Two meromorphic mappings sharing 2n + 2 hyperplanes regardless of multiplicity

TL;DR

This paper extends value sharing results for meromorphic mappings in several complex variables, showing that sharing hyperplanes under certain conditions implies algebraic degeneracy of the combined map.

Contribution

It generalizes previous results by establishing algebraic degeneracy when two meromorphic mappings share hyperplanes with specific multiplicity conditions.

Findings

Sharing 2n+1 hyperplanes ignoring multiplicity leads to algebraic degeneracy.

Sharing an additional hyperplane with truncated multiplicity n+1 influences the map's algebraic properties.

Abstract

Nevanlinna showed that two non-constant meromorphic functions on $\mathbb C$ must be linked by a M\"{o}bius transformation if they have the same inverse images counted with multiplicities for four distinct values. After that this results is generalized by Gundersen to the case where two meromorphic functions share two values ignoring multiplicity and share other two values with multiplicities trucated by 2. Previously, the first author proved that for $n\ge 2,$ there are at most two linearly nondegenerate meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ sharing $2n+2$ hyperplanes ingeneral position ignoring multiplicity. In this article, we will show that if two meromorphic mappings $f$ and $g$ of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ share $2n+1$ hyperplanes ignoring multiplicity and another hyperplane with multiplicities trucated by $n+1$ then the map $f\times g$ is algebraically degenerate.