A generalized and systematic approach to the problem of dead-beat response to polynomial time-domain inputs is presented. Necessary and sufficient conditions for the impulse response coefficients of the system, in the form of a set of linear equations, are derived. For the case of minimum prototype systems, a formula for the explicit computation of the overall pulse transfer function of the system is deduced and their properties are further studied, together with their ability to track complex inputs. It is also presented an analytical study of the system's control sequence, which plays an important role for the inter sample activity of the plant. By eliminating possible oscillations that may be present in this sequence, necessary and sufficient conditions for ripple-free dead-beat response are derived.