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I understand that Diophantine Analysis is an enormous field! Without first determining the solution set, suppose I'd like to calculate the number of non-negative integer solutions $(x,y,z)$ of \begin{eqnarray} ax^{p} + by^{q} \leq cz^{r} \end{eqnarray} for positive integers $a,b,c,p,q$ and $r$ (as a function of $z$). What general methods are available for such enumeration of this or more general Diophantine inequalities, including more variables or nonlinear forms? (Is our understanding of such enumeration limited to a select set of very specific equation types?) If no general methods exist, I'm sure that there are methods of bounding the number of solutions from below and above with error that can be estimated. Which is the best known?

References are welcome. Thanks!

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It seems that our understanding is limited to a select few equation types. For instance, see this: – PEV Jan 16 '11 at 8:12
Hey Trevor, thanks for the link! – user02138 Jan 16 '11 at 8:52
Well there will be infinitely many triples $(x,y,z)$ since you can just make $z$ arbitrarily large. I think the question you are looking for is the number of solutions $$ax^p +by^q \leq z$$ as a function of $z$. This is then well approximated by the area of this shape. – Eric Naslund Oct 31 '11 at 14:18
Agreed. Edited. – user02138 Oct 31 '11 at 15:57
Upvoted Eric's comment. – mick Aug 8 '13 at 22:49

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