  We investigate the problem of showing that the values of a given polynomial
are smooth (i.e., have no large prime factors) a positive proportion of the
time. Although some results exist that bound the number of smooth values of a
polynomial from above, a corresponding lower bound of the correct order of
magnitude has hitherto been established only in a few special cases. The
purpose of this paper is to provide such a lower bound for an arbitrary
polynomial. Various generalizations to subsets of the set of values taken by a
polynomial are also obtained.
