Let $P(X,Y)\in \mathbf Z[X,Y]$ be an irreducible polynomial and let $A$ denote the quotient ring $\mathbf Z[X,Y]/(P)$.


What is known about the group of units of $A$?
It's not even clear to me that why it is a finitely generated group. If so, can we say something about its rank?


This question is motivated by Dedekind's theorem, stating that with one variable less, the group of units of $\mathbf Z[X]/(P(X))$ is finitely generated and expressing its rank in terms of the number of real and complex embeddings.



Units of $\mathbf Z[X,Y]/(P(X,Y))$
The group of units of any finitely generated $\mathbb{Z}$-algebra is finitely generated: this is a result of Samuel, *À propos du théorème des unités.*
Bull. Sci. Math. (2) 90 (1966), 89-96. In such generality it is of course impossible to give a meaningful statement about its rank; I don't know if one can do better in your specific case.


