A generalization of Stoll's theorem for moving targets
Manabu Shirosaki · Tohoku Mathematical Journal · 1989
Introduction.In [4], Stoll obtained the defect bound n(n+1) of holomorphic map /: C^P\C) for slow moving targets, while it is well-known that the defect bound for constant targets is n +1 and this is best possible.In this paper we will show a generalization of StolΓs result which interpolates the above two results.The author thanks S. Mori for notifing his simplification [3] of the original proof of StolΓs theorem.His proof was a great help to the author in deriving the result of this paper. Preliminaries.First we introduce the notation and situations used throughout in this paper.We denote the homogeneous coordinates system of P\C) by the notation (w 0 ::w n ).Let /: C^P\C) be a holomorphic map and let (/ 0 , *,/"): C^>C n + ί be its reduced representation, i.e.,/ 0 , ,/" are holomorphic functions without common zeros and/(z) = (/ 0 (z)::/ π (z)) for all zeC.We fix one reduced representation of / and define ||/(z)|| 2 : = |/ 0 (z)| 2 + +|/ M (z)| 2 for all zeC.For/=1, -9 q, wegiverc+1 meromorphic functions a j 0 , , a{ without common zeros and common poles, where q>n + 2.Here we define the characteristic functions.