  The heavy-quark contribution to the polarisation function $\Pi$ at higher
perturbative orders is presently only known approximately. We scrutinise the
accuracy of state-of-the-art approximations at three- and four-loop order. At
three loops, we present for the first time a result with arbitrary numerical
precision for general kinematics and compare to the best Pad\'e estimate. At
four loops, we calculate the fourth (inverse) moment of the non-singlet
heavy-quark vacuum polarisation in order to test the prediction for this moment
based on Pad\'e approximation.
