6
2. Nonsingularity and Resolution of Singularities
as n — oc, and thus /(t
m
,p(t)) = 0. Thus
(2.6)
y
=
^ 6 ^
is a branch of the curve / = 0. This expansion is called a Puiseux series
(when ro = mult(/)), in honor of Puiseux, who introduced this theory into
algebraic geometry.
Remark 2.2. Our proof of Theorem 2.1 is not valid in positive character
istic, since we cannot conclude (2.4). Theorem 2.1 is in fact false over fields
of positive characteristic. See Exercise 2.4 at the end of this section.
Suppose that / G if[[x,y]] is irreducible, and that we have found a
solution y = p(x™) to f(x,y) = 0. We may suppose that m is the smallest
natural number for which it is possible to write such a series, y — p(x™)
divides / in R\ — if[[xm,i/]], Let u b e a primitive mth root of unity in K.
Since / is invariant under the iiTalgebra automorphism fi of R\ determined
by #m — u;xm and y —• y, it follows that y — p{u)^x^)) \ f in R\ for all j ,
and thus y — p(u)ix™) is a solution to /(#,y) = 0 for all j . These solutions
are distinct for 0 j m — 1, by our choice of ra. The series
m—1
9=
n
(?/P(^J^))
is invariant under (/, so y E if[[x,y]] and y  / in iiT[[x,y]], the ring of
invariants of R\ under the action of the group Z
m
generated by 0. Since /
is irreducible,
(ral
3=0
where u is a unit in if[[x,y]].
Remark 2.3. Some letters of Newton developing this idea are translated
(from Latin) in [18].
After we have defined nonsingularity, we will return to this algorithm
in (2.7) of Section 2.5, to see that we have actually constructed a resolution
of singularities of a plane curve singularity.
Exercise 2.4.
1. Construct a Puiseux series solution to
f{x, y) =
yA

2x3y2

4x5y
+
x6

x7
= 0
over the complex numbers.