The above results are the best results on the algebraic dependences of meromorphic
mappings sharing moving hyperplanes available at the present. Actually, there are many authors
consider the multiple values for meromorphic mappings sharing hyperplanes, i.e., consider only
the intersecting points of the mappings fi and the hyperplanes aj with the multiplicity not exceed
a certain number kj < +1. In 2016, L. N. Quynh [7] generalized and improved Theorems A by
considering the multiple values problem and reducing the number of hyperplanes. Moreover, the
condition (b) is very important role in above authors’ proof. In 2012, H. H. Giang, L. N. Quynh
and S. D. Quang [8] introduced new techniques to treat the case where the condition (b) is replaced
by a more general. Our purpose in this paper is to generalize Quynh’s result by considering the
general case where codim of the intersections of inverse images of arbitrary d, (d ≥ 1) hyperplanes
be at least two . Namely, we will prove the following.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2017-0028
Mathematical and Physical Sci., 2017, Vol. 62, Iss. 8, pp. 23-31
This paper is available online at
ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS SHARING
MOVING HYPERPLANES WITHOUT COUNTING MULTIPLICITIES
Ha Huong Giang
Faculty of Fundamental Sciences, Electric Power University
Abstract. In this article, we will prove a algebraic dependences theorem for meromorphic
mappings sharing moving hyperplanes with different multiple and a general condition on
the intersections of the inverse images of these hyperplanes.
Keywords: Algebraic dependence, meromorphic mapping, truncated, multiplicity.
1. Introduction
In 1989, W. Stoll [1] studied the theory on algebraic dependences of meromorphic mappings
in several complex variables into the complex projective spaces for fixed targets. In 2011, M. Ru [2]
generalized W. Stoll’s result to the case of holomorphic curves into the complex projective
spaces sharing moving hyperplanes. Recently, by using the new second main theorem given by
Thai-Quang [3], P. D. Thoan, P. V. Duc and S. D. Quang in [4], [5] and [6] gave some improvements
of the results of W. Stoll and M. Ru. In order to state some of their result, we first recall the
following.
A meromorphic mapping of Cm into Pn(C)∗ be called moving hyperplane in Pn(C). Let
a1, ..., aq (q ≥ n + 1) be q moving hyperplanes with reduced representations aj = (aj0 : ... :
ajn) (1 ≤ j ≤ q). We say that a1, ..., aq are in general position if det(ajkl) 6≡ 0 for any 1 ≤ j0 <
j1 < ... < jn ≤ q.
Let fi : Cm → Pn(C) (1 ≤ i ≤ λ) be meromorphic mappings with reduced
representations fi := (fi0 : ... : fin). Let aj (1 ≤ j ≤ q) be moving hyperplanes in
Pn(C) in general position with reduced representations aj := (aj0 : ... : ajn). Put (fi, aj) :=∑n
s=0 fisajs 6= 0 for each 1 ≤ i ≤ λ, 1 ≤ j ≤ q and assume that min{1, ν0(f1,aj),≤kj} = ... =
min{1, ν0(fλ,aj),≤kj}.
In, P. D. Thoan, P. V. Duc and S. D. Quang [5] proved an better algebraic dependences
theorem as follows.
Received September 11, 2017. Accepted September 30, 2017.
Contact Ha Huong Giang, e-mail: hhgiang79@yahoo.com.
23
Ha Huong Giang
Theorem A.
Let f1, ..., fλ : Cm → Pn(C) be non-constant meromorphic mappings. Let ai (1 ≤
i ≤ q) be moving hyperplanes in Pn(C) in general position such that T (r, ai) =
o(max1≤i≤λT (r, fj)) (1 ≤ i ≤ q) and (fi, aj) 6≡ 0 for 1 ≤ i ≤ λ, 1 ≤ j ≤ q. Assume
that the following conditions are satisfied.
(a) min{1, ν(f1,aj)} = ... = min{1, ν(fλ,aj)} for each 1 ≤ j ≤ q,
(b) dim{z| (f1, ai)(z) = (f1, aj)(z) = 0} ≤ m− 2 for each 1 ≤ i < j ≤ q,
(c) there exists an integer number l, 2 ≤ l ≤ λ, such that for any increasing sequence
1 ≤ j1 < ... < jl ≤ λ, fj1(z) ∧ ... ∧ fjl(z) = 0 for every point z ∈
⋃q
i=1(f1, gi)
−1{0}.
