Branching theorem
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 formula_1 and formula_2 be Riemann surfaces, and let formula_3 be a non-constant holomorphic map. Fix a point formula_4 and set formula_5. Then there exist formula_6 and charts formula_7 on formula_1 and formula_9 on formula_2 such that This theorem gives rise to several definitions: