A [theorem of Alon and Füredi](http://www.cs.tau.ac.il/%7Enogaa/PDFS/Publications/Covering%20the%20cube%20by%20affine%20hyperplanes.pdf) 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?



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> What is the smallest possible degree of a *homogeneous* polynomial $P(x,y)$ vanishing on all, but exactly one point of the grid $A\times B$?
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The underlying field $\mathbb F$ can be assumed to have characteristic $2$.



Alon-Füredi for homogeneous polynomials
In general homogeneity does not improve the bound. Take $A=\{1,q,q^2,\ldots,q^{a-1}\}$, $B=\{1,q,q^2,\ldots,q^{b-1}\}$, $f(x,y)=\prod\_{i=-(b-1)}^{a-2}(x-q^iy)$.


