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. The approach allows us 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. (C) 2003 Elsevier Ltd. All rights reserved.