MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
—————————–
Pham Duc Thoan
ON THE RELATION DEFECT AND THE ALGEBRAIC
DEPENDENCES OF MEROMORPHIC MAPPINGS
Specialized: Geometry and Topology
Code: 62.46.10.01
SUMMARY DOCTOR OF PHILOSOPHY IN MATHEMATICS
Hanoi, 01-2011
2
Thesis was completed at: Hanoi national University of Education
Science instructor: Prof. Dr. Do Duc Thai
Rewier 1: Prof. Dr. Nguyen Van Mau, Hanoi University of Science-
Vietnam National University
Rewier 2: Prof. Dr. Le Hung Son, Hanoi University of Technology.
Rewier 3: Prof. Dr. Nguyen Van Khue, Hanoi National University
of Education.
Thesis will be approved by School committee at
hour date month year
Thesis can be found at: -Viet Nam national library
-Library of Hanoi National University of
Education
1
Introduction
1. Reasons for selecting topics
In the late 20’s last century, Nevanlinna foundated the value
distribution theory of the meromorphic function of a variable. Over
the next decade many mathematicians in the world such as H.
Cartan, W. Stoll, PA Griffiths, L. Carlson, P. Vojta, J. Noguchi
interest in research and develop on Nevanlinna theory for more

unique problems of meromorphic mapping, the authors often proved
directly and through the second fundamental theorem. Here, we ap-
proach the problem with the theory of ”algebraic dependence” of the
meromorphic mappings of several complex variables that W. Stoll
proposed.
4. The results of the thesis
Among the theorems that Nevanlinna proved, the theorem about
the relationship of defect to keep a special role. Namely, the theorem
is stated the following:
Theorem A. If f be a nonconstant meromorphic function on
C then

a∈P
1
(C)
δ(a, f)  2.
Theorem A was proved for the class meromorphic functions of
complex variables. For example, theorem Cartan-Nochka said that
if f : C → P
n
(C) be a linearly nondependence holomorphic function
and {H
j
}
q−1
j=0
be hyperplanes in N-subgeneral position in P
n
(C)
then

, f) > 0 (1 ≤ j ≤ q) and
q

j=1
δ
[n]
(H
j
, f) = 2N − n + 1.
Then one of the following two statements holds:
(I) There are at least

2N − n + 1
n + 1

+ 1 of the hyperplanes H
j
at which f has deficiency value 1, i.e δ(H
j
, f) = 1,
(II) {H
j
}
q
j=1
has a Borel distribution.
Continue the above research , in the first two chapters of the thesis
we study the class meromorphic mappings which has a maximal de-
fect sum. Namely, in Chapter 1 we showed the necessary condition
for the class meromorphic function has a maximal defect function,

}
+∞
i=1
⊂ Z
+
such that

a∈C
δ(a, h
n
i
) =
2, ∀i ≥ 1.
4
Theorem 1.3.2. Let f : C
m
→ P
1
(C) be a meromorphic func-
tion of finite order satisfying
λ := ρ
f
/∈ Z and

a∈C
δ(a, f) = 2.
Denote by A the set of all nonconstant meromorphic functions
h : C
m
→ P

m
−→ P
n
(C) be a nonconstant
meromorphic mapping, and let {a
i
}
q−1
i=0
be ”small” (with respect
to f) meromorphic mappings of C
m
into P
n
(C) in N− subgeneral
position such that f is linearly nondegenerate over R({a
i
}
q−1
i=0
),
where 1 ≤ n < N and 2N − n + 1 < q < +∞. Suppose further
that f has nonzero deficiency value at a
i
for each 0 ≤ i ≤ q − 1
and

q−1
j=0
δ (a

, · · · , a
5
then f ≡ g. In the context of the review
theorem 5 point of Nevanlinna for meromorphic function of several
complex variables into complex projective space, in 1975 H. Fujimoto
proved the following important theorem.
Theorem C. Let H
i
(1 ≤ i ≤ 3N + 2) be 3N + 2 hyperplanes
in general position in P
N
(C), f and g be two non-constant
meromorphic mappings from C
n
to P
N
(C) such that f(C
n
) 
H
i
, g(C
n
)  H
i
. Assume that v
(f,H
i
)
= v

