An algebraic branch point of order 
is a branch point at
which analytic continuation around a sufficiently
small loop returns a chosen branch
to itself after exactly 
turns, and the branch has a convergent
Puiseux series with finitely many negative powers,
 |
where 
is an integer and 
is the smallest possible positive denominator
(Knopp 1996, Part II, p. 143; Duval 1989, p. 122). Substituting 
gives a Laurent series
in 
.
The 
sheets are cyclically permuted around 
, and 
is the ramification index
of the corresponding point on the Riemann surface.
For example, 
has such a point at the origin when 
is an integer and 
and 
are relatively prime.
For an algebraic function defined by an irreducible polynomial 
, the simultaneous equations 
and 
locate candidate branch
points with finite 
. At a singular point of
the algebraic curve, coincident roots
need not be permuted by analytic continuation.
For example, 
has polynomial discriminant 
. Its two branches 
are analytic
near the origin, so 
is not a branch point. In
contrast, 
is an algebraic branch point of order 2.
