3 problems

A ring of algebraic integers $O$ is the integral closure of $\mathbb Z$ in a finite field extension of $\mathbb Q$. The Diophantine problem in $O$, denoted $\mathcal{D}(O)$, asks w…

Let $O$ be a ring of algebraic integers, that is, the integral closure of $\mathbb{Z}$ in a number field. A subset is e-definable if it is definable by an existential system of pol…

Let $K$ be an algebraic number field, that is, a finite extension of $\mathbb{Q}$, and let $\mathcal{O}_K$ denote its ring of integers. The unit sum number $u(R)$ of a ring $R$ is…