Recently several new LMI conditions for stability of linear systems have been proposed, introducing additional slack variables to reduce the gap between conservative convex quadratic stability conditions and intractable non-convex robust stability conditions. In this paper we show that these improved LMI conditions can be derived with the help of some basic results on positive polynomial matrices, providing a clear interpretation of the role of the additional variables. The approach allows to derive in a unifying way results in the state-space and polynomial frameworks. Applications to robust stability analysis and robust stabilization of systems with multi-linear parametric uncertainty are fully described.