A density property of Henselian valued fields

TL;DR

This paper provides an elementary proof of a density property for Henselian valued fields, crucial for understanding the topology and geometry of algebraic subsets within these fields.

Contribution

It introduces a new elementary proof of a density property for Henselian valued fields, enhancing the understanding of their topological and geometric structure.

Findings

Density property established for Henselian valued fields

Elementary proof of a version of the implicit function theorem

Implications for algebraic subsets in valued fields

Abstract

We give an elementary proof of a version of the implicit function theorem over Henselian valued fields $K$. It yields a density property for such fields (introduced in a joint paper with J. Koll{\'a}r), which is indispensable for ensuring reasonable topological and geometric properties of algebraic subsets of $K^{n}$.