  We study directed polymers subject to a quenched random potential in d
transversal dimensions. This system is closely related to the
Kardar-Parisi-Zhang equation of nonlinear stochastic growth. By a careful
analysis of the perturbation theory we show that physical quantities develop
singular behavior for d to 4. For example, the universal finite size amplitude
of the free energy at the roughening transition is proportional to (4-d)^(1/2).
This shows that the dimension d=4 plays a special role for this system and
points towards d=4 as the upper critical dimension of the Kardar-Parisi-Zhang
problem.
