A Lower Bound for ∣{a+b:a∈A, b∈B, and P(a,b)≠0}∣

Let A and B be two finite subsets of a field F. In this paper, we provide a non-trivial lower bound for ∣{a+b:a∈A, b∈B, and P(a,b)≠0}∣ where P(x,y)∈F[x,y].