On value sets of polynomials over a field

Let F   be any field. Let p(F)p(F) be the characteristic of F if F   is not of characteristic zero, and let p(F)=+∞p(F)=+∞ otherwise. Let A1,…,AnA1,…,An be finite nonempty subsets of F, and letf(x1,…,xn)=a1x1k+⋯+anxnk+g(x1,…,xn)∈F[x1,…,xn] with k∈{1,2,3,…}k∈{1,2,3,…}, a1,…,an∈F∖{0}a1,…,an∈F∖{0} and deggkn>k, we propose a further conjecture which extends the Erdős–Heilbronn conjecture in a new direction.