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Branching theorem

From Wikipedia, the free encyclopedia

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.

References

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  • Ahlfors, Lars (1953), Complex analysis (3rd ed.), McGraw Hill (published 1979), ISBN 0-07-000657-1.