# The closedness theorem over Henselian valued fields

## TL;DR

This paper proves the closedness theorem over Henselian valued fields, extending previous results to a broader class of fields and enabling further developments in definable functions and algebraic geometry.

## Contribution

It generalizes the closedness theorem to Henselian valued fields using fiber shrinking and relative quantifier elimination, broadening its applicability.

## Key findings

- Established the closedness theorem over Henselian valued fields.
- Enabled derivation of ojasiewicz inequality and curve selection.
- Facilitated extension of hereditarily rational functions and development of regulous functions.

## Abstract

We prove the closedness theorem over Henselian valued fields, which was established over rank one valued fields in one of our recent papers. In the proof, as before, we use the local behaviour of definable functions of one variable and the so-called fiber shrinking, which is a relaxed version of curve selection. Now our approach applies also relative quantifier elimination for ordered abelian groups due to Cluckers--Halupczok. Afterwards the closedness theorem will allow us to achieve i.a. the \L{}ojasiewicz inequality, curve selection and extending hereditarily rational functions as well as to develop the theory of regulous functions and sheaves.