Polynomial support vector machines have shown a competitive performance for the problem of handwritten digit recognition. However, there is a large gap in performance vs. computing resources between the linear and the quadratic approach. By computing the complete quadratic classifier out of the quadratic support vector machine, a pivot point is found to trade between performance and effort. Different selection strategies are presented to reduce the complete quadratic classifier, which lower the required computing and memory resources by a factor of more than ten without affecting the generalization performance.