The second main theorem for moving targets
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.
scientific article; zbMATH DE number 4210499
scientific article