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Linear residual generation for differential-algebraic equation (DAE) systems is considered within a polynomial framework where a complete characterization and parameterization of all residual generators is presented. Further, a condition for fault detectability in DAE systems is given. Based on the characterization of all residual generators, a design strategy for residual generators for DAE systems is presented. The design strategy guarantees that the resulting residual generator is sensitive to all the detectable faults and also that the residual generator is of lowest possible order. In all results derived, no assumption about observability or controllability is needed. In particular, special care has been devoted to assure the lowest-order property also for non-controllable systems