Nash maps over large fields

TL;DR

This paper explores properties of Nash maps over large fields, establishing inverse and implicit function theorems, characterizations, and connections to logical tameness and field extensions, thereby advancing the understanding of algebraic and analytic structures in large fields.

Contribution

It introduces inverse and implicit function theorems for Nash maps over large fields and characterizes large fields via these theorems, linking algebraic, topological, and logical properties.

Findings

Inverse and implicit function theorems hold for Nash maps over large fields.

Large fields are characterized by satisfaction of inverse or implicit function theorems.

Nash functions provide a natural proof that large fields are existentially closed in formal Laurent series fields.

Abstract

In this short note we give some corollaries of the polynomial inverse function theorem for large fields. We prove inverse and implicit function theorems for Nash maps over large fields, characterize large fields as fields satisfying inverse or implicit function theorems, give inverse and implicit function theorems for gt-henselian field topologies, and show that definable functions in various logically tame fields of characteristic zero are generically Nash. We prove a version of Krasner's lemma for large fields and describe how Nash functions give a natural proof of the well-known fact that a large field $K$ is existentially closed in $K(\!(t)\!)$.