Is there (conjectural) upper bound for the largest solution of Diophantine equation with finitely many solutions? Let $F(x_1,\ldots,x_n)$ be polynomial with integer coefficients
and $x_i$ integers.
System of Diophantine equations can be brought in this form
via sum of squares.
Assume $F=0$ has finitely many integer solutions.
Let $M$ be the largest solution: $M=\max\{|a| : F(\ldots,a,\ldots)=0\}$.

Q1 Is there (conjectural) upper bound on $M$ in terms of the coefficients
  of $F$,$\deg(F)$ and $n$?

Or is it known $M$ can't be bounded this way?
For Thue equations, there is enormous upper bound (probably far from sharp).
If I remember correctly, there was Arxiv paper with many revisions 
giving upper bound in different notation.
 A: Please read the article:
A. Tyszka, A hypothetical way to compute an upper bound for the heights
of solutions of a Diophantine equation with a finite number of solutions,
Proceedings of the 2015 Federated Conference on Computer Science and
Information Systems (eds. M. Ganzha, L. Maciaszek, M. Paprzycki); Annals
of Computer Science and Information Systems, vol. 5, 709-716, IEEE Computer
Society Press, 2015,
http://dx.doi.org/10.15439/2015F41
Please read the preprint (updated on January 2017):
A. Tyszka, A conjecture which implies that there exists a computable upper
bound for the heights of solutions of a Diophantine equation with a finite
number of solutions,
https://arxiv.org/abs/1109.3826
Let $f(1)=2$, $f(2)=4$, and $f(n+1)=f(n)!$ for every integer $n \geq 2$. We conjecture that
if a system $S \subseteq \{x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in \{1,...,n\}\}$ has only
finitely many solutions in positive integers $x_1,...,x_n$, then each such solution
$(x_1,...,x_n)$ satisfies $x_1,...,x_n \leq f(2n)$.
The conjecture implies that there exists an algorithm which takes as input a Diophantine equation,
returns an integer, and this integer is greater than the heights of integer (non-negative integer,
positive integer, rational) solutions, if the solution set is finite.
