Computational methods of filter and observer design are presented for a class of polynomial systems with L2- bounded disturbance via convex optimization. A measurement and estimated state dependent polynomial filter gain stabilizes the origin of the error dynamics in an invariant set. In addition to the stability of the error dynamics, a polynomial observer gain guarantees a stability of the origin of the closedloop system in another invariant set for a given polynomial dependent estimated state feedback law. To compute the filter and observer gains and the invariant sets, matrix sum of squares relaxation and semidefinite programming are effectively applied. Numerical examples illustrate the design methods of the paper.