M. OKA
KODAI MATH. J.
33 (2010), 1–62
NON-DEGENERATE MIXED FUNCTIONS
Mutsuo Oka
Abstract
Mixed functions are analytic functions in variables z
1
; ...; z
n
and their conjugates
z
1
; ...; z
n
. We introduce the notion of Newton non-degeneracy for mixed functions and
develop a basic tool for the study of mixed hypersurface singularities. We show the
existence of a canonical resolution of the singularity, and the existence of the Milnor
fibration under the strong non-degeneracy condition.
1. Introduction
Let f ðzÞ be a holomorphic function of n-variables z
1
; ...; z
n
such that
f ð0Þ¼0. As is well-known, J. Milnor proved that there exists a positive
number e
0
such that the argument mapping f =jf j : S
2n1
e
nK
e
! S
1
is a locally
trivial fibration for any positive e with e a e
0
where K
e
¼ f
1
ð0ÞV S
2n1
e
([12]).
In the same book, he proposed to study the links coming from a pair of real-
valued real analytic functions gðx; yÞ, hðx; yÞ where z ¼ x þ yi. Namely putting
f ðx; yÞ :¼ gðx; yÞþihðx; yÞ : R
2n
! C, he proposed to study the condit ion for
f =jf j : S
2n1
e
nK
e
! S
1
to be a fibration. This is an interesting problem. In
fact, if one can find such a pair of analytic functions g, h , it may give an
interesting link variety K
e
whose complement S
2n1
e
nK
e
is fibered over S
1
where
K
e
cannot come from any complex analytic links. The di‰culty is that for an
arbitrary choice of g, h, it is usually not a fibration. A breakthrough is given by
the work of Ruas, Seade and Verjovsky [20]. After this work, many examples of
pairs fg; hg which give real Milnor fibrations hav e been investigated. However
in most papers, certain restricted types of functions are mainly considered ([5, 6,
22, 19, 11, 18, 3]).
The purpose of this paper is to propose a wide class of pairs fg; hg such that
the corresponding map ping f ¼ g þ ih defines a Milnor fibration. We consider
a complex valued analytic function f expanded in a convergent power series of
variables z ¼ðz
1
; ...; z
n
Þ and z ¼ðz
1
; ...; z
n
Þ
f ðz;
zÞ¼
X
n; m
c
n; m
z
n
z
m
1
2000 Mathematics Subject Classification. 14J70, 14J17, 32S25.
Key words and phrases. Mixed function, polar weighted polynomial, Milnor fibration.
Received November 26, 2008; revised May 21, 2009.
where z
n
¼ z
n
1
1
z
n
n
n
for n ¼ðn
1
; ...; n
n
Þ (respectively z
m
¼ z
m
1
1
z
m
n
n
for m ¼
ðm
1
; ...; m
n
Þ) as usual. Here z
j
is the complex conjugate of z
j
. We call
f ðz; zÞ a mixed analytic function (or a mixed polynomial, if f ðz; zÞ is a polynomial)
of z
1
; ...; z
n
. We are interested in the topology of the hypersurface V ¼
fz A C
n
j f ðz; zÞ¼0g, which we call a mixed hypersurface. Here we use the
terminology hypersurface in order to point out the similarity with complex
analytic hypersurfaces. We will see later that codim
R
V ¼ 2ifV is non-
degenerate (Theorem 19). We denote the set of mixed functions of variables
z,
z by Cfz; zg. This approach is equivalent to the original one. In fact, writing
z ¼ x þ iy with z
j
¼ x
j
þ iy
j
j ¼ 1; ...; n, and using real variables x ¼ðx
1
; ...; x
n
Þ
and y ¼ðy
1
; ...; y
n
Þ, and dividing f ðz; zÞ in the real and the imaginary parts so
that f ðx; yÞ¼gðx; yÞþihðx; yÞ where g :¼<f , h :¼=f , we can see that V is
defined by two real-valued analytic functions gðx; yÞ, hðx; yÞ of 2n-variables
x
1
; y
1
; ...; x
n
; y
n
. Conversely, for a given real analytic variety W ¼fgðx; yÞ¼
hðx; yÞ¼0g which is defined by two real-valued analytic functions g, h, we can
consider W as a mixed hypersurface by introducing a mixed function f ðz;
zÞ¼0
where
f ðz;
zÞ :¼ g
z þ z
2
;
z z
2i
þ ih
z þ z
2
;
z z
2i
:
The advantage of our view point is that we can use rich techniques of complex
hypersurface singularities. For complex hypersurfaces defined by holomorphic
functions, the notion of the non-degeneracy in the sense of the Newton boundary
plays an important role for the resolution of singularities and the determination
of the Milnor fibration ([10, 23, 14, 15, 16]). We will introduce the notion
of non-degeneracy for mixed functions or mixed polynomials and prove basic
properties in §2 and §3.
