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This paper presents a novel Model Predictive Control strategy for input-saturated nonlinear systems having a polynomial structure. The method is based on algebraic geometry and viability theory arguments for computing the sets of states that can be steered in a finite number of steps, via a general control law, to a given robust positive invariant set. A first key aim is to present Sum-of-Squares conditions under which off-line controlled-invariant sets for nonlinear polynomial systems can be derived and their relevant properties analyzed. Then, a second relevant contribution is to describe an online MPC strategy that leads to less conservative performance w.r.t. most existing methods based on global linearization approaches. Finally, an illustrative example is provided to show the effectiveness of the proposed SOS-based algorithm.