g
(r)}) (0 ≤ j ≤
7) and f(z) = a
j
(z) ⇔ g(z) = a
j
(z) then f ≡ g.
Continue the above research , in the chapters 3 of the thesis, we
have showed some unicity theorem for meromorphic mappings of
several variables to complex projective space through the study of
6
the algebraic dependence of their mapping. The results that we
achieved a significant expansion for the theorems of M. Ru. Namely,
we proved the following theorem:
Theorem 3.2.4. Let f
1
, · · · , f
λ
: C
m
→ P
n
(C) be nonconstant
meromorphic mappings. Let g
i
: C
m
→ P
n
(C) (1 ≤ i ≤ q) be

, g
i
)(z) = (f
1
, g
j
)(z) = 0} ≤ m − 2 for each
1 ≤ i < j ≤ q,
iii) There exists an integer number l, 2 ≤ l ≤ λ, such that for
any increasing sequence 1 ≤ j
1
< · · · < j
l
≤ λ, f
j
1
(z) ∧ · · · ∧
f
j
l
(z) = 0 for every point z ∈ ∪
q
i=1
(f
1
, g
i
)
−1
{0}.

λ − l + 1
,
then f
1
, · · · , f
λ
are algebraically dependent over C.
iii) If f
i
, 1 ≤ i ≤ λ are linearly nondegenerate over C, g
i
(1 ≤
i ≤ q) are constant mappings and (q − n − 1)((λ − 1)(κ − 1) +
q(λ − l + 1)) ≤ qnλ, then f
1
, · · · , f
λ
are algebraically dependent
7
over C
m
.
Theorem 3.3.1 Let f
1
, f
2
: C
m
→ P
n

2
,g
j
)
} for each z ∈ C
m
and
1 ≤ j ≤ q
ii) dim{(f
1
, g
i
)
−1
{0} ∩ (f
1
, g
j
)
−1
(z)} ≤ n − 2 for each 1 ≤ i <
j ≤ q
iii) f
1
(z) = f
2
(z) for each z ∈ ∪
q
j=1
(f

Theorem of classical Nevanlinna deficiency relation was pointed
out that if f : C
m
−→ P
1
(C) be a meromorphic function then

a∈P
1
(C)
δ(a, f)  2.
A question naturally arises: What can be said about the class of
meromorphic functions f is

a∈P
1
(C)
δ(a, f) = 2? This problem has
research interests of many mathematicians, such as N. Toda, J. Lu
and Y. Yasheng
As stated in the introduction, the purpose of this chapter is to
continue to study the problem on the meromorphic function of
the fixed target. Namely, we show the necessary conditions of
the maximality of defect sum. Later, we show that the class of
meromorphic functions with maximal defect sum is very thin in
the sense that deformations of meromorphic functions with maximal
defect sum by small meromorphic functions are not meromorphic
functions with maximal defect sum. Further more, we can measure
the deviation of defect sum of meromorphic functions before and
after by constant.

meromorphic function of finite order. Then, T
D
f
(r)  2T
f
(r) +
O(log rT
f
(r)) and hence, ρ
D
f
 ρ
f
.
Lemma 1.2.5.
Let f : C
m
→ P
1
(C) be a nonconstant meromorphic function.
Define g = f
n
, where n ∈ Z
+
and h = f + a with a ∈ C. Then
ρ
g
= ρ
f
, ρ

(ii) ρ
D
1
f
= ρ
1
f
.
Lemma 1.2.10. Let f : C
m
→ P
1
(C) be a nonconstant
10
meromorphic function such that δ(∞, f) = 0. Then

a∈C
δ(a, f) =

a∈C
δ(a, f)  2δ(0, D
f
).
By the same argument in Lemma 1.2.3, we have the following:
Lemma 1.2.11. Let f, g : C
m
→ P
1
(C) be nonconstant
meromorphic functions of finite order satisfying ρ


. Put g = f + h. Then δ(∞, g) = 0.
Lemma 1.2.14. Let f : C
m
→ P
1
(C) be a nonconstant
meromorphic function. Define g = f
n
, where n ∈ Z
+
. Then
T
D
g
(r) 
n+1
n
T
g
(r) + O(log rT
f
(r)).
Lemma 1.2.15. Let f : C
m
→ P
1
(C) be a nonconstant
meromorphic function of finite order. Then, there exists a
meromorphic function f

