Multivariate Hensel Lemma for valuated fields

To appear.
On arXiv ArXiv 2301.06546 .
with MariEmi Alonso and Stefan Neuwirth.

Abstract
The Multivariate Hensel Lemma for local rings is usually proved as a consequence of the Grothendieck version of Zariski's Main Theorem. This ZMT deals with a more general situation that is a priori much more difficult. In this paper we give a direct constructive proof of the Multivariate Hensel Lemma for ultrametric valuated fields, without using ZMT. In classical mathematics this implies the Lemma for rank-one valued fields.