Page 65 - Geometric Modeling and Algebraic Geometry

62 P. H. Johansen et al.
Both Proposition 6 and Corollary 7 were known to Rohn, who stated these results
only in the case d =4, but said they could be generalized to arbitrary d [15, p. 60].
For the rest of the section we will assume k = . It turns out that we can find
a real normal form for the singularities other than O. The complex singularities of
type A n come in several real types, with normal forms x ± x ± x n+1 . Varying the
2
2
1 2 3
± gives two types for n =1 and n even, and three types for n ≥ 3 odd. The real
type with normal form x − x + x n+1 is called an A singularity, or of type A ,
−
−
2
2
n
1 2 3
and is what we find on real monoids:
Proposition 8. On a real monoid, all singularities other than O are of type A .
−
Proof. Assume p =(0:0:1) is a singular point on Z(F) and set g =
F(x 0 ,x 1 ,x 2 , 1) as in the proof of Proposition 6.
First note that u −1 g = x 0 (x 1 − ϕ(x 2 )) + f d (x 1 ,x 2 )u −1 is an equation for the
n
singularity. We will now prove that u −1 g is right equivalent to ±(x − x + x ), for
2
2
0 1 2
some n, by constructing right equivalent functions u −1 g =: g (0) ∼ g (1) ∼ g (2) ∼
n
g (3) ∼±(x − x + x ). Let
2
2
0 1 2
g (1) (x 0 ,x 1 ,x 2 )= g (0) (x 0 ,x 1 + ϕ(x 2 ),x 2 )
= x 0 x 1 + f d (x 1 + ϕ(x 2 ),x 2 )u −1 (x 1 + ϕ(x 2 ),x 2 )
= x 0 x 1 + ψ(x 1 ,x 2 )
where ψ(x 1 ,x 2 ) ∈ [[x 1 ,x 2 ]]. Write ψ(x 1 ,x 2 )= x 1 ψ 1 (x 1 ,x 2 )+ ψ 2 (x 2 ) and
define
g (2) (x 0 ,x 1 ,x 2 )= g (1) (x 0 − ψ 1 (x 1 ,x 2 ),x 1 ,x 2 )= x 0 x 1 + ψ 2 (x 2 ).
The power series ψ 2 (x 2 ) can be written on the form
n
ψ 2 (x 2 )= sx (a 0 + a 1 x 2 + a 2 x + ... )
2
2 2
where s = ±1 and a 0 > 0. We see that g (2) is right equivalent to g (3) = x 0 x 1 + sx n
2
since

n
a 0 + a 1 x 2 + a 2 x + ... .
2
g (2) (x 0 ,x 1 ,x 2 )= g (3) x 0 ,x 1 ,x 2
2
Finally we see that
n
g (4) (x 0 ,x 1 ,x 2 ):= g (3) (sx 0 − sx 1 ,x 0 + x 1 ,x 2 )= s(x − x + x )
2
2
0 1 2
n
proves that u −1 g is right equivalent to s(x − x + x ) which is an equation for an
2
2
0 1 2
n
A n−1 singularity with normal form x − x + x .
2
2
0 1 2
Note that for d =3, the singularity at O can be an A singularity. This happens
+
1
for example when f 2 = x + x + x .
2
2
2
0 1 2