Meromorphic function fields closed by partial derivatives

Abstract

We characterize meromorphic function fields closed by partial derivatives in n variables. 1 2

Degenerate abelian function fields

“…Therefore, Γ ⊂ Γ K ⊂ Γ K , where Γ K is the period group of K. Hence we obtain the following sequence of homomorphisms: The period group of σ * (Mer(A)| A ) is Γ. Then τ 1 : A = C π /Γ −→ C π /Γ K is also an isogeny by Proposition 3 in [5]. Therefore, τ 2 • τ 1 : A −→ C π /Γ is an isogeny.…”

“…For a subfield K of Mer(C n ) we denote by Γ K := f ∈K Γ f the period group of K. A subfield K is said to be non-degenerate if it has a non-degenerate meromorphic function. We stated the following lemma without proof in [5]. Here we give its proof for the convenience of readers.…”

“…[5]), we haveMer(A)| A ∼ = Mer(B)| B and Mer(B)| B ∼ = Mer (C π /Γ K ) C π /ΓK . Hence we obtain K ∼ = σ * Mer(A)| A .…”

A transcendence result on elliptic functions using functions of two variables

“…Recently, the author determined meromorphic function fields closed by partial derivatives ( [1]). It suggests how we may construct meromorphic functions with periods satisfying the condition in Theorem 2, except abelian functions.…”