Design and analysis of residual generators for polynomial systems is considered. This paper presents a systematic procedure, given an input-output description of system dynamics, to design residual generators for fault diagnosis. The design procedure is based on standard elimination theory. The design procedure is applied in a simulation study on a non-linear system, where it is showed how multiplicative and additive faults are detected and isolated. The example also shows how a fault detectability/isolability analysis can be made during the design.