The second main theorem for moving targets
type: Item
instance of: scholarly article
title: The second main theorem for moving targets
zbMATH Open document ID: 0732.30025
DOI: 10.1007/BF02938116
author: Min Ru
author: W. Stoll
published in: The Journal of Geometric Analysis
publication date: 1991-01-01T00:00:00Z
review text: Let G be a finite set of meromorphic functions g: \({\mathbb{C}}\to {\mathbb{P}}(V^*)\), where \(V^*\) is the dual vector space of the complex vector space V. It exists one and only one smallest field \(R_ G\) such that each \(g\in G\) is dependent on \(R_ G\). The authors make the following assumptions: (A1) Let V be a hermitian vector space of dimension \(n+1>1.\) (A2) Let f: \({\mathbb{C}}\to {\mathbb{P}}(V)\) be a transcendental holomorphic map. (A3) Let G be a finite set of holomorphic maps g: \({\mathbb{C}}\to {\mathbb{P}}(V^*)\) with \(k:=\#G\geq n+1.\) (A4) Assume that \(T_ g(r,s)/T_ f(r,s)\to 0\) for \(r\to 0\) for all \(g\in G.\) (A5) Assume that G is in general position. (A6) Assume that f is linearly nondegenerate over the field \(R_ G.\) The main result is the following theorem (the second main theorem for moving targets): Assume (A1)-(A6) hold. Take \(s>0\) and \(\epsilon >0\). Then \[ \sum_{g\in G}m_{f,g}(r) {\underset{.} \leq} (n+1+\epsilon)T_ f(r,s), \] where \({\underset{.}\leq}\) means the inequality holds except for a finite measure subset \(E_ s\subset {\mathbb{R}}^+\). The authors also obtain the defect relation: Theorem. Assume (A1)-(A6) hold, then \[ \sum_{g\in G}\delta (f,g)\leq n+1. \] In this connection the full historical details are included.
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 32H30
zbMATH DE Number: 4210499
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02938116
OpenAlex ID: W2041275311
description: scientific article; zbMATH DE number 4210499