CLASSIFICATION OF POLYNOMIALS FROM C

Description

CLASSIFICATION OF POLYNOMIALS FROM C 2 TO C WITH ONE CRITICAL VALUE ARNAUD BODIN 1. Introdu tion Let f : C 2 ! C be a polynomial map. The bifur ation set B is the minimal set of points of C su h that f : C 2 n f 1 (B) ! C n B is a lo ally trivial bration. We an des ribe B as follows: let B a = f(x; y) j grad f (x; y) = (0; 0) be the set of aÆne riti al values. The set B a is a subset of B but is not ne essarily equal to B. The value 2 C is regular at innity if there exists a disk D entered at and a ompa t set K of C 2 with a lo ally trivial bration f : f 1 (D) nK ! D. There is only a nite number of non-regular values at innity: the riti al values at innity olle ted in B 1 . The bifur ation set B is now: B = B a [ B 1 : For 2 C , we denote the ber f 1 ( ) by F .

• redu ed polynomial


• no riti


• polynomial map


• has genus


• indu ed


• topologi al las


• map oming


• situation has

Sujets

• Suzuki
• Polynomial

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 22
• Langue: English

Extrait

CLASSIFICA
TION
OF
POL
olynomial.
)
olo
YNOMIALS
=
FR
2
OM
f
C
=
2
d
TO

C
is
WITH
and
ONE
et
CRITICAL
"
V
1
ALUE
6
ARNA
,
UD

BODIN
Q
1.
x
Intr
o
oduction

Let
diagram
f
C
:
e
C
r
2
,
!
s
C
e
b
)
e
m
a
d
p
=
olynomial
1
map.
and
The
x
bifur
)

y
ation
<
set
to
B
and
is
e
the
there
minimal
that
set
2
of
2
p
C
oin
2
ts

of
p
C
al

0
h
(
that
+
f
L
:
(
C
g
2
=1
n
1
f

1
let
(
r
B

)
d
!
=1
C
B
n
f
B
=
is
?
a
r
lo
or

(
trivial
1
bration.
f
W
n
e
)

),
describ
en
e
t
B
olynomials
as
are
follo
al
ws:
(
let
)
B
homeomorphisms
a

=
follo

utes:
f/
(
x;
y/
)
Theorem.
j
:
grad
C
f
r
(
d
x;
denote
y
q
)
prime
=
ers,
(0
2
;
1
0)
=

y
b
s
e
,
the
0
set
n
of
let
ane
)

p
al
x
values
n
.
x
The
i
set
m
B
2
a

is
;
a
r
subset
e
of

B
olynomial
but
to
is
r
not
x

n
equal
x
to

B
=
.
?
The
x
v
B
alue
0

1
2
{
C

is
d
r
,
e

gular
i
at
p
innity
p
if
n
there
{
exists
x
a
0
disk
=1
D
iy

(
tered
<
at


giv
and
up
a
homeomorphisms:

w
set
p
K
f
of
g
C
top
2

with
ly
a
quivalent
lo
f

g
trivial
if
bration
exists
f

:

f
h
1
the
(
wing
D

)
C
n

K/
!
D
C
.
g
There
is

only/
a
:
nite
L
n
f
um
C
b
!
er
b
of
a
non-regular
e
v
e
alues
p
at
We
innit
by
y:
and
the
two

elatively
al
natur
values
numb
at
";
innity
0

f
in
;
B
g
1

.

The
x;
bifurcation
)
set
x
B
y
is
1
no
(
w:
>
B
).
=
et
B
>
a
and
[
g
B
x
1
b
:
the
F
olynomial
or
(

)
2
Q
C
i
,
(
w
i
e
m
denote
with
the
6
b
1
er
m
f
6
1

(
6

n
)
and
b
g
y
e
F
b

the
.
e
If
e
s
p
=
asso
2
d
B
g
,
g
then
e
the
(
b
)
er
Q
F
i
s
(
is
i

.
a
If

a
b
B
er
=
and
then
is

denoted
,
F
if
gen
a
.
f
The
g
aim
B
of
=
this
then
pap
f
er
y
is
g
to
e
describ
(
e
)
the
{

f
of
x

n
p
=1
olyno-
x
mial
iy
maps
(if
with
=
one
then

>
v
),
alue,
or
that

is,
"
for
"

Q
v
i
enience,
(
B
p
=
q
f
,
0
1
g
p
.
q
The
1