Alphabeta Math
RemarkRemark: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-28
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Argument-principle multiplicity agrees with the earlier local degree

Remark

Fix a nonconstant holomorphic map f and a point a of local degree m=deg⁡af. The earlier local-sheet theorem A local degree-m holomorphic map has m nearby sheets says that every nearby value other than f(a) has exactly m nearby preimages. On the other hand, the argument-principle preimage count on a small circle around a counts the zeros of f−w in that same disc, with multiplicity, and therefore gives the same integer m.

So the multiplicity seen analytically by the argument principle agrees with the local degree already built from the normal form. This is an agreement remark, not a replacement of the earlier construction.

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