If q > n(2n+1)λ−(n−1)(λ−1)λ−l+1 , then f1 ∧ ... ∧ fλ ≡ 0.
The above results are the best results on the algebraic dependences of meromorphic
mappings sharing moving hyperplanes available at the present. Actually, there are many authors
consider the multiple values for meromorphic mappings sharing hyperplanes, i.e., consider only
the intersecting points of the mappings fi and the hyperplanes aj with the multiplicity not exceed
a certain number kj < +∞. In 2016, L. N. Quynh [7] generalized and improved Theorems A by
considering the multiple values problem and reducing the number of hyperplanes. Moreover, the
condition (b) is very important role in above authors’ proof. In 2012, H. H. Giang, L. N. Quynh
and S. D. Quang [8] introduced new techniques to treat the case where the condition (b) is replaced
by a more general. Our purpose in this paper is to generalize Quynh’s result by considering the
general case where codim of the intersections of inverse images of arbitrary d, (d ≥ 1) hyperplanes
be at least two . Namely, we will prove the following.
Theorem 1.1. Let f1, ..., fλ : Cm → Pn(C) be non-constant meromorphic mappings.
Let {aj}qj=1 be q moving hyperplanes in Pn(C) in general position satisfying T (r, aj) =
o(max1≤i≤λ T (r, fi)) (1 ≤ j ≤ q). Let kj (1 ≤ j ≤ q) be positive integers or +∞. Assume
that (fi, aj) 6≡ 0 for 1 ≤ i ≤ λ, 1 ≤ j ≤ q, and the following conditions are satisfied.
(a) min{1, ν0(f1,aj),≤kj} = ... = min{1, ν0(fλ,aj),≤kj} for each 1 ≤ j ≤ q,
(b) dim
(⋂d+1
i=1 f
−1
1 (aji)
)
≤ m− 2 for each 1 ≤ j1 < ... < jd+1 ≤ q,
(c) there exists an integer number l, 2 ≤ l ≤ λ, such that for any increasing sequence
1 ≤ i1 < ... < il ≤ λ, fi1(z) ∧ ... ∧ fil(z) = 0, for every point z ∈
⋃q
j=1 f
−1
1 (aj).
We assume further that rankR{aj}f1 = ... = rankR{aj}fλ = N + 1, whereN is a positive
integer. If
q∑
j=1
1
kj + 1−N <
q
N(2n−N + 2) −
λdq
q(d(λ− l) + 1) + λ(N − 1) .
then f1 ∧ ... ∧ fλ ≡ 0.
24
Algebraic dependences of meromorphic mappings sharing moving hyperplanes...
2. Some second main theorems and lemmas
2.1. We set ||z|| = (|z1|2 + · · ·+ |zm|2)1/2 for z = (z1, . . . , zm) ∈ Cm and define B(r) := {z ∈
Cm : ||z|| < r}, S(r) := {z ∈ Cm : ||z|| = r} (0 < r < +∞). Define
vn−1(z) := (ddc||z||2)n−1 and
σn(z) := d
c log ||z||2 ∧ (ddc log ||z||2)n−1 on Cm\0.
2.2. For a divisor ν on Cm, we denote byN(r, ν) the counting function of the divisor ν as usual in
Nevanlinna theory (see [9]). For a positive integerM orM =∞, we define the truncated divisors
of ν by
ν[M ](z) = min{M,ν(z)}, ν[M ]≤k (z) :=
{
ν[M ](z) if ν[M ](z) ≤ k
0 if ν[M ](z) > k.
Similarly, we define ν[M ]>k . We will write N
[M ](r, ν), N
[M ]
≤k (r, ν), N
[M ]
>k (r, ν) for N(r, ν
[M ]),
N(r, ν
[M ]
≤k ), N(r, ν
[M ]
>k ) as respectively.
