# A localized version of the basic triangle theorem

## TL;DR

This paper presents a localized version of the basic triangle theorem to facilitate the study of hyperlogarithms, providing direct access to rings of scalars and simplifying the analysis over various function fields.

## Contribution

It introduces a localized version of the basic triangle theorem, enabling more direct and simplified analysis of hyperlogarithms over different function fields.

## Key findings

- Provides a localized theorem version for hyperlogarithm independence
- Avoids using fraction fields like meromorphic functions
- Simplifies the analysis of hyperlogarithms over various rings

## Abstract

In this short note, we give a localized version of the basic triangle theorem, first published in 2011 (see [4]) in order to prove the independence of hyperlogarithms over various function fields. This version provides direct access to rings of scalars and avoids the recourse to fraction fields as that of meromorphic functions for instance.