Polynomial values free of large prime factors

For F ∈ Z [ X], let &PSgr; F (x, y) denote the number of positive integers n not exceeding x such that F(n) is free of prime factors > y. Our main purpose is to obtain lower bounds of the form &PSgr; (x, y) >> x for arbitrary F and for y equal to a suitable power of x. Our proofs rest on some results and methods of two articles by the third author concerning localization of divisors of polynomial values. Analogous results for the polynomial values at prime arguments are also obtained.