Cubic Diophantine equations

Question

We know how to solve Diophantine equations of first degree (linear) and of the second degree. What is available for solving Diophantine equations of the third degree? Let us say that we are given a two-variable polynomial equation P(x,y)=0 with integer coefficients, of third degree in both variables. What known methods or theories are used to solve such a general Diophantine equation of third degree? Added: I am assuming the polynomial is genuine third degree (i.e. is irreducible) and that all powers of x and y may be present (alas second powers may be always eliminated by appropriate substitutions).

Answer 1

I'll assume $P(x,y)$ is irreducible over the rationals. The first thing you'll want to look at is its genus. In many cases the genus will be $1$, meaning this is an elliptic curve. Especially if it is Weierstrass form, there are ways to find the integer solutions (e.g. with Sage). If it isn't in Weierstrass form, you can put it in Weierstrass form with a rational transformation, look at the rational points of that transformed curve, and try to see which correspond to integer solutions of the original.