On functions given by algebraic power series over Henselian valued fields

TL;DR

This paper explores algebraic power series over Henselian valued fields, establishing key theorems on implicit functions, algebraic series, and limits, with applications to definable functions and closedness properties.

Contribution

It extends classical theorems to Henselian valued fields and applies these results to analyze definable functions and their limits in this context.

Findings

Theorems on implicit functions over Henselian valued fields.

Results on algebraic power series and their properties.

Application of fiber shrinking and limit existence in definable functions.

Abstract

This paper provides, over Henselian valued fields, some theorems on implicit function and of Artin--Mazur on algebraic power series. Also discussed are certain versions of the theorems of Abhyankar--Jung and Newton--Puiseux. The latter is used in analysis of functions of one variable, definable in the language of Denef--Pas, to obtain a theorem on existence of the limit, proven over rank one valued fields in one of our recent papers. This result along with the technique of fiber shrinking (developed there over rank one valued fields) were, in turn, two basic tools in the proof of the closedness theorem.