In §4, we will give a canonical resolution of mixed hypersurface singularities.
First we take an admissible toric modification
^
pp : X ! C
n
. This does not resolve
the singularities but it turns out that we only need a real modification or a polar
modification after the toric modification to complete the resolution (Theorem 24).
In §5, we consider the Milnor fibration of a given mixed function f ðz;
zÞ.It
turns out that the non-degeneracy is not enough for the existence of the Milnor
fibration of f . We need the strong non-degeneracy of f ðz;
zÞ which guarantees
the existence of the Milnor fibrat ion (Theorem 33, Theorem 29). We show that
the Milnor fibrations of the first type and of the second type,
f =jf j : S
e
nK
e
! S
1
and f : qEðr; dÞ
! S
1
d
;
are equivalent (Theorem 36). We also show that for a polar weighted homo-
geneous polynomial, the global fibration is equivalent to the above two fibrations
(Theorem 33).
In §6, we will see that the mixed singularities are much more complicated
than the complex singularities and that the topological equivalence class is not a
combinatorial invariant even in the easiest case of plane curves.
2 mutsuo oka
In §7, we discuss Milnor fibrations for non-isolated mixed singularities under
the super strong non-degeneracy condition (Theorem 52).
In §8, we give an A’Campo type formula for the zeta function of the Milnor
fibration in the case of mixed curves (Theorem 60).
This paper is a continuation of the previous one [17] and we use the same
notations. This paper consists of the following sections. We hope this paper
provides a systematical method to study mixed singularities.
Contents
Section 1. Introduction
Section 2. Newton boundary and non-degeneracy of mixed functions
Section 3. Isolatedness of the singularities
Section 4. Resolution of the singularities
Section 5. Milnor fibration
Section 6. Curves defined by mixed functions
Section 7. Milnor fibration for mixed polynomials with non-isolated singularities
Section 8. Resolution of a polar type and the zeta function
Below are notations we use frequently in this paper:
S
2n1
r
, S
r
¼fz ¼ðz
1
; ...; z
n
Þ A C
n
jkzk¼rg, (sphere of the radius r)
kzk¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jz
1
j
2
þþjz
n
j
2
q
B
2n
r
, B
r
¼fz ¼ðz
1
; ...; z
n
Þ A C
n
jkzka rg (ball of the radius r)
C
I
¼fz ¼ðz
1
; ...; z
n
Þjz
j
¼ 0; j B I g, B
I
r
¼fz A C
I
jkzka rg
C
I
¼fz ¼ðz
1
; ...; z
n
Þjz
j
¼ 0 , j B I g
C
n
¼ C
I
, B
n
¼ B
I
with I ¼f1; ...; ng
R
þn
¼fðx
1
; ...; x
n
Þ A R
n
jx
j
b 0; j ¼ 1; ...; ng
ðz; wÞ¼z
1
w
1
þþz
n
w
n
: hermitian inner product
<ðz; wÞ¼<ðz
1
w
1
þþz
n
w
n
Þ : real Euclidean inner product
DðdÞ :¼fh A C jjhja dg, DðdÞ
:¼fh A C j0 < jhja dg
S
1
d
:¼fh A C jjhj¼dg.
