Trisha Shetty (Editor)

Branching theorem

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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

Let X and Y be Riemann surfaces, and let f : X Y be a non-constant holomorphic map. Fix a point a X and set b := f ( a ) Y . Then there exist k N and charts ψ 1 : U 1 V 1 on X and ψ 2 : U 2 V 2 on Y such that

  • ψ 1 ( a ) = ψ 2 ( b ) = 0 ; and
  • ψ 2 f ψ 1 1 : V 1 V 2 is z z k .
  • This theorem gives rise to several definitions:

  • We call k the multiplicity of f at a . Some authors denote this ν ( f , a ) .
  • If k > 1 , the point a is called a branch point of f .
  • If f has no branch points, it is called unbranched. See also unramified morphism.
  • References

    Branching theorem Wikipedia


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