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We provide an algorithm for establishing the nonexistence of a common quadratic Lyapunov function for switched LTI systems under arbitrary switching. We show that this nonex-istence question is equivalent to the emptiness of an associated semi-algebraic set. The celebrated Positivstellensatz from real algebraic geometry provides a complete characterization of when this set is empty. Finally, we obtain the desired certificates of set emptiness using sum of squares programming.