Fundamentals of the Theory of Analytic Sets

Abstract

Let a function f be holomorphic in a domain V = V’ × {|z_n| < R}, where V’ is a neighborhood of the coordinate origin 0’ in ℂⁿ⁻¹, and let also f(0’,z_n) ≠ 0 in the disc |z_n| < R. Let r < R be such that f (0’,z_n) does not have zeros on the circle |z_n| = r, and let k be the number of its zeros in the disk U_n: |z_n| < r, counted with multiplicities. Then f can, in a certain neighborhood U = U’ × U_n ⊂ V of the coordinate origin in ℂⁿ, be represented in the form

$$f(z) = \left( {z_n^k + {c_1}\left( {z'} \right)z_n^{k - 1} + \cdots + {c_k}\left( {z'} \right)} \right)\phi (z)$$

(*)

where the functions c_j(z’) are holomorphic in U’, while ϕ is holoniorphic and zero free in U.