Suppose $C$ is a smooth closed non-intersecting curve in the plane. Think of quadratic or cubic if you will. I need to know bounds for the number of rational points of $C$.
Disclaimer: My knowledge in Algebraic geometry (real or complex) is very limited.
We assume we are working with smooth projective curves defined over $\Bbb Q$. There are three cases here, depending on the genus of the curve.
If the genus is 0, the curve has either zero or infinitely many points, as a result of using any point and the collection of lines through that point to parameterize solutions.
If the genus is 1, this corresponds to the elliptic curve case, and both the finite and infinite cases occurr. In fact, it's conjectured that the chance of either finitely many or infinitely many rational points occurs with probability one-half. The precise distribution of the classes that occur in the infinitely-many-points case is an active area of current research: you may wish to search for "distribution of ranks of elliptic curves".
If the genus is 2 or more, there are finitely many points, by Falting's theorem, though we do not know of any general algorithm for listing all such rational points. There's a conjecture of Lang which would imply that there's an upper bound depending only on the genus, but I am not up to date on recent progress there, and I believe it is far from being settled.