Universally defining finitely generated subrings of global fields

Abstract

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

Universally defining Z$\mathbb {Z}$ in Q$\mathbb {Q}$ with 10 quantifiers

“…By [4, Corollary 4.12], we have 𝐷 is definable by a positive-existential  ring (𝐾)-formula with 𝑚 quantifiers, that is, a formula which is logically equivalent to ∃𝑌 1 , … , 𝑌 𝑚 𝜓 for some  ring (𝐾)formula 𝜓 built up from atomic  ring (𝐾)-formulae using only conjunctions and disjunctions (no negations). By [4,Remark 3.4], this implies 𝐷 can be described as in (2) for certain 𝑟 and 𝑓 1 , … , 𝑓 𝑟 , where one may choose 𝑟 = 1 when 𝐾 is not algebraically closed. □…”

“…Consider 𝑥 ∈ 𝔪 𝑤 for some 𝑤 ∈  𝐾 ⧵ 𝑆. As in the proof of [2, Lemma 6.6], we can find (𝑎, 𝑏) ∈ Φ 𝑆 𝑢 such that Δ[𝑎2 , 𝑏𝜋) 𝐾 = 𝑆 ∪ {𝑤} and 𝑤(1 + 4𝑎 2 ) = 0. We must then have 𝑤(𝑏𝜋) is odd by Proposition 3.4, and since 𝑤(𝜋) is even, this implies 𝑤(𝑏) is odd.…”

Universally defining $\mathbb{Z}$ in $\mathbb{Q}$ with $10$ quantifiers

“…In [Koe16;Par13;EM18] the number of quantifiers was not counted; according to an older preprint of Koenigsmann, his technique leads to a universal defintion with 418 quantifiers [Koe10, Theorem 1]. In [Daa21] it was shown that rings of S-integers in global fields have a universal definition with 37 quantifiers; in the case of Z in Q, this was further refined by Sun and Zhang to 32 quantifiers in [ZS21].…”

“…Quaternion algebras (and quadratic forms) over global fields have played a historic role in establishing existential definability of subrings of global fields. Hence, in Section 3 we survey some algebraic ingredients regarding global fields and quaternion algebras -more details can be found in [Daa21,Sections 3 and 4].…”

“…In fact, we show something more. Work by Park [Par13], Eisenträger and Morrison [EM18] and the author [Daa21] revealed that Koenigsmann's method can be applied more generally to show that in any global field K, any ring of S-integers has a universal first-order definition. By a global field we mean either a number field or a function field in one variable over a finite field.…”