On analytic functions in an ordered field with an infinite rank valuation

TL;DR

This paper investigates analytic functions over a non-Archimedean ordered field with infinite rank valuation, establishing an invertibility theorem using Banach fixed point theorem, contributing to the understanding of such functions in this unique setting.

Contribution

It introduces a local invertibility theorem for analytic functions on a valued field with infinite rank valuation, expanding the theory in non-Archimedean analysis.

Findings

Established a local invertibility theorem for analytic functions.

Applied Banach fixed point theorem in a non-Archimedean context.

Demonstrated the compatibility of topology, order, and ultrametric in the field.

Abstract

Let K be the scalar field of the first orthomodular (or Form Hilbert) space, described by H. Keller in 1980. It has a non-Archimedean order, an infinite rank valuation compatible with the order as well as an explicitly defined ultrametric, all of which induce the same topology. We study analytic functions defined on valued field K, and we will establish an invertibility local theorem for these functions as an application of Banach fixed point theorem.