The purpose of this chapter is to prove the following two results:
Theorem 1.3.1. Let f : C → P
1
(C) be a meromorphic function
of finite order. For each n ≥ 1, define g
n
(z) = f(z
n
), ∀z ∈ C and
11
h
n
(z) = f
n
(z), ∀z ∈ C. Then we have necessarily λ := ρ
f
∈ Z
+
and λ equals the lower order of f
1) If there exists n
0
≥ 2 such that

a∈C
δ(a, g
n
0
) = 2.
2) If there exists a sequence {n
i

(C) such that
T
h
(r) = o

T
f
(r)

, T
D
h
(r) = o

T
D
f
(r)

.
Then, for each h ∈ A, we have

a∈C
δ(a, f +h)  2−2k(λ) < 2,
where k(λ) is a positive constant which depends only on λ.
12
Chapter 2
Meromorphic mapping with maximal with
defect sum for moving targets
This chapter is based on the article [3]. The chapter for the study of

}
q−1
i=0
).
Then

q−1
j=0
δ (a
j
, f) ≤ 2N − n + 1.
Thus there is a question naturally arises:What we can say about the
functions f which is the number of defects for maximal? In other
words, we can extend the results of N. Toda for the meromorphic
mappings of several variables that have maximal defect sum with
moving targets or not? The main purpose of this chapter is to answer
that question.
2.2 The initial results
13
First we recall two of the Nochka weight Lemma for moving targets.
How to prove they are repeated all the claims corresponding to the
fixed hyperplane.
Lemma 2.2.1. Let {a
i
}
i∈Q
be q moving targets in P
n
(C) in N-
subgeneral position, and assume that q > 2N −n+1. Then there

ω
j
≤ rank{a
i
}
i∈R
.
The above ω
j
are called Nochka weights, and
˜
ω the Nochka constant.
We will denote θ = ˜ω
−1
for later convenience.
Lemma 2.2.2. Let q > 2N − n + 1, and let {a
i
}
i∈Q
be q
moving targets in P
n
C in N-subgeneral position. Let {ω
j
}
j∈Q
be its Nochka weights. Let E
j
≥ 1, j ∈ Q be arbitrarily given
numbers. Then for every subset R ⊂ Q with 0 < |R| ≤ N + 1,

(C) with a reduced representation f = (f
0
: · · · : f
n
). Consider
N > n and q be any integer satisfying 2N −n+1 < q < +∞. Put
14
Q = {0, 1 . . . , q − 1}. Let X = {a
j
: j ∈ Q} be the set of ”small”
(with respect to f) meromorphic mappings from C
m
into P
n
(C)
in N-subgeneral position. Assume that f is nondegenerate over
R({a
i
}
q−1
i=0
) and any function ω : Q → (0, 1] satisfy the condition
(iv) in Lemma 2.2.1, we have

q−1
j=0
ω(j) · δ (a
j
, f) ≤ n + 1.
2.3 The meromorphic mapping with maximal defi-

and

q−1
j=0
δ (a
j
, f) = 2N − n + 1.
Then one of the following two statements holds.
(I) There are at least [
2N − n + 1
n + 1
] + 1 of the moving targets a
j
at
which f has deficiency value 1, i.e δ(a
j
, f) = 1 ,
(II) n is odd and the family {a
j
}
q−1
j=0
has a Borel distribution.
15
Chapter 3
The algebraic dependences of meromorphic
mappings and applications
This chapter for the study of algebraic dependence of the mero-
morphic mapping from C
m

1
, . . . , g
q
are located in general position if
det(g
j
k
l
) ≡ 0 for any 1 ≤ j
0
< j
1
< < j
N
≤ q.
Let M
n
be the field of all meromorphic functions on C
m
. Denote
by R


g
j

q
j=1

⊂ M

0
, . . . , f
N
are linearly independent
16
over R


g
j

q
j=1

.
Let f
t
: C
m
→ P
n
(C) (1  t  λ) be meromorphic mappings
with reduced representations f
t
:= (f
t0
: · · · : f
tn
). Let g
j

