This paper studies robust stability of linear systems with polytopic uncertainty. New necessary and sufficient conditions for the existence of an a. ne parameter-dependent Lyapunov function assuring the Hurwitz or the Schur stability of a polytopic system are presented. These conditions are composed of a family of linear matrix inequality conditions of increasing precision. At each step, a set of linear matrix inequalities provides sufficient conditions for the existence of the a. ne parameter-dependent Lyapunov function, and necessity is asymptotically attained. Compared with the existing results in the literature, it is shown that the new stability conditions provide less conservative tests at each step. Numerical examples are given to illustrate the effectiveness of the new results.