TITLE: Polynomial with infinitely many zeros.
QUESTION [18 upvotes]: 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?

REPLY [4 votes]: Any non constant polynomial in one variable has at least one root in $\mathbb C$.
If $P(x,y)\in\mathbb C[x,y]$ is nonconstant and $a$ is a root of the polynomial $P(x,0)$ then $(a,0)$ is a root of $P(x,y)$...
Consider the polynomials $P(x,0),P(x,1),P(x,2),P(x,3),\ldots$