Geodesic
Universally defining finitely generated subrings of global fields
Documenta mathematica, Tome 26 (2021), pp. 1851-1869.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 11U99, 11R52
Keywords: Diophantine set, definability, quaternion algebra, Hilbert 10th problem 
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     author = {Daans, Nicolas},
     title = {Universally defining finitely generated subrings of global fields},
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     year = {2021},
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     url = {http://geodesic.mathdoc.fr/item/DOCMA_2021__26__a6/}
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Daans, Nicolas. Universally defining finitely generated subrings of global fields. Documenta mathematica, Tome 26 (2021), pp. 1851-1869. http://geodesic.mathdoc.fr/item/DOCMA_2021__26__a6/