)
−1
{0} = · · · = (f
λ
, g
j
)
−1
{0}. Put A
j
= (f
1
, g
j
)
−1
{0} for
each 1  j  q. Assume that every analytic set A
j
has the irriducible
decomposition as follows A
j
= ∪
t
j
i=1
A
ji
(1  t
j

)}, we define ρ(z) = {j|z ∈ A
j
}.
Then ρ(z) ≤ n. For any positive number r > 0, define ρ(r) =
sup{ρ(z)||z| ≤ r}, where the supremum is taken over all z ∈
C
m
\{∪
β∈T [n+1,q]
{z|g
β(1)
(z)∧· · ·∧g
β(n+1)
(z) = 0}∪A∪∪
λ
i=1
I(f
i
)}.
Then ρ(r) is a decreasing function. Let
d := lim
r→+∞
ρ(r).
Then d ≤ n. If for each i = j, dim{A
i
∩ A
j
} ≤ m − 2, then d = 1.
We state the following M. Ru’ theorems
Theorem A. Let f

−1
{0} = · · · =
(f
λ
, g
j
)
−1
{0} for each 1 ≤ j ≤ q. Denote A = ∪
q
j=1
A
j
. Let
17
l, 2 ≤ l ≤ λ, be an integer such that for any increasing sequence
1 ≤ j
1
< · · · < j
l
≤ λ, f
j
1
(z) ∧ · · · ∧ f
j
l
(z) = 0 for every point
z ∈ A. If q >
dn
2

in the fisrt part of chapter 3 we extend to the result of M. Ru
decreased significantly by moving targets.
The unicity theorems with truncated multiplicities of meromorphic
mappings of C
m
into the complex projective space P
n
(C) sharing a
finite set of fixed (or moving) hyperplanes in P
n
(C) have received
much attention in the last few decades. We now state a recent result
of Do Duc Thai and Si Duc Quang which is the one of the best results
available at present.
Theorem C. Let f : C
m
→ P
n
(C) be a meromorphic mapping.
Let κ be a positive integer. Let {a
j
}
q
j=1
be ”small” (with respect
to f) meromorphic mappings of C
m
into P
n
(C) in general po-

(i) min (v
(f,a
j
)
, κ) = min (v
(g,a
j
)
, κ) (1 ≤ j ≤ q),
18
(ii) f(z) = g(z) on

q
j=1
{z ∈ C
n
: (f, a
j
)(z) = 0}.
Then we have
(i) If q = 2n
2
+ 4n and n ≥ 2, then  F(f, {a
j
}
q
j=1
, 1) = 1,
where denote by  S the cardinality of the set S.
(ii) If q =

be ”small”
(with respect to f) meromorphic mappings of C
m
into P
n
(C) in
general position such that (f, a
j
) ≡ 0 (1 ≤ j ≤ q) and dim{z ∈
C
m
: (f, a
i
)(z) = (f, a
j
)(z) = 0} ≤ m − 2 (1 ≤ i < j ≤ q).
We denote G(f, {a
j
}
q
j=1
, κ) be the set of all meromorphic maps
g : C
m
→ P
n
(C) satisfying the conditions:
(i) min (v
(f,a
j

In the last of chapter 3, we give the answer for the above problem.
19
Our approach is based on the theorem ”algebraic dependences” of
meromorphic mappings.
3.1 The some auxiliary results
The following, Do Duc Thai and Si Duc Quang have proved the
Second Main Theorem for moving targets.
Theorem 3.1.2. Let f : C
m
→ P
n
(C) be a meromorphic
mapping. Let {a
1
, , a
q
} (q = 2) be a set of q meromorphic
mappings of C
m
into P
n
(C) located in general position such
that f is linearly nondegenerate over R


g
j

q
j=1

.
Theorem 3.1.3. Let f : C
m
→ P
n
(C) be a meromorphic
mapping. Let A = {a
0
, , a
q−1
} (q ≥ 2n + 1) be a set of
q meromorphic mappings of C
m
into P
n
(C) located in general
position such that (f, a
i
) ≡ 0 for each 1 ≤ i ≤ q. Then




