# Bounds on solutions to Diophantine equations of the form p(x)=q(y)

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, 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?

 By Bilu's and Tichy's paper, 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?

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Which question did you ask? – Igor Rivin Aug 1 '13 at 21:08
The previous question is here. – Dylan Airey Aug 1 '13 at 21:35
You should e-mail Mike Zieve, at the University of Michigan. – David Speyer Aug 1 '13 at 23:06
The statement of the Bilu-Tichy result in your post isn't correct, for instance $x^3=-y^3$ has infinitely many integer solutions. A more subtle issue is that, since $y^2=2x^2+1$ has infinitely many integer solutions, therefore also $h(y^2)=h(2x^2+1)$ has infinitely many integer solutions, for any integer polynomial $h$. – Michael Zieve Aug 2 '13 at 2:11
You're right, I forgot to mention when $p$ and $q$ are equal to $h\circ f \circ \mu$ and $h\circ g\circ \lambda$ with $f$ and $g$ a standard pair of the second kind and $\mu$ and $\lambda$ are linear. I should have said that by Bilu and Tichy's paper, there are finitely many solutions, or the solutions are exponential in nature. – Dylan Airey Aug 2 '13 at 2:33

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.

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I don't know exactly what the problem you're working on is, you want a bound in the case that it is finite? Or you want to prove that there are finitely many solutions?

The papers I'd recommend are Bilu and Tichy's The Diophantine Equation p(x) = q(y)" or Avanzi and Zannier'sGenus one curves defined by separated variable polynomials and the polynomial Pell equation."

Cheers,

Ben

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I have edited the question. I used Bilu and Tichy's results to show there are finitely many solutions, but now I would like to find explicit bounds on the size of solutions. I will look over Avanzi and Zannier's paper. – Dylan Airey Aug 1 '13 at 21:41

A bound in terms of what? If you take $p(x) = p(y) (q(y)+1),$ then every integral zero of $q$ will be a solution. Granted, you can bound those in terms of the coefficients of $q,$ but is that your input?

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I mean a bound on the maximum values of $x$ and $y$ integers so that $p(x)=q(y)$ in terms of some properties of the polynomials. For example, in Tengely's thesis, he uses the degrees and coefficients of the two polynomials to find a bound in a restricted case. – Dylan Airey Aug 2 '13 at 2:35