Algebraic variety in $\mathbb{C^{2}}$

Question

would somebody algebraic-geometry-savvy please help me with this problem, as I am pretty new to algebraic geometry: I have to find smallest algebraic variety (irreducible algebraic set) in $\mathbb{C^{2}}$ that contains all the points with integer coordinates. I don't really know how to start, so at least an idea, hint or even better sketch of the solution would be much appreciated. Thank you very much.

Answer 1

Extended hint: Let $$ F(X,Y)=\sum_{i=0}^n F_i(Y)X^i $$ be a non-zero polynomial from $\Bbb{C}[X,Y]$. Assume that it vanishes at all the points where $(X,Y)\in\Bbb{Z}^2$. The polynomial $F_n(Y)$ is non-zero as the leading coefficient. It can only have finitely many zeros. Therefore there exists an integer $m$ such that $F_n(m)\neq0$. Thus $$ G(X):=F(X,m)=\sum_{i=0}^nF_i(m)X^i\in\Bbb{C}[X] $$ is non-zero.

Answer 2

The only such is the plane itself. Note that there can be no algebraic curve passing through all those points since then it would have an infinite number of intersections with a line, which is impossible.