Next case: Degree 3

$$F(x,y) = ax^3 + bx^2y + cxy^2 + dy^3 + ex^2 + fxy + gy^2 + hx + iy +j = 0.$$

This is getting complicated!

Open Problem: Give an algorithm (a ``procedure'' that on any input terminates in a finite number of steps and outputs ``yes'' or ``no'') that decides whether or not $F=0$ has any rational solutions.

Open Problem: If there is a solution, describe all the solutions?

These are HARD PROBLEMS that are at the core of the frontiers of number theory.

If $F(x,y)=0$ has a rational solution, then the curve it defines is called an elliptic curve.