Polynomial with infinitely many zeros. Can a polynomial in $ \mathbb{C}[x,y] $ have infinitely many zeros? This is clearly not true in the one-variable case, but what happens in two or more variables?

For variety, only the zero polynomial has infinitely many zeroes for $y$.
That is, if there are infinitely many values $b$ such that $f(x,b) = 0$, then $f$ is the zero polynomial.
But this is a very different question than asking about the zeroes in $x$ and $y$ together: that is, pais $(a,b)$ such that $f(a,b)=0$.
