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In this paper, we propose a novel approach for robust state reconstruction of multi-output discrete-time systems with polynomial nonlinear dynamics and measurements in presence of unknown but bounded disturbances corrupting both the state and measurement equations. The proposed recursive algorithm is based on ellipsoidal state bounding techniques and consists of a two-step prediction-correction procedure, with each step requiring the solution of a convex optimization problem under sum of squares rather than linear matrix inequality constraints. The presented approach reduces the conservatism in the calculation of the state bounding ellipsoid when compared to existing robust filtering approaches. The applicability and effectiveness of the new algorithm is exemplarily shown in numerical simulations of a benchmark system.