6
2. Non-singularity 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 m-th root of unity in K.
Since / is invariant under the iiT-algebra 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,
(ra-l
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 non-singularity, 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.
