There's no algorithm that correctly decides if a Diophantine equation does or doesn't have a solution. Still, many equations can be successfully analyzed, and I'm wondering if anyone wrote down a "cookbook" for dealing with Diophantine equations of various shapes and forms, including the higher-degree, higher-dimensionality ones.
Given a system of polynomial equations with integer coefficients, we may wish to determine if there are any solutions in integers, and if so, whether there are finitely or infinitely many, and whether they can be explicitly described; we may also wish to determine if there are any solutions in rational numbers, and if so, whether there are finitely many, etc.
- If the system is linear, do this (easy).
- If there is just one variable, do that (easy).
- If there's one quadratic equation in two variables (or a homogenous one in three variables), there's again an explicit procedure: check if there's a singularity, determine if there are integer solutions at all (Hasse Minkowski), parametrize the curve, etc. I think all questions can be effectively answered in the case of genus 0 curves.
- If it's an elliptic curve, follow these steps... (I don't think all questions can be algorithmically answered, at present).
- Higher genus curve? What do you do? Find the Jacobian? What else?
- Higher dimensional surfaces and varieties? What do you do? Which heuristics do you try, what are some useful families of equations that can be attacked?
All of those pieces are well covered in the literature - I'm just wondering if there's a good resource that succinctly describes the various alternatives that we may be able to handle.
Note: this older question has similar goals, but it stops short of giving details on how to handle genus 0 and genus 1 and says nothing much about higher genera and higher dimensional varieties.