Second main theorem for meromorphic mappings with moving hypersurfaces in subgeneral position

TL;DR

This paper extends the second main theorem in Nevanlinna theory to meromorphic mappings with moving hypersurfaces in subgeneral position, providing explicit bounds and an improved understanding of value distribution.

Contribution

It generalizes the second main theorem to include moving hypersurfaces in subgeneral position with explicit bounds on the truncation level M.

Findings

Established a second main theorem for meromorphic mappings with moving hypersurfaces.

Provided explicit estimate for the truncation level M.

Extended previous results to more general hypersurface configurations.

Abstract

Let $f$ be an algebraically nondegenerate meromorphic mapping from $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ and let $Q_1,...,Q_q$ be $q$ hypersurfaces in $\mathbb P^n(\mathbb C)$ of degree $d_i$, in $N-$subgeneral position. In this paper, we will prove that, for every $\epsilon >0$, there exists a positive integer $M$ such that $$||\ (q-(N-n+1)(n+1)-\epsilon) T_f(r)\le\sum_{i=1}^q\frac{1}{d_i}N^{[M]}(r,f^*Q_i)+o(T_f(r)).$$ Moreover, an explicit estimate for $M$ is given. Our result is an extension of the previous second main theorem for the mappings and moving hyperplanes or moving hypersurfaces.