For $x, y \in GF[2^n]$, consider the two-parameter polynomial $P(x,y) = x \cdot y + f(x) + g(y)$, where $f$ and $g$ are arbitrary polynomials on $GF[2^n]$.
Can we say anything about the number of zeros of $P$? Intuitively one would expect that the product $x \cdot y$ does not coincide with the sum of two one-parameter polynomials $f$ and $g$ at many points, but is there a (simple) way to show this?