Puiseux's theorem states that the local branches of a complex algebraic curve admit convergent Puiseux
expansions. If 
is a nonzero polynomial with polynomial
degree 
in 
,
each local solution near 
can be written
 |
(1)
|
where 
is a positive integer and 
is an integer. The series
converges for sufficiently small nonzero 
on a chosen branch of 
(McMullen 2026, pp. 44-45).
Thus, after 
,
a branch with finite limit 
has a power
series parametrization
 |  |  |
(2)
|
 |  |  |
(3)
|
Branches tending to infinity have a Laurent series with finitely many negative powers instead.
