CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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A local degree-m holomorphic map has m nearby sheets

Statement

Let $f : Ω \to C$ be nonconstant and holomorphic on a complex domain $Ω$ , let $a \in Ω$ , and put $m = \deg_{a} f$ . After shrinking around $a$ , every nearby value other than $f (a)$ has exactly $m$ distinct preimages.

Precisely, for every neighbourhood $N$ of $a$ in $Ω$ , there are an open neighbourhood $V$ of $a$ with $V \subseteq N$ and a real $ρ > 0$ such that, for every $w$ with $0 < ∣ w - f (a) ∣ < ρ^{m}$ , the equation $f (z) = w$ has exactly $m$ distinct solutions in $V$ . The value $f (a)$ has the single preimage $a$ in $V$ , counted with multiplicity $m$ .

Facts & Assumptions

Given: A nonconstant holomorphic function $f : Ω \to C$ on a complex domain, a point $a \in Ω$ , the positive natural $m = \deg_{a} f$ (Local degree of a nonconstant holomorphic map), and an arbitrary neighbourhood $N$ of $a$ in $Ω$ . A biholomorphism is bijective with holomorphic inverse (Biholomorphic maps between complex domains).

[L1]

If $f : Ω \to C$ is nonconstant and holomorphic on a complex domain, $a \in Ω$ , and $m = \deg_{a} f$ , then near $a$ there is a biholomorphic coordinate $ϕ$ with $ϕ (a) = 0$ and $f (z) - f (a) = ϕ (z)^{m}$ (Local normal form of a nonconstant holomorphic map).

[L2]

Every nonzero complex number has exactly $m$ distinct $m$ th roots when $m \geq 1$ , while $0$ has the single $m$ th root $0$ (The $n$ -th roots of a complex number and the $n$ distinct roots of unity for every $n \geq 1$ ).

Proof

technique · direct

1.1

Take a complex domain $V_{0}$ and biholomorphic coordinate $ϕ$ from [L1]. Since $N$ is a neighbourhood of $a$ , choose an open set $O$ with $a \in O \subseteq N$ . The set $ϕ [V_{0} \cap O]$ is open and contains $0$ , so choose $ρ > 0$ with $D (0, ρ) \subseteq ϕ [V_{0} \cap O]$ and put $V : = ϕ^{- 1} [D (0, ρ)] \subseteq N$ .

L1givenchoose

2.1

If $0 < ∣ w - f (a) ∣ < ρ^{m}$ , then [L2] gives exactly $m$ distinct roots $u$ of $u^{m} = w - f (a)$ , and each satisfies $∣ u ∣ = ∣ w - f (a) ∣^{1 / m} < ρ$ .

step 1.1L2

3.1

Since $ϕ$ is bijective, its inverse transports those roots to exactly $m$ distinct points $z \in V$ satisfying $f (z) = w$ . At $w = f (a)$ , [L2] says the only coordinate root is $0$ , so the only point is $a = ϕ^{- 1} (0)$ , and the normal form records multiplicity $m$ .

step 1.1step 2.1L2given

4.1

Thus every noncentral value in the stated target disc has exactly $m$ distinct preimages in $V$ , while the central value has the one preimage of multiplicity $m$ .

step 2.1step 3.1∎

Depends on

Used by

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Guide to Cultivating Complex Analysis, Theorem 5.1.3 (standard reference, not scraped)
B. V. Shabat, Introduction to Complex Analysis, §1.2 (standard reference, not scraped)