In this paper, an explicit nonlinear model predictive controller (ENMPC) for the stabilization of nonlinear systems is investigated. The proposed ENMPC is constructed using tensored polynomial basis functions and samples drawn from low-discrepancy sequences. Solutions of a finite-horizon optimal control problem at the sampled nodes are used 1) to learn an inner and outer approximation of the feasible region of the ENMPC using support vector machines, and 2) to construct the ENMPC control surface on the computed feasible region using regression or sparse-grid interpolation, depending on the shape of the feasible region. The attractiveness of the proposed control scheme lies in its tractability to higher-dimensional systems with feasibility and stability guarantees, significantly small online computation times, and ease of implementation.