# Diophantine problems in rings and algebras: undecidability and reductions to rings of algebraic integers

Journal Article

We study systems of equations in different families of rings and algebras. In each such structure \$R\$ we interpret by systems of equations (e-interpret) a ring of integers \$O\$ of a global field. The long standing conjecture that \$\mathbb{Z}\$ is always e-interpretable in \$O\$ then carries over to \$R\$, and if true it implies that the Diophantine problem in \$R\$ is undecidable. The conjecture is known to be true if \$O\$ has positive characteristic, i.e. if \$O\$ is not a ring of algebraic integers. As a corollary we describe families of structures where the Diophantine problem is undecidable, and in other cases we conjecture that it is so. In passing we obtain that the first order theory with constants of all the aforementioned structures \$R\$ is undecidable.

### Cited Authors

• Garreta, A; Miasnikov, A; Ovchinnikov, D