On value sets of polynomials over a field

Let F be any field. Let p(F) be the characteristic of F if F is not of characteristic zero, and let p(F)=+∞ otherwise. Let A1,…,An be finite nonempty subsets of F, and let with k {1,2,3,…}, a1,…,an F {0} and deggk, we propose a further conjecture which extends the Erdős–Heilbronn conjecture in a new direction