Let ϕ : Cm −→ C be a meromorphic function. Define
Nϕ(r) = N(r, νϕ), N
[M ]
ϕ (r) = N
[M ](r, νϕ),
N
[M ]
ϕ,≤k(r) = N
[M ]
≤k (r, νϕ),
N
[M ]
ϕ,>k(r) = N
[M ]
>k (r, νϕ).
We will omit the character [M ] ifM =∞.
2.3. Let f : Cm → Pn(C) be a meromorphic mapping. For arbitrarily fixed homogeneous
coordinates (w0 : ... : wn) on Pn(C), we take a reduced representation f = (f0 : ... : fn),
which means that each fi is a holomorphic function on Cm and f(z) = (f0(z) : ... : fn(z))
outside the analytic set I(f) = {f0 = ... = fn = 0} of codimension ≥ 2. Let a be a moving
hyperplane in Pn(C) with reduced representation a = (a0 : ... : an). We denote by T (r, f) the
characteristic function of f and bymf,a(r) the proximity function of f with respect to a (see [10]).
If (f, a) 6≡= 0, then the first main theorem for moving targets in value distribution theory
states
T (r, f) + T (r, a) = mf,a(r) +N(f,a)(r).
Theorem 2.4. (The First Main Theorem for general position [1]). Let fj : Cm → Pn(C), (1 ≤
j ≤ λ) be meromorphic mappings located in general position. Assume that 1 ≤ k ≤ n. Then
N(r, µf1∧...∧fλ) +m(r, f1 ∧ ... ∧ fλ) ≤
∑
1≤i≤λ
T (r, fi) +O(1).
Here, by µf1∧...∧fλ we denote the divisor associated with f1 ∧ ... ∧ fλ. We also denote by
Nf1∧...∧fλ(r) the counting function associated with the divisor µf1∧...∧fλ .
Let V be a complex vector space of dimension n ≥ 1. The vectors {v1, ..., vk} are said to
be in general position if for each selection of integers 1 ≤ i1 < ... < ip ≤ k with p ≤ n, then
25
Ha Huong Giang
vi1 ∧ ... ∧ vip 6= 0. The vectors {v1, ..., vk} are said to be in special position if they are not in
general position. Take 1 ≤ p ≤ k, then {v1, ..., vk} are said to be in p-special position if for each
selection of integers 1 ≤ i1 < ... < ip ≤ k, the vectors {vi1 , ..., vip} are in special position.
Theorem 2.5. (The SecondMain Theorem for general position [1]). LetM 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 fj : M → P(V ), 1 ≤ j ≤ k, be meromorphic mappings. Assume that f1, ..., fk are in
general position. Also assume that f1, ..., fk are in p-special position on A. Then we have
µf1∧...∧fk ≥ (k − p+ 1)νA.
Here by νA we denote the reduced divisor whose support is the set A.
The following is a new second main theorem given by S. D. Quang [10], which is an
improvement the second main theorem of Thai-Quang in [3].
Theorem 2.6.(The Second Main Theorem for moving target [10]). Let f : Cm → Pn(C) be a
meromorphic mapping. Let {ai}qi=1 (q = 2n−N+2) be moving hyperplanes in Pn(C) in general
position such that (f, ai) 6≡ 0 (1 ≤ i ≤ q), where N + 1 = rankR{ai}(f). Then we have
|| q
2n−N + 2Tf (r) ≤
q∑
i=1
N
[N ]
(f,ai)
(r) + o(Tf (r)) +O(max1≤i≤qTai(r)).
As usual, by the notation ” || P " we mean the assertion P holds for all r ∈ [0,∞) excluding a
Borel subset E of the interval [0,∞) with ∫E dr <∞.
Remark: With the assumption of Theorem 2.6, we see that
|| Tf (r) ≤ N(2n−N + 2)
q
q∑
i=1
N
[1]
(f,ai)
(r) + o(Tf (r)) +O( max
1≤i≤q
Tai(r))
≤ n(n+ 2)
q
q∑
i=1
N
[1]
(f,ai)
(r) + o(Tf (r)) +O( max
1≤i≤q
Tai(r)). (2..1)
3. Proof of Theorem 1.1.
It suffices to prove Theorem 1.1 in the case of λ ≤ n+ 1. We set N = rankR{aj}(fi)− 1.
