Sur la d\'efinissabilit\'e existentielle de la non-nullit\'e dans les anneaux

TL;DR

This paper explores the existential definability of non-null elements in rings, establishing conditions under which this property holds, especially in Noetherian domains and Henselian rings, and addresses a question on strong approximation.

Contribution

It characterizes when the set of nonzero elements is positive-existential in certain rings, providing new insights into Henselian domains and answering a question by Popescu.

Findings

Nonzero elements are positive-existential in non-local Henselian Noetherian domains.

Such sets are not positive-existential in excellent local Henselian domains that are not fields.

Provides an answer to Popescu's question on strong approximation for Henselian pairs.

Abstract

We investigate the rings in which the set of nonzero elements is positive-existential (i.e. a finite union of projections of "algebraic" sets). In the case of Noetherian domains, we prove in particular that this condition is satisfied whenever the ring in question is not local Henselian, while it is not satisfied for any excellent local Henselian domain which is not a field. As a byproduct, we obtain an answer to a question of Popescu on strong approximation for Henselian pairs.