A closed-form state estimator for some polynomial nonlinear systems is derived in this paper. Exploiting full Taylor series expansion we first give exact matrix expressions to compute mean and covariance of any random variable distribution that has been transformed through a polynomial function. An original discrete-time Kalman filtering implementation relying on this exact polynomial transformation is proposed. The important problem of chaotic synchronization of Chebyshev maps is then considered to illustrate the significance of these results. Mean Square Error (MSE) between synchronized signals and consistency criteria are chosen as performance measures under various signal-to-noise ratios (SNR). Comparisons to the popular Extended Kalman Filter (EKF) and to the recent Unscented Kalman Filter (UKF) are also conducted to show the pertinence of our filtering formulation.