry14Pxx, 14Mxxtext