2. Newton boundary and non-degeneracy of mixed functions
2.1. Polar weighted homogeneous polynomials
2.1.1. Radial degree and polar degree. Let M ¼ z
n
z
m
be a mixed monomial
where n ¼ðn
1
; ...; n
n
Þ, m ¼ðm
1
; ...; m
n
Þ and let P ¼
t
ðp
1
; ...; p
n
Þ be a weight
vector. We define the radial degree of M, rdeg
P
M and the polar degree of M,
pdeg
P
M with respect to P by
rdeg
P
M ¼
X
n
j¼1
p
j
ðn
j
þ m
j
Þ; pdeg
P
M ¼
X
n
j¼1
p
j
ðn
j
m
j
Þ:
3non-degenerate mixed functions
2.1.2. Weighted homogeneous polynomials. Recall that a polynomial hðzÞ
is called a weighted homogeneous polynomial with weights P ¼
t
ðp
1
; ...; p
n
Þ if
p
1
; ...; p
n
are integers and there exists a positive integer d so that
f ðt
p
1
z
1
; ...; t
p
n
z
n
Þ¼t
d
f ðzÞ; t A C:
The integer d is called the degree of f with respect to the weight vector P.
A mixed polynomial f ð z ;
zÞ¼
P
l
i¼1
c
i
z
n
i
z
m
i
is called a radial ly weighted
homogeneous polynomial if there exist integers q
1
; ...; q
n
b 0 and d
r
> 0 such
that it satisfies the equality:
f ðt
q
1
z
1
; ...; t
q
n
z
n
; t
q
1
z
1
; ...; t
q
n
z
n
Þ¼t
d
r
f ðz; zÞ; t A R
:
Putting Q ¼
t
ðq
1
; ...; q
n
Þ, this is equivalent to rdeg
Q
z
n
i
z
m
i
¼ d
r
for i ¼ 1; ...; l
with c
i
0 0. Write f ¼ g þ ih so that g, h are polynomials with real coe‰cients
of 2n-variables ðx
1
; y
1
; ...; x
n
; y
n
Þ.Iff is a radially weighted homogeneous
polynomial of type ðq
1
; ...; q
n
; d
r
Þ, gðx; yÞ and hðx; yÞ are weighted homogeneous
polynomials of type ðq
1
; q
1
; ...; q
n
; q
n
; d
r
Þ (i.e., deg x
j
¼ deg y
j
¼ q
j
).
A polynomial f ðz; zÞ is called a polar weighted homogeneous polynomial if
there exists a weight vector ðp
1
; ...; p
n
Þ and a non-zero integer d
p
such that
f ðl
p
1
z
1
; ...; l
p
n
z
n
; l
p
1
z
1
; ...; l
p
n
z
n
Þ¼l
d
p
f ðz; zÞ; l A C
; jlj¼1
where gcdðp
1
; ...; p
n
Þ¼1. Usually we assume that d
p
> 0. This is equivalen t
to
pdeg
P
z
n
i
z
m
i
¼ d
p
; i ¼ 1; ...; l:
Here the weight p
i
can be zero or a negative integer. The weight vector
ðp
1
; ...; p
n
Þ is called the polar weights and d
p
is called the polar degree
respectively. This notion was first introduced by Ruas-S eade-Verjovsky [20]
and Cisneros-Moli na [4]. In [17], we have assumed that a polar weighted
homogeneous polynomial is also a radially weighted homogeneous polynomial.
Although it is not necessary to be assumed, we will only consider such
polynomials in this paper.
Recall that th e radial weights and polar weights define R
-action and S
1
-
action on C
n
respectively by
t z ¼ðt
q
1
z
1
; ...; t
q
n
z
n
Þ; t z ¼ðt
q
1
z
1
; ...; t
q
n
z
n
Þ; t A R
l z ¼ðl
p
1
z
1
; ...; l
p
n
z
n
Þ; l z ¼ l z; l A S
1
H C
In other words, this is an R
S
1
action on C
n
.
