Branching theorem
Encyclopedia
In mathematics
, the branching theorem is a theorem
about Riemann surface
s. Intuitively, it states that every non-constant holomorphic function
is locally a polynomial
.
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
This theorem gives rise to several definitions:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the branching theorem is a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
about Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s. Intuitively, it states that every non-constant holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
is locally a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
.
Statement of the theorem
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.