Proposition 3.1.[see [7]] For every 1 ≤ i ≤ λ, 1 ≤ j ≤ q and 1 ≤ N ≤ n, we have
N
[N ]
(fi,aj),≤kj (r) ≥
kj + 1
kj + 1−NN
[N ]
(fi,aj)
(r)− N
kj + 1−N T (r, fi).
Proposition 3.2.[see [7]] Let hi : Cm → Pn(C) (1 ≤ i ≤ p ≤ n + 1) be meromorphic
mappings with reduced representations hi := (hi0 : ... : hin). Let ai (1 ≤ i ≤ n + 1) be
moving hyperplanes in Pn(C) with reduced representations ai := (ai0 : ... : ain). Put h˜i :=
((hi, a1) : ... : (hi, an+1)). Assume that a1, ..., an+1 are located in general position such that
(hi, aj) 6≡ 0 (1 ≤ i ≤ p, 1 ≤ j ≤ n+ 1). Let S be a pure (m− 1)-dimensional analytic subset of
26
Algebraic dependences of meromorphic mappings sharing moving hyperplanes...
Cm such that S 6⊂ (a1∧...∧an+1)−1{0}. Then h1∧...∧hp = 0 on S if and only if h˜1∧...∧h˜p = 0
on S.
Suppose that f1 ∧ ... ∧ fλ 6≡ 0. For λ indices 0 = j0 < j1 < ... < jλ−1 ≤ n such that
(f1, aj0) ... (fλ, aj0)
(f1, aj1) ... (fλ, aj1)
... ... ...
(f1, ajλ−1) ... (fλ, ajλ−1)
is nondegenerate.
Put J = {j0, ..., jλ−1}, Jc = {1, ..., q}\J and
BJ =
(f1, aj0) ... (fλ, aj0)
(f1, aj1) ... (fλ, aj1)
... ... ...
(f1, ajλ−1) ... (fλ, ajλ−1)
Proposition 3.3. If BJ is nondegenerate, i.e., detBJ 6≡ 0 then
1
d
∑
j∈J
(min1≤i≤λ{ν0(fi,aj),≤kj} −min{1, ν0(f1,aj),≤kj})
+
1
d
q∑
j=1
(d(λ− l) + 1)min{1, ν0(f1,aj),≤kj} ≤ µf˜1∧...∧f˜λ ,
on the set Cm\
(
A ∪⋃λi=1 I(fi) ∪ (aj0 ∧ ... ∧ ajλ−1)−1(0)) where f˜i := ((fi, aj0) : ... :
(fi, ajλ−1)) and A =
⋃
1≤i<j≤q Z(f1, ai) ∩ Z(f1, aj).
Proof. Indeed, putA := ⋃j∈J Z(f1, aj), Ac := ⋃j∈Jc Z(f1, aj). We consider the following two
cases.
* Case 1. Let z0 ∈ A\
(
A ∪⋃λi=1 I(fi) ∪ (aj0 ∧ ... ∧ ajλ−1)−1(0)) be a regular point of
A. Then z0 is a zero of one of the meromorphic functions {(f1, aj)}j∈J . Without loss of
generality we may assume that z0 is a zero of (f1, aj0). Let S be an irreducible component of
A containing z0. Suppose that U is an open neighborhood of z0 in Cm such that U ∩ {A\S} = ∅.
Choose a holomorphic function h on an open neighborhood U ⊂ U of z0 such that νh(z) =
1
d
min1≤i≤λ{ν0(fi,aj0 ),≤kj0 (z)} if z ∈ S and νh(z) = 0 if z 6∈ S. Then (fi, aj0) = aih (1 ≤ i ≤
λ), where ai are holomorphic functions. Therefore, the matrix
(f1, aj1) ... (fλ, aj1)... ... ...
(f1, ajλ−1) ... (fλ, ajλ−1)
is
of rank ≤ λ− 1. Hence, there exist λ holomorphic functions b1, ..., bλ, not all zeros, such that
λ∑
i=1
bi.(fi, ajk) = 0 (1 ≤ k ≤ λ− 1).
