  The strong-coupling regime of Kardar-Parisi-Zhang surface growth driven by
short-ranged noise has an upper critical dimension d_> less or equal to four
(where the dynamic exponent z takes the value z (d_>) = 2). To derive this, we
use the mapping onto directed polymers with quenched disorder. Two such
polymers coupled by a small contact attraction are shown to form a bound state
at all temperatures below the roughening temperature of a single polymer.
Comparing the singularities of the localization length at and below the
critical temperature yields d_> \leq 4.
