Non-Archimedean second main theorem sharing small functions

TL;DR

This paper proves a new second main theorem for non-Archimedean meromorphic functions involving small functions and demonstrates that sharing five small functions implies the functions are identical, improving previous results.

Contribution

It introduces a novel second main theorem for non-Archimedean functions with truncated counting, and establishes a uniqueness result when sharing five small functions.

Findings

Established a second main theorem for non-Archimedean meromorphic functions.

Proved that sharing five small functions implies the functions are identical.

Improved upon previous uniqueness results in non-Archimedean value theory.

Abstract

In this paper, we establish a new second main theorem for meromorphic functions on a non-Archimedean field and small functions with counting functions truncated to level $1.$ As an application, we show that two meromorphic functions on a non-Archimedean field must coincide to each other if they share $q\, (q\geq 5)$ distinct small functions ignoring multiplicities. In particular, if two non-Archimedean meromorphic functions share $5$ small functions ignoring multiplicities, they must be identical. Thus, our work improves the results in \cite{EY}.