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This note presents a novel model predictive control (MPC) strategy for input-saturated nonlinear systems having a polynomial structure. The results here proposed are a significant generalization w.r.t. similar existing algorithms which are tailored only for linearized or multi-model plant descriptions. A first key aim is to present sum-of-squares (SOS) conditions under which off-line one-step controllable sets for nonlinear polynomial systems can be derived. A second relevant contribution is to describe an on-line MPC strategy that leads to less conservative performance w.r.t. most existing methods based on global linearization approaches. An illustrative example is finally provided in order to show the effectiveness of the proposed SOS-based MPC algorithm.