Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets
The purpose of our paper is twofold. Our first aim is to prove a uniqueness theorem for meromorphic mappings of $C^n$ into $P^N(C)$ sharing $2N + 2$ hyperplanes in the general position with truncated multiplicities, where all common zeros with multiplicities more than a certain number do not need to...
Gespeichert in:
| Datum: | 2019 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2019
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1448 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507203252781056 |
|---|---|
| author | Pham, Hoang Ha Si, Duc Quang Фам, Хоанг Га Сі, Дук Куанг |
| author_facet | Pham, Hoang Ha Si, Duc Quang Фам, Хоанг Га Сі, Дук Куанг |
| author_sort | Pham, Hoang Ha |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:55:13Z |
| description | The purpose of our paper is twofold. Our first aim is to prove a uniqueness theorem for meromorphic mappings of $C^n$ into
$P^N(C)$ sharing $2N + 2$ hyperplanes in the general position with truncated multiplicities, where all common zeros with
multiplicities more than a certain number do not need to be counted. Second, we consider the case of mappings sharing
less than $2N + 2$ hyperplanes. These results are improvements of some recent results. |
| first_indexed | 2026-03-24T02:05:35Z |
| format | Article |
| fulltext |
UDC 517.5
Pham Hoang Ha (Hanoi Nat. Univ. Education, Vietnam),
Si Duc Quang (Hanoi Nat. Univ. Education; Thang Long Inst. Math. and Appl. Sci., Hanoi, Vietnam)
UNICITY THEOREMS WITH TRUNCATED MULTIPLICITIES
OF MEROMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES
FOR FEW FIXED TARGETS*
ТЕОРЕМИ ЄДИНОСТI З ОБРIЗАНИМИ КРАТНОСТЯМИ
ДЛЯ МЕРОМОРФНИХ ВIДОБРАЖЕНЬ ЗА КIЛЬКОМА ЗМIННИМИ
ДЛЯ НЕВЕЛИКОЇ КIЛЬКОСТI ОБ’ЄКТIВ
The purpose of our paper is twofold. Our first aim is to prove a uniqueness theorem for meromorphic mappings of \BbbC n into
\BbbP N (\BbbC ) sharing 2N + 2 hyperplanes in the general position with truncated multiplicities, where all common zeros with
multiplicities more than a certain number do not need to be counted. Second, we consider the case of mappings sharing
less than 2N + 2 hyperplanes. These results are improvements of some recent results.
Робота має двi основнi мети. По-перше, доведено теорему єдиностi для мероморфних вiдображень з \BbbC n в \BbbP N (\BbbC ), що
подiляють 2N+2 гiперплощини загального положення з обрiзаними кратностями, де всi спiльнi нулi з кратностями,
що перевищують деяке число, можна не враховувати. По-друге, розглянуто випадок, коли вiдображення подiляють
менше, нiж 2N + 2 гiперплощини. Отриманi результатi покращують деякi вiдомi новi результати.
1. Introduction. In 1926, R. Nevanlinna showed that two distinct nonconstant meromorphic func-
tions f and g on the complex plane \BbbC cannot have the same inverse images for five distinct values.
This result is usually called the Nevanlinna’s five values theorem, which is the first theorem on the
uniqueness problem of meromorphic mappings. After that, the uniqueness problems with truncated
multiplicities of meromorphic mappings of \BbbC n into the complex projective space \BbbP N (\BbbC ) sharing a
finite set of hyperplanes in \BbbP N (\BbbC ) has been studied intensively by many authors such as H. Fujimoto
[5], L. Smiley [11], S. Ji [7], Z.-H. Tu [16], G. Dethloff, T. V. Tan [3], D. D. Thai, S. D. Quang
[10, 14], Z. Chen, Q. Yan [2] and others.
To state some of them, first of all we recall the following.
(a) Let f be a nonconstant meromorphic mapping of \BbbC n into \BbbP N (\BbbC ) with a reduced representa-
tion f = (f0 : . . . : fN ). Let H be a hyperplane in \BbbP N (\BbbC ) given by H = \{ a0\omega 0+ . . .+aN\omega N\} . We
set (f,H) =
\sum N
i=0
aifi . Then we can define the corresponding divisor \nu (f,H) which is rephrased as
the intersection multiplicity of the image of f and H at f(z). Let k be a positive integer or k = \infty .
For every z \in \BbbC n, we set
\nu (f,H),\leq k(z) =
\left\{ 0 if \nu (f,H)(z) > k,
\nu (f,H)(z) if \nu (f,H)(z) \leq k,
and
\nu (f,H),\geq k(z) =
\left\{ 0 if \nu (f,H)(z) < k,
\nu (f,H)(z) if \nu (f,H)(z) \geq k.
