Here :
https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem
it is stated that Hilbert's tenth problem is already undecideable, if we allow $9$ unknowns.
I wonder whether the case of two variables is known :
Is the solveability of a diophantine equation in two unknowns decideable ?
For degree $1$, this is clearly true (even for arbitary many unknowns) and for $2$, if I am informed right, this is always the case. Is it known whether it is decideable in general ? And if not, is it known for which degree it is undecideable ?