Is Hilbert's tenth problem decidable for degree $2$? Hilbert's tenth problem is the problem to determine whether a given multivariate polyomial with integer coefficients has an integer solution. It is well known that this problem is undecidable and that it is decidable in the linear case. In the quadratic case (degree $2$) , the case with $2$ variables is decidable.

Is the case of degree $2$ decidable ? And if yes, can we always find the complete solution set ?

I ask this because many diophantine equations turn out to be solvable despite of the negative result of the problem. Wikipedia gives upper bounds for the degree and for the number of variables sufficient to make the problem undecidable, but I nowhere found a classification of the cases known to be solvable , apart from the elliptic curves and the cases I mentioned above.

The OP asked for further inputs on the two-variable case of Hilbert's Tenth Problem.
One can check out the discussion and answers to this closely related MO question: Connection between the two-variable case of Hilbert's Tenth Problem and Roth's Theorem..
I quote Felipe Voloch:


"(answer) $\ldots$ The case of diophantine equation of two variables is generally believed to be decidable. Poonen has a couple of expository articles on this (check his web page) that you might find useful."




"(comment) The state of the art today is that two variable problems can in practice be solved and systematically are. We just can't yet prove that we will always succeed. Look, e.g., at the papers of M. Stoll and his collaborators."


