  The multiplicative anomaly associated with the zeta-function regularized
determinant is computed for the Laplace-type operators $L_1=-\lap+V_1$ and
$L_2=-\lap+V_2$, with $V_1$, $V_2$ constant, in a D-dimensional compact smooth
manifold $ M_D$, making use of several results due to Wodzicki and by direct
calculations in some explicit examples. It is found that the multiplicative
anomaly is vanishing for $D$ odd and for D=2. An application to the one-loop
effective potential of the O(2) self-interacting scalar model is outlined.
