Product representations of polynomials

For a fixed polyomial f ∈ Z [ X ] , let ρ k ( N ) denote the maximum size of a set A ⊂ { 1 , 2 , … , N } such that no product of k distinct elements of A is in the value set of f . In this paper, we determine the asymptotic behaviour of ρ k ( N ) for a wide class of polynomials. Our results generalize earlier theorems of Erdős, Sós and Sárközy.