q
2n+1
·
T (r, f) ≤

q
i=1

f
1
∧···∧f
λ
)+m(r, f
1
∧· · ·∧f
λ
) ≤

1≤i≤λ
T (r, f
i
)+O(1).
Theorem 3.1.5 [The Second Main Theorem for general
position]. Let M be a connected complex manifold of dimension
m. Let A be a pure (m−1)-dimensional analytic subset of M. Let
V be a complex vector space of dimension n + 1 > 1. Let p and
k be integers with 1 ≤ p ≤ k ≤ n + 1. Let f
j
: M → P (V ), 1 ≤
j ≤ k, be meromorphic mappings. Assume that f
1
, , f
k
are
in general position. Also assume that f
1
, , f
k

: C
m
→ P
n
(C) (1 ≤ i ≤ q)
be moving targets located in general position and T (r, g
i
) =
o(max
1≤j≤λ
T (r, f
j
)) (1 ≤ i ≤ q). Assume that (f
i
, g
j
) ≡ 0
for 1 ≤ i ≤ λ, 1 ≤ j ≤ q and A
j
:= (f
1
, g
j
)
−1
{0} = · · · =
(f
λ
, g
j

m
, if q >
dn(2n + 1)λ
λ − l + 1
.
Theorem 3.2.3. In addition to the assumption in Theorem
3.2.1 we assume further that f
i
, 1 ≤ i ≤ λ, are linearly nonde-
generate over R

{g
j
}
q
j=1

. Then f
1
, · · · , f
λ
are algebraically de-
pendent over C, i.e. f
1
∧ · · · ∧ f
λ
≡ 0 on C
m
, if q >
dn(n + 2)λ

1 ≤ i ≤ λ, 1 ≤ j ≤ q. Let κ be a positive integer or κ = +∞
and κ = min{κ, n}. Assume that the following conditions are
satisfied.
21
i) min{κ, v
(f
1
,g
j
)
} = · · · = min{κ, v
(f
λ
,g
j
)
} for each 1 ≤ j ≤ q,
ii) dim{z|(f
1
, g
i
)(z) = (f
1
, g
j
)(z) = 0} ≤ m − 2 for each
1 ≤ i < j ≤ q,
iii) There exists an integer number l, 2 ≤ l ≤ λ, such that for
any increasing sequence 1 ≤ j
1

∧ · · · ∧ f
λ
≡ 0 on C
m
.
ii) If f
i
, 1 ≤ i ≤ λ are linearly nondegenerate over R

{g
j
}
q
j=1

and
q >
n(n + 2)λ − (κ − 1)(λ − 1)
λ − l + 1
,
then f
1
, · · · , f
λ
are algebraically dependent over C.
iii) If f
i
, 1 ≤ i ≤ λ are linearly nondegenerate over C, g
i
(1 ≤

) = o(max
1≤i≤2
{T (r, f
i
)}), 1 ≤
j ≤ q and (f
i
, g
j
) ≡ 0 for 1 ≤ i ≤ 2, 1 ≤ j ≤ q. Let κ be a pos-
22
itive integer or κ = +∞ and κ = min{κ, n}. Assume that the
following conditions are satisfied.
i) min{κ, v
(f
1
,g
j
)
(z)} = min{κ, v
(f
2
,g
j
)
} for each z ∈ C
m
and
1 ≤ j ≤ q
ii) dim{(f

2
.
23
Conclusion and recommentdations
Conclusions
The main results of the thesis
• Pointed out some necessary condition for meromorphic function
of a class of maximal deficiency and showed that the class
of meromorphic function with maximal deficiency sum is very
”thin” in the sense that if the ”noise” of the meromorphic
function of the maximal sum meromorphic by the meromorphic
function of ”small”, they no longer are meromorphic function of
the maximal deficiency sum of shortcomings as well.
• Pointed out some theorem about mappings of several complex
variable with maximal deficiency sum for moving targets.
• Proved three theorems about algebraic dependence of the mero-
morphic mapping from C
m
to P
n
(C) with moving targets in
general position.
• Proved the unicity theorem with truncated multiplicities of
meromorphic mappings in several complex variables with targets
of q < 4n
2
+ 2n in situation no assumption about the non
degenerate linear mapping of the meromorphic of f : C
m
→