Thanks to responses in a previous question I asked, I was able to show that if $p$ and $q$ are two distinct polynomials with equal degree greater than 2 and coefficients in $\mathbb{N}$ that the number of integer solutions to $p(x)=q(y)$ is finite.


This is enough for a part of the research problem I am working on, but I'd really like to be able to have a bound on these solutions, or have some algorithm to compute the largest solutions. I have found [this paper](http://www.math.klte.hu/~tengely/tsz/uploads/Main/TengelyThesis.pdf), which solves the problem for when p and q are monic and p(x)-q(y) is irreducible over $\mathbb{Q}[X,Y]$.


Do you know of any other results on bounding solutions to Diophantine equations of this form?


[Edit] By [Bilu's and Tichy's paper](http://matwbn.icm.edu.pl/ksiazki/aa/aa95/aa9534.pdf), there are either finitely many solutions, or the solutions are exponential in nature (as in Pell's equation) when $p \neq q$ and $\deg(p)=\deg(q)\geq2$. The question I'm asking now is given that there are finitely many solutions, are there results similar to those in Tengely's thesis linked above that give explicit bounds on the size of these solutions?



Bounds on solutions to Diophantine equations of the form p(x)=q(y)
Tengely's thesis is a good place to learn the state of the art for bounding these solutions in the irreducible case. All the main methods appear there. In the reducible case there aren't any known bounds, but there are Galois-theoretic results about reducibility of $p(x)-q(y)$ (for instance Theorem 8.1 in the Bilu-Tichy paper) which conceivably might enable one to prove bounds on the solutions.


