Product representations of polynomials

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