* The research of Pham Hoang Ha was supported in part by a NAFOSTED grant of Vietnam (No. 101.04-2018.03).
c\bigcirc PHAM HOANG HA, SI DUC QUANG, 2019
412 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
UNICITY THEOREMS WITH TRUNCATED MULTIPLICITIES OF MEROMORPHIC MAPPINGS . . . 413
Take a meromorphic mapping f of \BbbC n into \BbbP N (\BbbC ) which is linearly nondegenerate over \BbbC , a
positive integer d, a positive integer k or k = \infty and q hyperplanes H1, . . . ,Hq in \BbbP N (\BbbC ) located
in general position with
\mathrm{d}\mathrm{i}\mathrm{m}
\bigl\{
z \in \BbbC n : \nu (f,Hi),\leq k(z) > 0 and \nu (f,Hj),\leq k(z) > 0
\bigr\}
\leq n - 2, 1 \leq i < j \leq q,
and consider the set \scrF (f, \{ Hj\} qj=1, k, d) of all linearly nondegenerate meromorphic maps g : \BbbC n \rightarrow
\rightarrow \BbbP N (\BbbC ) satisfying the conditions
(a) \mathrm{m}\mathrm{i}\mathrm{n} (\nu (f,Hj),\leq k, d) = \mathrm{m}\mathrm{i}\mathrm{n} (\nu (g,Hj),\leq k, d), 1 \leq j \leq q,
(b) f(z) = g(z) on
\bigcup q
j=1
\bigl\{
z \in \BbbC n : \nu (f,Hj),\leq k(z) > 0
\bigr\}
.
Denote by \sharp S the cardinality of the set S.
In 1983, L. Smiley [11] proved the following, which is usually called the unicity theorem for
meromorphic mapping sharing few hyperplanes regardless of multiplicity.
Theorem A [11]. If q \geq 3N + 2, then \sharp \scrF
\bigl(
f, \{ Hi\} qi=1,\infty , 1
\bigr)
= 1.
In 2006, D. D. Thai and S. D. Quang [14] improved slightly the result of Smiley for the case of
N \geq 2 to the following.
Theorem B [14]. If N \geq 2, then \sharp \scrF
\bigl(
f, \{ Hi\} 3N+1
i=1 ,\infty , 1
\bigr)
= 1.
In 2009, Z. Chen and Q. Yan [2] showed that the above unicity theorems are still valid for the
case of meromorphic mapping sharing 2N + 3 hyperplanes. They proved the following theorem.
Theorem C [2]. \sharp \scrF
\bigl(
f, \{ Hi\} 2N+3
i=1 ,\infty , 1
\bigr)
= 1.
Recently, S. D. Quang [10] improved the above result of Chen – Yan by omitting all zeros with
multiplicities larger than a certain number. He proved the following theorem.
Theorem D [10]. If k >
N(4N2 + 11N + 4)
3N + 2
- 1, then \sharp \scrF
\bigl(
f, \{ Hi\} 2N+3
i=1 , k, 1
\bigr)
= 1.
Then a natural question arise here: Are there any unicity theorems with truncated multiplicities
in the case where q \leq 2N + 2?
This question is first considered in [10] by S. D. Quang. He gave the following theorem.
Theorem E [10]. Let f1 and f2 be two linearly nondegenerate meromorphic mappings of \BbbC n
into \BbbP N (\BbbC ), N \geq 2, and let H1, . . . ,H2N+2 be hyperplanes in \BbbP N (\BbbC ) located in general position
such that
\mathrm{d}\mathrm{i}\mathrm{m}
\bigl\{
z \in \BbbC n : \nu (f1,Hi)(z) > 0 and \nu (f1,Hj)(z) > 0
\bigr\}
\leq n - 2
for every 1 \leq i < j \leq 2N + 2. Assume that the following conditions are satisfied:
(a) \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu (f1,Hj),\leq N , 1\} = \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu (f2,Hj),\leq N , 1\} and \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu (f1,Hj),\geq N , 1\} = \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu (f2,Hj),\geq N , 1\} ,
1 \leq j \leq 2N + 2,
(b) f1(z) = f2(z) on
\bigcup 2N+2
j=1
\bigl\{
z \in \BbbC n : \nu (f1,Hj)(z) > 0
\bigr\}
.
Then f1 \equiv f2 .
However, we see that in the above theorem all zeros of the functions (f,Hi)’s are counted.
