Zeta-function regularization, the multiplicative anomaly and the Wodzicki residue

The multiplicative anomaly associated with the zeta-function regularized determinant is computed for the Laplace-type operators L-1 = -Delta+V-1 and L-2 = -Delta+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.