# Second main theorems with weighted counting functions and its applications

## TL;DR

This paper generalizes second main theorems in complex analysis for meromorphic mappings, incorporating weighted counting functions and truncated multiplicities, and applies these results to algebraic dependence theorems involving moving hyperplanes.

## Contribution

It introduces a generalized framework for second main theorems with weighted, truncated counting functions and extends algebraic dependence results to more complex hyperplane configurations.

## Key findings

- Generalized second main theorems with weighted counting functions
- Improved algebraic dependence theorems for meromorphic mappings
- Enhanced understanding of hyperplane configurations in complex projective space

## Abstract

The purpose of this article has two fold. The first is to generalize some recent second main theorems for the mappings and moving hyperplanes of $\P^n(\C)$ to the case where the counting functions are truncated multiplicity (by level $n$) and have different weights. As its application, the second purpose of this article is to generalize and improve some algebraic dependence theorems for meromorphic mappings having the same inverse images of some moving hyperplanes to the case where the moving hyperplanes involve the assumption with different roles.