Experiment design for system identification has seen significant progress in the last decade. One contribution has been to derive convex relaxations of such problems. Consider that only a scalar function of the system parameters is of interest. A standard step in such a case is to first linearize this function with respect to the estimated parameters. The objective of this contribution is twofold: firstly, to examine if there are cases where the linearized approximation is inadequate, and secondly to explore how to improve upon this approximation. By way of examples we show that it is not difficult to construct examples where linearization is insufficient. Furthermore, we introduce the use of higher order approximations and we formally show that this leads to polynomial optimization problems under Gaussian assumptions. We propose the use of cylindrical algebraic decomposition as a method to obtain exact solutions for this type of problems. Numerical examples are provided.