27
Ha Huong Giang
Without loss of generality, we may assume that the set of common zeros of {bi}λi=1 is an analytic
subset of codimension ≥ 2. Then there exists an index i1, 1 ≤ i1 ≤ λ, such that S 6⊂ b−1i1 {0}. We
may assume that i1 = λ. Then for each z ∈ (U ′ ∪ S)\b−1λ {0}, we have
f˜1(z) ∧ ... ∧ f˜λ(z) = f˜1(z) ∧ ... ∧ f˜λ−1(z) ∧
(
f˜λ(z) +
λ−1∑
i=1
bi
bλ
f˜i(z)
)
= f˜1(z) ∧ ... ∧ f˜λ−1(z) ∧ (V (z)h(z))
= h(z)∆(f˜1(z) ∧ ... ∧ f˜λ−1(z) ∧ V (z)),
where V (z) := (aλ +
∑λ−1
i=1
bi
bλ
ai, 0, ..., 0).
By the assumption and by the Proposition 3.2, for any increasing sequence 1 ≤ i1 < ... <
il ≤ λ − 1, we have f˜i1 ∧ ... ∧ f˜il = 0 on S. This implies that the family {f˜1, ..., f˜λ−1, V } is in
(l + 1)-special position on S. By using The Second Main Theorem for general position [1], we
have
µf˜1∧...∧f˜λ−1∧V (z) ≥ λ− l, ∀ z ∈ S.
Hence
µf˜1∧...∧f˜λ−1(z) ≥ νh(z) + λ− l =
1
d
min1≤i≤λ{ν0(fi,aj0 ),≤kj0 (z)}+ λ− l,
for all z ∈ (U ′ ∪ S)\b−1i1 {0}. This implies that
1
d
∑
j∈J
(min1≤i≤λ{ν0(fi,aj),≤kj (z0)} −min{1, ν0(f1,aj),≤kj (z0)})
+
1
d
q∑
j=1
(d(λ− l) + 1)min{1, ν0(f1,aj),≤kj (z0)}
=
1
d
min1≤i≤λ{ν0(fi,aj0 ),≤kj0 (z0)}+ λ− l
≤ µf˜1∧...∧f˜λ(z0).
* Case 2. Let z0 ∈ Ac\
(
A ∪⋃λi=1 I(fi) ∪ (aj0 ∧ ... ∧ ajλ−1)−1(0)) be a regular point of Ac.
Then z0 is a zero of (f1, aj), j ∈ Jc. By the assumption and by the Proposition 3.2, the family
{f˜1, ..., f˜λ} is in l-special position on each irreducible component of Ac containing z0. By using
The Second Main Theorem for general position [1], we have
µf˜1∧...∧f˜λ(z0) ≥ λ− l + 1.
Hence
1
d
∑
j∈J
(min1≤i≤λ{ν0(fi,aj),≤kj (z0)} −min{1, ν0(f1,aj),≤kj (z0)})
+
1
d
q∑
j=1
(d(λ− l) + 1)min{1, ν0(f1,aj),≤kj (z0)}
= λ− l + 1 ≤ µf˜1∧...∧f˜λ(z0).
28
Algebraic dependences of meromorphic mappings sharing moving hyperplanes...
From the above two cases we get the desired inequality of the Proposition.
Proof Theorem 1.1.
For each j, 1 ≤ j ≤ q, we set
Nj(r) =
λ∑
i=1
N
[N ]
(fi,aj),≤kj (r)− ((λ− 1)N + 1)N
[1]
(f1,aj),≤kj (r).
For each permutation I = (j1, ..., jq) of (1, ..., q), we set
TI = {r ∈ [1,+∞);Nj1(r) ≥ ... ≥ Njq(r)}.
It is clear that
⋃
I TI = [1,+∞). Therefore, there exists a permutation, for instance it is I0 =
(1, ..., q), such that
∫
TI0
dr = +∞. Then we have
N1(r) ≥ N2(r) ≥ ... ≥ Nq(r), for all r ∈ TI0 .
