A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables

TL;DR

This paper develops a new version of the second main theorem for meromorphic mappings in several complex variables, using Casorati determinants, leading to improved defect relations, uniqueness results, and Picard-type theorems.

Contribution

It introduces a novel second main theorem for meromorphic mappings with hyperorder less than one, replacing the Wronskian with Casorati determinants, and derives related value distribution results.

Findings

Established a second main theorem without truncated multiplicity

Derived a defect relation for meromorphic mappings

Proved a difference analogue of the Picard theorem

Abstract

Let $c\in \mathbb{C}^{m},$ $f:\mathbb{C}^{m}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly nondegenerate meromorphic mapping over the field $\mathcal{P}_{c}$ of $c$-periodic meromorphic functions in $\mathbb{C}^{m}$, and let $H_{j}$ $(1\leq j\leq q)$ be $q(>2N-n+1)$ hyperplanes in $N$-subgeneral position of $\mathbb{P}^{n}(\mathbb{C}).$ We prove a new version of the second main theorem for meromorphic mappings of hyperorder strictly less than one without truncated multiplicity by considering the Casorati determinant of $f$ instead of its Wronskian determinant. As its applications, we obtain a defect relation, a uniqueness theorem and a difference analogue of generalized Picard theorem.