The Relationship Between the Growth of an Entire Function and the Growth of its Zero Set

Abstract

The problem of constructing a holomorphic function of one complex variable with a given zero set was solved by Weierstrass in the middle of the nineteenth century. He showed that if Ω is a domain in the complex plane, if {a _v } is a sequence of points without limit point in Ω, and if {m _v } is a sequence of positive integers, then there exists a function f(z)∈ ℋ(Ω) which has a zero of order exactly m _v at every point a _v . The equivalent problem for several complex variables is Cousin’s Second Problem, which we state as follows: if Ω is a domain in ℂⁿ, then for every zero set defined locally in Ω, does there exist a global holomorphic function which defines the same zero set? More specifically, if U _i is an open covering of Ω and f _i ∈ ℋ(U _i ) are such f _i f _j ⁻¹ ∈ ℋ (U _i ⋂U _j ) and f _j f _i ⁻¹ ∈ ℋ(U _i ⋂U _j ) for all pairs i,j, does there exist f ∈ ℋ(Ω) such that f f _i ⁻¹ ∈ ℋ(U _i ) and f⁻¹ f _i ∈ ℋ(U _i ) for all i? The answer is in general negative, even when Ω is a domain of holomorphy, and depends upon the topological as well as the complex analytic properties of Ω (cf. [A, B]); however, when Ω is a simply connected domain of holomorphy (as in the case of ℂⁿ), the answer is always affirmative.