Second main theorems with weighted counting functions and algebraic dependence of meromorphic mappings

The purpose of this article is twofold. The first is to prove a new second main theorem for meromorphic mappings and moving hyperplanes of P n ( C ) P^n(C) , where the counting functions are truncated multiplicities and have different weights. Our result is an extension of previous second main theorems for moving hyperplanes with the truncated (to level n n ) counting functions. As its application, the second purpose of this article is to prove a new algebraic dependence theorem for meromorphic mappings having the same inverse images of some moving hyperplanes, where the moving hyperplanes involve the assumption with different roles.