Alphabeta Math
DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Local degree of a nonconstant holomorphic map

Definition

Let Ω be a complex domain, let f:Ω→C be nonconstant and holomorphic, and let a∈Ω. The local degree of f at a is deg⁡af:=ord⁡a(f−f(a)).

This is a positive natural number. Indeed, f−f(a) is not identically zero by Identity theorem for holomorphic functions, while it vanishes at a. Its zero at a is therefore isolated by Zeros of a nonzero holomorphic function are isolated, so it does not vanish on a neighbourhood of a. The equivalence in The order of a zero is the exponent in its local holomorphic factorization then rules out infinite order, and the convention of The order of a zero of a holomorphic function makes the remaining finite order positive. The local degree is also called the multiplicity of f at a.

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Sources