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Nombre de lectures | 24 |
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NON
REALITY
AND
NON
giv
n
CONNECTIVITY
umerical
OF
is
COMPLEX
y
POL
YNOMIALS
)
ARNA
=
UD
olynomial
BODIN
the
Abstra
Rudolph
Using
w
the
let
same
n
metho
alues
d
ersion
w
to
e
(ev
pro
0
vide
ert
negativ
for
e
true
answ
A.
ers
to
to
w
the
of
follo
len
wing
;
questions:
ectiv
Is
er
it
the
p
b
ossible
w
to
-constan
nd
[7
real
to
equations
[
for
[
germ
p
equiv
oly-
;
nomials
een
in
and
t
theory
w
question
o
[10
v
e
ariables
olo
up
not
to
al
top
2.
ological
problem.
equiv
ed
some
(Lee
top
Rudolph)?
for
Can
;
t
1
w
m
o
of
top
the
ologically
v
equiv
at
alen
er
t
,
p
v
olynomials
B
b
e
e
^
w
b
ossible
y
app
a
hapter
a
tin-
2
uous
y
family
in
of
y
top
that
ologically
f
equiv
0)
alen
t
t
(
p
This
olynomials?
has
1.
used
Intr
A'Camp
oduction
(see
Tw
in
o
divides.
p
ed
olynomials
it
f
p
;
W
g
a
2
er:
C
to
[
al
x;
it
y
p
℄
r
are
quations
top
p
olo
e
with
al
[5
ly
pro
e
a
quivalent
olynomials
,
t
and
are
w
equiv
e
More
will
p
denote
=
f
B
#
g
#
,
e
if
teger
there
exist
ane
homeomorphisms
b
um-
:
ane
C
the
2
b
!
y
C
um
2
and
innit
n
:
of
C
(with
!
a
C
).
ha
h
global
that
the
g
ujam
Æ
theorem:
it
=
p
(see
Æ
℄
f
endix
.
They
1)
are
nd
algebr
p
f
al
R
ly
x;
e
℄
quivalent
en
,
Z
and
x;
w
℄
e
h
will
the
denote
(
f
=
;
g
is
,
alen
if
to
w
germ
e
C
ha
0).
v
prop
e
y
b
2
widely
Aut
b
C
N.
2
o
and
others
℄
=
example)
id.
the
It
of
is
Lee
alw
ask
a
the
ys
whether
p
is
ossible
for
to
olynomials
nd
℄
real
e
equations
e
for
negativ
germs
answ
of
Theorem
plane
Up
top
es
up
e
to
e
top
is
ological
always
equiv
ossible
nd
In
e
e
the
for
pro
omplex
of
olynomials.
is
W
as
no
follo
deal
ws:
another
the
In
top
℄
ological
e
t
v
yp
that
e
family
of
p
a
with
germ
of
n
plane
data
all
e
ologically
(
a-
C
t.
;
precisely
0)
a
is
olynomial
determined
m
b
(
y
#
the
a
;
B
pairs
;
of
B
the
b
Puiseux
the
expansions
ulti-in
of
resp
the
ely
osed
the
hes
Milnor
and
um
b
er,
y
n
the
b
in
of
tersection
m
alues,
ultiplicities
Milnor
b
um
et
er
w
innit
een
,
these
n
b
hes.
of
Then
v
w
at
e
y
the
um
ho
er
ose
the
alues
B
B
ts
[
of
1
the
Then
Puiseux
e
expansions
v
in
a
R
v
(ev
of
en
L
in
e-Raman
Z
).
t
No
12
ARNA
UD
BODIN
P
0
Theorem
Theorem
line
([5
function
℄
e
.
top
L
et
Morse
(
(
f
[2
t
in
)
The
t
s
2
s
[0
to
;
olynomials
1]
b
enables
e
Let
a
family
Cogolludo
of
degree
(
omplex
homeomorphic
p
based
olynomials
e
in
that
two
of
variables
in
whose
ery
to
o
s
together
ar
s
e
an
p
(0).
olynomials
prop
in
y
t
ro
.
a
Supp
k
ose
J.
that
ples
the
;
multi-
v
inte
2)
ger
C
m
0
(
orien
t
in
)
b
and
the
(
de
p
gr
=
e
C
e
ends
deg
There
f
h
t
Moreo
do
ologically
not
dep
0
end
e
on
t
equiv
2
f
[0
2
;
1]
parameters,
.
is
Then
n
the
p
v
olynomials
reality
f
=
0
x
and
k
f
2
1
do
ar
with
e
that
top
825.
olo
E.
and
al
e
ly
pro
e
es
quivalent.
0
It
that
is
true
(
that
the
t
2
w
and
o
;
top
are
ologically
y
equiv
homeomorphism.
alen
d
t
note
p
the
olynomials
w
ha
v
e
a
the
s
same
C
m
ulti-in
f
tegers
is
m
t
.
.
A
line
natural
a
question
C
is:
enough
that
t
olynomial
w
tially
o
er
top
olynomial
ologically
alen
equiv
s
alen
alen
t
f
p
where
olynomials
y
b
t
e
F
the
b
top
y
t
a
the
is
tin
on
uous
f
family
W
of
ose
top
with
ologically
equiv
f
alen
a
t
C
p
1
olynomials
?
are
Theorem
to
B.
the
Ther
4.
e
f
exist
y
two
(
top
y
olo
)
;
al
e
ly
of
e
+
quivalent
Ther
p
not
olynomials
olynomial
f
e
0
;
f
degree
1
In
that
℄
Artal,
annot
Carmona
b
L.
e
giv
exam-
onne
of
d
by
C
a
C
family
of
of
6
e
ha
quivalent
e
p
equations
olynomials.
Q
That
p
me
but
ans
pairs
that
P
for
;
e
)
ach
(
2
ontinuous
C
family
)
(
not
f
b
t
an
)
tation-preserving
t
3.
2
metho
[0
used
;
this
1]
is
ther
on
e
relationship
exists
et
a
een
o-
2
and
℄
equiv
;
w
1[
set
family
that
f
f
)
2
is
of
not
olynomials
top
h
olo
(
s
al
0)
ly
a
e
arrangemen
quivalent
in
to
2
f
One
0
the
.
dep
It
on
parameter
b
2
e
.
are
that
lines
the
order
answ
er
p
is
is
p
essen
ositiv
unique.
e
v
for
ev
p
equiv
top
equiv
Tw
t
o
f
is
equiv
equiv
alen
t
t
a
p
s
olynomials
,
s
b
ma
e
b
dieren
b
from
y
.
or
equiv
parameters
alen
p
t
are
p
ologically
olynomials
alen
since
all
Aut
and
C
function
2
s
is
a
function
b
C
y
n
Jung's
1
theorem.
(0).
e
h
ho
kinds
our
of
ter-examples
problems
ha
for
v
h
e
exam