In the first part of this paper, we will improve Theorem E by omitting the zeros with multiplicity
larger than a certain number k . Namely, we will prove the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
414 PHAM HOANG HA, SI DUC QUANG
Theorem 1.1. Let f1 and f2 be two linearly nondegenerate meromorphic mappings of \BbbC n into
\BbbP N (\BbbC ), N \geq 2, and let H1, . . . ,H2N+2 be hyperplanes in \BbbP N (\BbbC ) located in general position such
that
\mathrm{d}\mathrm{i}\mathrm{m}
\bigl\{
z \in \BbbC n : \nu (f1,Hi),\leq k(z) > 0 and \nu (f1,Hj),\leq k(z) > 0
\bigr\}
\leq n - 2
for every 1 \leq i < j \leq 2N + 2, where k be a positive integer such that
k > 6(m - 2)m(N2 - 1) + 2N - 1 with m =
\biggl(
2N + 2
N + 1
\biggr)
.
Assume that the following conditions are satisfied:
(a) \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu (f1,Hj)(z),\leq N , 1\} = \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu (f2,Hj)(z),\leq N , 1\} and \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu (f1,Hj),\geq N (z), 1\} =
= \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu (f2,Hj),\geq N (z), 1\} for all z \in \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} (\nu (f1,Hi),\leq k) \cup \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} (\nu (f1,Hj),\leq k), 1 \leq j \leq 2N + 2,
(b) f1(z) = f2(z) on
\bigcup 2N+2
j=1 \{ z \in \BbbC n : \nu (f1,Hj),\leq k(z) > 0\} .
Then f1 \equiv f2 .
We would like to emphasize that our paper is a apart of the doctoral thesis of the first author at
Hanoi National University of Education with some slight improvements. Recently, motivated by our
method, H. Z. Cao and T. B. Cao [6] (Theorem 1.9) have proved a result similar to Theorem 1.1,
where the condition “z \in \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} (\nu (f1,Hi),\leq k) \cup \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} (\nu (f1,Hj),\leq k)” is replaced by “\nu (f1,Hj)(z) \equiv
\equiv \nu (f2,Hj)(z)(\mathrm{m}\mathrm{o}\mathrm{d} T )” with a large enough positive integer T . The proof of Theorem 1.1 is
presented in Section 3.
(b) In the last part of this paper, we will consider the uniqueness problem for the case meromor-
phic mappings sharing less than 2N + 2 hyperplanes. In fact, we will prove a uniqueness theorem
for the case where the mappings share q hyperplanes with N +3 \leq q < 2N +2. To state our results,
we give the following.
Take a meromorphic mapping f of \BbbC n into \BbbP N (\BbbC ) which is linearly nondegenerate over \BbbC , a
positive integer d, a positive integer k or k = +\infty and q hyperplanes H1, . . . ,Hq in \BbbP N (\BbbC ) located
in general position with
\mathrm{d}\mathrm{i}\mathrm{m}
\bigl\{
z \in \BbbC n : \nu (f,Hi),\leq k(z) > 0 and \nu (f,Hj),\leq k(z) > 0
\bigr\}
\leq n - 2, 1 \leq i < j \leq q.
With the above notations, we have the following definition.
Definition 1.1. We denote by \scrG (f, \{ Hj\} qj=1, k, d) the set of all linearly nondegenerate mero-
morphic maps g : \BbbC n \rightarrow \BbbP N (\BbbC ) satisfying the conditions:
(a) \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu (f,Hj),\leq k, d\} = \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu (g,Hj),\leq k, d\} , 1 \leq j \leq q.
(b) Let f = (f0 : . . . : fN ) and g = (g0 : . . . : gN ) be reduced representations of f and g,
respectively. Then, for each 0 \leq j \leq N and for each \omega \in
\bigcup q
i=1\{ z \in \BbbC n : \nu (f,Hi),\leq k(z) > 0\} , the
following two conditions are satisfied:
(i) if fj(\omega ) = 0, then gj(\omega ) = 0,
(ii) if fj(\omega )gj(\omega ) \not = 0, then \scrD \alpha
\biggl(
fi
fj
\biggr)
(\omega ) = \scrD \alpha
\biggl(
gi
gj
\biggr)
(\omega ) for each n-tuple \alpha = (\alpha 1, . . . , \alpha n) of
nonnegative integers with | \alpha | = \alpha 1+. . .+\alpha n < d and for each i \not = j, where \scrD \alpha =
\partial | \alpha |
\partial \alpha 1z1 . . . \partial \alpha nzn
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
UNICITY THEOREMS WITH TRUNCATED MULTIPLICITIES OF MEROMORPHIC MAPPINGS . . . 415
Remark that the condition (b) does not depend on the choice of reduced representations.
