Two meromorphic mappings having the same inverse images of moving hyperplanes

Abstract

In this paper, we will show that if two meromorphic mappings $f$ and $g$ of ${\mathbb{C}}^m$ into ${\mathbb{P}}^n(\mathbb{C})$ have the same inverse images for $(2n+2)$ moving hyperplanes $\{a_i{\}}_{i=1}^{2n+2}$ with multiplicities counted to level $l_0$ then the map $f\times g$ must be algebraically degenerated over the field $\mathcal{R}\{a_i{\}}_{i=1}^{2n+2}$, where $l_0=3n^3(n+1)q(q-2)$ with $q=\left(\genfrac{}{}{0ex}{}{2n+2}{n+2}\right)$. Our result generalizes the previous result for fixed hyperplanes case of Fujimoto and also improves his result by giving an explicit estimate for the number $l_0$.

The result in this version is improved by giving an explicit estimate for the truncation level $l_0$. The method of the proof also is changed, different from the original method. This paper will appear in Complex Variable and Elliptic Equations