QCD Corrections from Top Quark to Relations between Electroweak Parameters to Order $\alpha_s^2$

The vacuum polarization functions $\Pi(q^2)$ of charged and neutral gauge bosons which arise from top and bottom quark loops lead to important shifts in relations between electroweak parameters which can be measured with ever-increasing precision. The large mass of the top quark allows approximation of these functions through the first two terms of an expansion in $M_Z^2/M_t^2$. The first three terms of the Taylor series of $\Pi(q^2)$ are evaluated analytically up to order $\as^2$. The first two are required to derive the approximation, the third can be used to demonstrate the smallness of the neglected terms. The paper improves earlier results based on the leading term $\propto G_F M_t^2 \as^2$. Results for the subleading contributions to $\dr$ and the effective mixing angle $\sineff$ are presented.