Universally defining finitely generated subrings of global fields

Abstract

It is a problem of general interest when a domain R is first-orderdefinable within its field of fractions K via a universal first-order formula. We show that this is the case when K is a global field and R is finitely generated. Hereby we recover Koenigsmann's result that the ring of integers has a universal first-order definition in the field of rational numbers, as well as the generalisations of Koenigsmann's result by Park for number fields and by Eisenträger and Morrison for global function fields of odd characteristic.

Existential rank and essential dimension of diophantine sets

“…The analogue of Koenigsmann's result for F p (t) was proven in [EM18,Daa18]. For further recent results on diophantine sets in fields see for example [Kol08,AF17,Dit18].…”