For each real number x, by [x] we denote the largest integer which does not exceed x. We will
prove the following, which is the last purpose of this paper.
Theorem 1.2. Let f1 and f2 be two meromorphic mappings in \scrG
\bigl(
f, \{ Hi\} qi=1, k, d
\bigr)
, where
d, k, q, q \geq N + 2, are positive integers.
(a) If q >
N + 3
2
+
\sqrt{}
2(N + 1)(N - d)
d
+
\biggl(
N + 3
2
\biggr) 2
and
k >
2d(Nq - 2N - 1) + 2q + 2N2 + 2N
dq(q - N - 3) - 2(N + 1)(N - d)
,
then there exist [q/2] + 1 indices i1, . . . , i[q/2]+1 such that
(f1, Hi1)
(f2, Hi1)
= . . . =
(f1, H[q/2])
(f2, H[q/2]+1)
.
In particular, if q \geq 2N, then f1 = f2 .
(b) If 1 \leq d \leq N, q >
(N + 1)(d+ 1)
2d
+
\sqrt{}
(N + 1)2(d+ 1)2
4d2
+
N2 - 1
d
and
k >
q(N - 1)(dq - N - 1) + (dq +N - 1)(N + 1)
dq2 - (N + 1)(d+ 1)q - N2 + 1
,
then f1 = f2.
(c) If d \geq N +1, q > N +1+
2N
d
and k >
d(Nq - q +N + 1) - 2N2 + 2N
d(q - N - 1) - 2N
, then f1 = f2.
We note that, in [15] we together with Do Duc Thai also proved a similar result for the case
where the meromorphic mappings sharing N +2 moving hyperplanes. Our this result deals with the
general case where the number q of fixed hyperplanes belongs to [N + 2; 2N + 2].
We distinguish here some cases where the assumptions of Theorem 1.2 are satisfied.
1. The assumptions of the assertion (a) satisfies in the following cases:
d = 1, q = 2N +3 and k >
6N2 + 4N + 4
3N + 2
(therefore, in this case Theorem 1.2 is improvement
of Theorems C and D);
2 \leq d \leq N, q = 2N + 4 - d and k >
d(4N2 + 4N - 2) + 2N2 + 2N
(N - d)(d(2N + 4 - d) - 2(N + 1)) + 2N + 4 - d
;
d \geq N + 1, q = N + 3 and k >
2d(N2 +N - 1) + 2N2 + 4N + 6
2(d - N)(N + 1)
.
2. The assumptions of the assertion (b) satisfies in the following cases:
d = N, q = N + 3, N \geq 2 and k >
N4 + 5N3 + 4N2 - 7N + 3
2(N - 1)
;
N = d = 1, q = 5 and k > 2 (therefore, in this case we have an improvement of the Nevanlinna’s
five values theorem).
3. The assumptions of the assertion (c) satisfies in the following cases:
N + 1 \leq d \leq 2N, q = N + 2 and k >
d(N2 + 3N - 2) - 2n2 + 2N
2(d - N)
;
d \geq 2N + 1, q = N + 2 and k >
d(N2 + 2N - 1) - 2n2 + 2N
d - 2N
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
416 PHAM HOANG HA, SI DUC QUANG
2. Basic notions in Nevanlinna theory. 2.1. Divisors on \BbbC n . We set \| z\| =
\bigl(
| z1| 2 + . . .
. . . + | zn| 2
\bigr) 1/2
for z = (z1, . . . , zn) \in \BbbC n and define B(r) := \{ z \in \BbbC n : \| z\| < r\} , S(r) := \{ z \in
\in \BbbC n : \| z\| = r\} , 0 < r < \infty .
Define d = \partial + \=\partial , dc =
i
4\pi
(\=\partial - \partial ) and
vn - 1(z) :=
\bigl(
ddc\| z\| 2
\bigr) n - 1
,
\sigma n(z) := dc \mathrm{l}\mathrm{o}\mathrm{g} \| z\| 2 \wedge
\bigl(
ddc \mathrm{l}\mathrm{o}\mathrm{g} \| z\| 2
\bigr) n - 1
on \BbbC n \setminus \{ 0\} .
