Page 134 - Geometric Modeling and Algebraic Geometry

7 Curve Parametrization Exploiting the Newton Polygon 135
s v 6 = v 7

v 8
Π(f) v 4 = v 5

v 3

r
v 1
v 2
Fig. 7.4. Newton polygon of example

7.7.1 Analysis of the singularities

The first chart U 1 contains two singular points, Q 1 with coordinates (u 1 ,v 1 )=
(−1, 0) and Q 2 with coordinates (u 1 ,v 1 )=(0, −1). The point Q 1 is also con-
tained in U 2 and Q 2 is contained in U 8 . These are already all singular points, so all
subsequent computations can be done in the first chart. For future reference we also
give the partial derivative by v 1 :

= −54v u − 12v u 1 +26v u +8v 1 u − 40v u 1 − 8v 1 u 2
d f 1
2 2
3
2 3
3
2
1 1 1 1 1 1 1 1
d v 1
2
+8v − 8v 1 u 1 +8v 1
1
According to the singularities we compute Puiseux expansions in u 1 +1 for Q 1
and in u 1 for Q 2 :
σ 1 (u 1 +1) = − (u 1 +1)+ γ 1 (u 1 +1) +( 608 1 − 51 )(u 1 +1) ...
γ
2
3
435
1
4 2432
σ 2 (u 1 )= −1 − u 1 + γ 2 u +( 21 γ 2 + 195 )u ...
2
3
5
1 1
2 4 16
Here γ 1 and γ 2 are elements of Q s.t. 1024γ + 516γ 1 +63 = 0 and 16γ +24γ 2 −
2
2
1 2
45 = 0. From these expansions we get the following two monomorphisms from the
function field into a field of Laurent series:
ϕ 1 :Q(Q[u 1 ,v 1 ]/f 1 ) → Q((t)) :

u 1 +1 → t
v 1 →− t + γ 1 t +( 608 1 − 51 )t ...
γ
435
3
1
2
4 2432
ϕ 2 :Q(Q[u 1 ,v 1 ]/f 1 ) → Q((t)) :

u 1 → t
v 1 +1 →− t + γ 2 t +( 21 γ 2 + 195 )t ...
5
3
2
2 4 16
These homomorphisms induce valuations ν i := ord t ◦ϕ i . Using these valuations we
are able to speak about a resolution π : C → C without constructing it explicitly.