Alphabeta MathHow 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 be a complex domain, let be nonconstant and holomorphic, and let . The local degree of at is
This is a positive natural number. Indeed, is not identically zero by Identity theorem for holomorphic functions, while it vanishes at . Its zero at is therefore isolated by Zeros of a nonzero holomorphic function are isolated, so it does not vanish on a neighbourhood of . 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 at .