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Branching theorem
From Wikipedia, the free encyclopedia
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In mathematics, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial.
Statement of the theorem
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Let and be Riemann surfaces, and let be a non-constant holomorphic map. Fix a point and set . Then there exist and charts on and on such that
- ; and
- is
This theorem gives rise to several definitions:
- We call the multiplicity of at . Some authors denote this .
- If , the point is called a branch point of .
- If has no branch points, it is called unbranched. See also unramified morphism.
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References
- Ahlfors, Lars (1979), Complex analysis (3rd ed.), McGraw Hill, ISBN 0-07-000657-1.
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