I am trying to prove the following claim:
Let $F \in \mathbb{Z}[X, Y]$ be a square-free binary form of degree $d \geq 4$, and let $G \in \mathbb{Z}[X, Y]$ be a polynomial of total degree $g \leq d − 3$. Prove that there are only finitely many pairs $(x, y) \in \mathbb{Z}^2$ with $F(x, y) = G(x, y)$ and $F(x, y) \neq 0.$
Here is what I have tried:
The general outline is to try to prove that I can find a solution $(x,y)$ such that I can apply Roth's theorem to $x/y.$ The main problem is that I can't get a good enough approximation.
I've tried to use the fact that $$|F(x, y)| \geq c(F, κ) \max(|x|, |y|)^{d-2-\epsilon}$$ and bound, but I've found that this bound isn't tight enough for Roth's theorem.