Exceptional values of entire functions of finite order in one of the variables

Abstract

Let F(z,w) be a holomorphic function in Cn×C of finite order in w with n≥2. Let Ω be the set of points z∈Cn where F(z,w) is a non-constant function omitting a value π(z). Near a finite accumulation point z0 of Ω, we prove in the main result (Theorem 1) that Ω is a local analytic set and π(z) is holomorphic, and show the existence of a proper globally analytic set Δ of Cn such that either Ω⊂Δ or Ω=Cn∖Δ, being possible in the last case to also determine F(z,w) in terms of π(z). We apply this result to several problems. First, we extend a Theorem due to Nishino about exceptional values when near z0 dimension of Ω is n and assure the existence of a meromorphic function α(z) in Cn such that π(z)=α(z) except at points where α(z) has poles or F(z,w) is constant (also being F(z,w) a polynomial in w if α(z) is ∞). After, we prove that Ω is a local analytic set in Cn and the existence of a proper analytic subset E of Cn such that Ω⊂E or Ω=Cn∖E. Finally, we generalize a Lelong-Gruman Theorem about the set of points z where π(z)=0.