  The vacuum--polarization contributions of ${\cal O}(b^{n-1} \al_s^n)$ to the
relation between the pole--mass $M$ of a quark and the \MS parameter $\mh (M)$
are evaluated by a straightforward method. They are found to approximate very
well the exact answer, known through ${\cal O}(\al_s^2)$, thus providing a
simple physical interpretation. Results are also given for the cases when the
vacuum--polarization contributions are defined by the pinch technique
prescription and specific background field gauges. Assuming that the terms $n
\geq 3$ are also dominant, we evaluate $M_t/\hat{m}_t(M_t)$, compare the
results with those of optimization methods, briefly discuss $M_b/\mh_b (M_b)$,
and estimate the irreducible errors in the perturbative series. Implications
for the electroweak amplitude $\Delta\rho$ are emphasized. An update of the QCD
corrections to this amplitude, including an estimate of the theoretical error,
is given.
