Nash maps over large 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 $K(\!(t)\!)$.