By the assumption for f1∧ ...∧ fλ 6≡ 0, there exist indices J = {j0, ..., jλ−1} with 0 = j0 < j1 <
... < jλ−1 ≤ n such that detBJ 6≡ 0. We note that
N1(r) = Nj0(r) ≥ Nj1(r) ≥ ... ≥ Njλ−1(r) ≥ Nn(r), for each r ∈ TI0 .
We see that min1≤i≤λai ≥
∑λ
i=1min{N, ai} − (λ − 1)N for every λ non-negative integers
a1, ..., aλ. Then the Proposition 3.3 implies that
1
d
∑
j∈J
(
λ∑
i=1
min{N, ν0(fi,aj),≤kj} − ((λ− 1)N + 1)min{1, ν0(f1,aj),≤kj}
)
+
1
d
q∑
j=1
(d(λ− l) + 1)min{1, ν0(f1,aj),≤kj} ≤ µf˜1∧...∧f˜λ ,
on the set Cm\
(
A ∪⋃λi=1 I(fi) ∪ (aj0 ∧ ... ∧ ajλ−1)−1(0)). Integrating both sides of this
inequality, we have
1
d
∑
j∈J
(
λ∑
i=1
N
[N ]
(fi,aj),≤kj (r)− ((λ− 1)N + 1)N
[1]
(f1,aj),≤kj (r)
)
+
1
d
q∑
j=1
(d(λ− l) + 1)N [1](f1,aj),≤kj (r) ≤ Nf˜1∧...∧f˜λ(r) = NdetBJ (r). (3..1)
Also, by Jensens formula, we have
NdetBJ (r) ≤
∫
S(r)
log |detBJ |σn +O(1) ≤
λ∑
i=1
T (r, fi) + o( max
1≤i≤λ
T (r, fi)). (3..2)
29
Ha Huong Giang
Set T (r) =
∑λ
i=1 T (r, fi). Combining (3..1) and (3..2), then for all r ∈ I0, we have
|| T (r) ≥ 1
d
λ−1∑
i=0
Nji(r) +
q∑
j=1
(λ− l + 1
d
)N
[1]
(f1,aj),≤kj (r) + o(T (r))
≥ λ
dq
q∑
j=1
Nj(r) +
q∑
j=1
(λ− l + 1
d
)N
[1]
(f1,aj),≤kj (r) + o(T (r))
=
q∑
j=1
(
λ− l + 1
d
− λ((λ− 1)N + 1)
dq
)
N
[1]
(f1,aj),≤kj (r)
+
q∑
j=1
λ
dq
λ∑
i=1
N
[N ]
(fi,aj),≤kj (r) + o(T (r))
≥
λ∑
i=1
q∑
j=1
(
λ
dq
+
d(λ− l) + 1
dλN
− (λ− 1)N + 1
Ndq
)
N
[N ]
(fi,aj),≤kj (r) + o(T (r))
≥
λ∑
i=1
q∑
j=1
q(d(λ− l) + 1) + λ(N − 1)
λNdq
(
N
[N ]
(fi,aj)
(r)− N
kj + 1−N T (r, fi)
)
+ o(T (r))
≥ q(d(λ− l) + 1) + λ(N − 1)
λNdq
λ∑
i=1
q∑
j=1
N
[N ]
(fi,aj)
(r)−
q∑
j=1
N
kj + 1−N T (r)
+ o(T (r))
≥ q(d(λ− l) + 1) + λ(N − 1)
λNdq
q
2n−N + 2 −
q∑
j=1
N
kj + 1−N
T (r)
+ o(T (r)).
Letting r → +∞, we have
1 ≥ q(d(λ− l) + 1) + λ(N − 1)
λNdq
q
2n−N + 2 −
q∑
j=1
N
kj + 1−N
.
Thus
q∑
j=1
1
kj + 1−N ≥
q
N(2n−N + 2) −
λdq
q(d(λ− l) + 1) + λ(N − 1) .
This is a contradiction. Thus, we have f1 ∧ ... ∧ fλ = 0. The theorem is proved.
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30
Algebraic dependences of meromorphic mappings sharing moving hyperplanes...
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