How does Hilbert's 10th problem imply that the growth of solutions to Diophantine equations is uncomputable?

Question

I'm working on an expository paper about Diophantine equations and elliptic curves, and one of the equations I'm discussing in the paper is covered in An unusual cubic representation problem. How does Hilbert's 10th problem imply that the growth of solutions to Diophantine equations is uncomputable as N increases? Alon Amit's answer states this, but I'm looking for a paper that explains it better and is also a source that I can cite.

Answer 1

This is pretty trivial. Suppose there was a computable function $f$ which takes a Diophantine equation and gives an upper bound for the size of an integer solution to it, if any integer solution exists (where here by "size" I mean the maximum of the absolute values of all the variables, say). This immediately gives an algorithm to determine whether a Diophantine equation has an integer solution: just feed the equation into $f$, and test all possible integer inputs up to the bound given by $f$. If you find a solution, then an integer solution exists. Otherwise, no integer solution exists.

But by the solution to Hilbert's 10th problem, no algorithm exists to determine whether a Diophantine equation has a solution. Thus no such computable function $f$ can exist.