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This paper provides finite dimensional convex conditions to construct homogeneous polynomially parameter- dependent Lur'e functions which ensure the stability of nonlinear systems with state-dependent nonlinearities lying in general sectors and which are affected by uncertain parameters belonging to the unit simplex. The proposed conditions are written as linear matrix inequalities parameterized in terms of the degree g of the parameter-dependent solution and in terms of the relaxation level d of the inequality constraints, based on an extension of Polya's Theorem. As g and d increase, progressive less conservative solutions are obtained. The results in the paper include as special cases existing conditions for robust stability analysis and for absolute stability. A convex solution for control design is also provided. Numerical examples illustrate the efficiency of the proposed conditions.