  The goal of this paper is to extend to higher dimensionality the methods and
computations of vacuum polarization effects in black hole spacetimes. We focus
our attention on the case of five dimensional Schwarzschild-Tangherlini black
holes, for which we adapt the general method initially developed by Candelas
and later refined by Anderson and others. We make use of point splitting
regularization and of the WKB approximation to extract the divergences occuring
in the coincidence limit of the Green function and, after calculating the
counter-terms using the Schwinger - De Witt expansion, we explicitly prove the
cancellation of the divergences and the regularity of the vacuum polarization
once counter-terms are added up. We finally handle numerically the renormalized
expression of the vacuum polarization. As a check on the method we also prove
the regularity of the vacuum polarization in the six dimensional case in the
large mass limit.