Lemma 1. Let f ðz;
zÞ be a radially weighted homogeneous polynomial, V ¼
fz A C
n
j f ðz; zÞ¼0g and V
¼ V V C
n
. Assume that VnfOg (respectively V
) is
smooth and codim
R
V ¼ 2. If the radial weight vector is strictly positive, namely
q
j
> 0 for any j ¼ 1; ...; n, the sphere S
r
intersects transversely with VnfOg (resp.
with V
) for any r > 0.
4 mutsuo oka
We are mainly considering the case that V nfOg has no mixed singularity in
the sense of §3.1.
Proof. This is essentially the same with Proposition 4 in [17]. In Prop-
osition 4, we have assumed that f ðz;
zÞ is polar weighted hom ogeneous but we
did not use this assumption in the proof. The radial action is enough as we will
see below. Assume that three vectors dg, dh, df are linearly dependent at
z
0
¼ðx
0
; y
0
Þ A V
, where f ðz; zÞ¼gðx ; yÞþihðx; yÞ and f ðx; yÞ¼
P
n
j¼1
ðx
2
j
þ y
2
j
Þ.
As V nfOg (resp. V
) is non-singular, we can find real numbers a, b so
that dfðx
0
; y
0
Þ¼a dgðx
0
; y
0
Þþb dhðx
0
; y
0
Þ. Here df, dg, dh are the respective
gradient vectors of the functions f, g, h. For example, dgðx; yÞ¼
qg
qx
1
;
qg
qy
1
...;
qg
qx
n
;
qg
qy
n
. Let l ðtÞ¼ðt x
0
; t y
0
Þ, t A R
þ
be the orbit of z
0
by
the radial action. Let v be the tangent vector of the orbit. Then we have:
lðtÞ¼ðt
q
1
x
01
; t
q
1
y
01
; ...; t
q
n
x
0n
; t
q
n
y
0n
Þ
d
dt
fðlðtÞÞj
t¼1
¼<ðdfðx
0
; y
0
Þ; vÞ¼2
X
n
i¼1
q
i
ðx
2
0i
þ y
2
0i
Þ > 0:
On the other hand, we also have the equality:
d
dt
fðlðtÞÞj
t¼1
¼ a<ðdgðx
0
; y
0
Þ; vÞþb<ðdhðx
0
; y
0
Þ; vÞ
¼ a
dgðlðtÞÞ
dt
t¼1
þ b
dhðlðtÞÞ
dt
t¼1
¼ 0:
This is an obvious contradiction to the above inequality. r
2.2. Newton boundary of a mixed function. Suppose that we are given a
mixed analytic function f ðz;
zÞ¼
P
n; m
c
n; m
z
n
z
m
. We always assume that c
0; 0
¼ 0
so that O A f
1
ð0Þ. We call the variety V ¼ f
1
ð0Þ the mixed hypersurface.
The radial Newton polygon G
þ
ðf ; z; zÞ (at the origin) of a mixed function f ðz; zÞ
is defined by the convex hull of
6
c
n; m
00
ðn þ mÞþR
þn
:
Hereafter we call G
þ
ðf ; z; zÞ simply the Newton polygon of f ðz; zÞ. The Newton
boundary Gðf ; z; zÞ is defined by the union of compact faces of G
þ
ðf Þ. Observe
that Gðf Þ is nothing but the ordinary Newton boundary if f is a complex
analytic function. For a given positive integer vector P ¼ðp
1
; ...; p
n
Þ, we asso-
ciate a linear function l
P
on Gðf Þ defined by l
P
ðnÞ¼
P
n
j¼1
p
j
n
j
for n A Gðf Þ and
let DðP; f Þ¼DðPÞ be the face where l
P
takes its minimal value. In other words,
P gives radial weights for variables z
1
; ...; z
n
by rdeg
P
z
j
¼ rdeg
P
z
j
¼ p
j
and
rdeg
P
z
n
z
m
¼
P
n
j¼1
p
j
ðn
j
þ m
j
Þ. To distinguish the points on the Newton bound-
ary and weight vectors, we denote by N the set of integer weight vectors and
5non-degenerate mixed functions