A theorem of Alon and Füredi says that if $A$ and $B$ are finite, nonempty subsets of the field $\mathbb F$, and if a polynomial $P(x,y)\in\mathbb F[x,y]$ vanishes on all, but exactly one point of the grid $A\times B$, then $\deg P\ge |A|+|B|-2$.
Can one improve this bound given that $P$ is homogeneous?
What is the smallest possible degree of a homogeneous polynomial $P(x,y)$ vanishing on all, but exactly one point of the grid $A\times B$?
The underlying field $\mathbb F$ can be assumed to have characteristic $2$.