Is there a list somewhere of which types of Diophantine equations are solvable, which types are not solvable, and which types are not known to be solvable or not? (When I say solvable, I mean that we can determine in a finite number of steps whether or not there exist solutions.) For instance, we know that linear Diophantine equations are solvable. But we know by Matiyasevich's theorem that there are some Diophantine equations that are not solvable.
I am interested in seeing what the borderline between solvable and unsolvable looks like.
Best Answer
Craig:
For a while, there was some research on improving bounds on the number of variables or degree of unsolvable Diophantine equations. Unfortunately, I never got around to cataloging the known results in any systematic way, so all I can offer is some pointers to relevant references, but I am not sure of what the current records are.
Perhaps the first paper to consider along these lines is
The most significant paper in the area is undoubtedly
Let me quote from Matiyasevich's review for MathReviews:
Here is a link to Jones's abstract; of note is the following part:
There are two other, more recent, papers I'm aware of:
I haven't seen this paper, but I believe it is in Chinese. Here is the Abstract:
Finally, there is the following (quoting from MathReviews):
Let me close by mentioning that work on the tenth problem is still very active, although it has moved from just the setting of ${\mathbb Z}$ to more general number rings and beyond. A great reference is the recent book by Shlapentokh, "Hilbert's tenth problem. Diophantine classes and extensions to global fields". New Mathematical Monographs, 7. Cambridge University Press, Cambridge, 2007.
[Edit (April 12, 2017)] The talk by Zhi-Wei Sun mentioned in the comments, "On Hilbert's tenth problem and related topics", a talk given at the City University of Hong Kong on April 14, 2000, is at his page.
Also, Sun has just posted to the arXiv a paper on precisely this topic, Further Results on Hilbert's Tenth Problem. Here is the abstract: