Polynomial Diophantine Equations

Question

So in general how does one decide if: $$ a_0 + a_1x + a_2x^2 ... a_{n_1}x^{n_1} = b_1y + b_2y^2 ... + b_{n_2}y^{n_k}$$

Has solutions for integers $x,y$ given real numbers $a_0, a_1. .. a_{n_1}, b_1 , b_2 ... b_{n_2}$

I noticed: In general deciding

$$ a_0 + a_1x + a_2x^2 ... a_{n_1}x^{n_1} = b_0y $$

Is really a matter of deciding if there exists integers $x$ such that:

$$a_1x + a_2x^2 ... a_{n_1}x^{n_1} \equiv -a_0 \mod b_0$$

Thus my question can also be rephrased as. What sort of abstract machinery (a generalization of modular arithmetic can be created) to decide for problems of the variety given earlier?

Answer 1

You can learn some stuff on Hensel's Lemma, which is quite strong in such problems.