Meromorphic functions together with their partial derivatives sharing zeros and poles in Cᵐ

In the paper, using Nevanlinna's value distribution theory of meromorphic functions in Cᵐ, we study the problem: Let f and g be two finite order non-constant meromorphic functions in Cᵐ such that f and g share $0$ and ∞ CM. If ∂zᵢ(f) and ∂zᵢ(g) share $0$ CM and if for one $n>1$, ∂ⁿzᵢ(f) and ∂ⁿzᵢ(g) share $0$ CM, where i=1,2,…,m, then f and g have special presentations. Our finding extends the result of Köhler (Meromorphic functions sharing zeros and poles and also some of their derivatives sharing zeros, Complex Var., 11 (1989), 39-48) into higher dimensions. Moreover, we show by an example that the restriction on the growth of f and g is necessary.