Universally defining finitely generated subrings of global fields

It is shown that any finitely generated subring of a global field has universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring integers and subsequent extensions to rings number fields $S$-integers function odd characteristic. In this article proof presented which uniform all fields, including characteristic two case, where entirely novel. Furthermore, proposed method results formulae requiring significantly fewer quantifiers than can be derived through previous approaches.