Let F be a nonzero holomorphic function on a domain \Omega in \BbbC n . For a multiindex \alpha = (\alpha 1, . . . , \alpha n),
we set | \alpha | = \alpha 1 + . . .+ \alpha n and \scrD \alpha F =
\partial | \alpha | F
\partial \alpha 1z1 . . . \partial \alpha nzn
. We define the mapping \nu F : \Omega \rightarrow \BbbZ by
\nu F (z) := \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
m : \scrD \alpha F (z) = 0 for all \alpha with | \alpha | < m
\bigr\}
, z \in \Omega .
We mean by a divisor on a domain \Omega in \BbbC n a mapping \nu : \Omega \rightarrow \BbbZ such that, for each a \in \Omega ,
there are nonzero holomorphic functions F and G on a connected neighborhood U of a (\subset \Omega ) such
that \nu (z) = \nu F (z) - \nu G(z) for each z \in U outside an analytic set of dimension \leq n - 2. Two
divisors are regarded as the same if they are identical outside an analytic set of dimension \leq n - 2.
For a divisor \nu on \Omega we set | \nu | := \{ z : \nu (z) \not = 0\} , which is a purely (n - 1)-dimensional analytic
subset of \Omega or empty.
Take a nonzero meromorphic function \varphi on a domain \Omega in \BbbC n . For each a \in \Omega , we choose
nonzero holomorphic functions F and G on a neighborhood U \subset \Omega such that \varphi =
F
G
on U and
\mathrm{d}\mathrm{i}\mathrm{m}(F - 1(0)\cap G - 1(0)) \leq n - 2, and we define the divisors \nu \varphi , \nu
\infty
\varphi by \nu \varphi := \nu F , \nu
\infty
\varphi := \nu G, which
are independent of choices of F and G. Hence, they are globally well-defined on \Omega .
2.2. Counting functions. For a divisor \nu on \BbbC n and for positive integers k,M (or M = \infty ),
we define the counting functions of \nu as follows. Set
\nu (M)(z) = \mathrm{m}\mathrm{i}\mathrm{n} \{ M,\nu (z)\} ,
\nu
(M)
\leq k (z) =
\left\{ 0 if \nu (z) > k,
\nu (M)(z) if \nu (z) \leq k,
\nu
(M)
>k (z) =
\left\{ \nu (M)(z) if \nu (z) > k,
0 if \nu (z) \leq k.
We define n(t) by
n(t) =
\left\{
\int
| \nu | \cap B(t)
\nu (z)vn - 1 if n \geq 2,\sum
| z| \leq t
\nu (z) if n = 1.
Similarly, we define n(M)(t), n
(M)
\leq k (t), n
(M)
>k (t).
Define
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
UNICITY THEOREMS WITH TRUNCATED MULTIPLICITIES OF MEROMORPHIC MAPPINGS . . . 417
N(r, \nu ) =
r\int
1
n(t)
t2n - 1
dt, 1 < r < \infty .
Similarly, we define N
\bigl(
r, \nu (M)
\bigr)
, N
\bigl(
r, \nu
(M)
\leq k
\bigr)
, N
\bigl(
r, \nu
(M)
>k
\bigr)
and denote them by N (M)(r, \nu ),
N
(M)
\leq k (r, \nu ), N
(M)
>k (r, \nu ), respectively.
Let \varphi : \BbbC n - \rightarrow \BbbC be a meromorphic function. Define
N\varphi (r) = N(r, \nu \varphi ), N (M)
\varphi (r) = N (M)(r, \nu \varphi ),
N
(M)
\varphi ,\leq k(r) = N
(M)
\leq k (r, \nu \varphi ), N
(M)
\varphi ,>k(r) = N
(M)
>k (r, \nu \varphi ).
For brevity we will omit the superscript (M) if M = \infty .
2.3. Characteristic and proximity functions. Let f : \BbbC n - \rightarrow \BbbP N (\BbbC ) be a meromorphic map-
ping. For arbitrarily fixed homogeneous coordinates (w0 : . . . : wN ) on \BbbP N (\BbbC ), we take a reduced
representation f = (f0 : . . . : fN ), which means that each fi is a holomorphic function on \BbbC n and
f(z) =
\bigl(
f0(z) : . . . : fN (z)
\bigr)
outside the analytic set \{ f0 = . . . = fN = 0\} of codimension \geq 2. Set
\| f\| =
\bigl(
| f0| 2 + . . .+ | fN | 2
\bigr) 1/2
.
The characteristic function of f is defined by
T (r, f) =
\int
S(r)
\mathrm{l}\mathrm{o}\mathrm{g} \| f\| \sigma n -
\int
S(1)
\mathrm{l}\mathrm{o}\mathrm{g} \| f\| \sigma n.
Let H be a hyperplane in \BbbP N (\BbbC ) given by H = \{ a0\omega 0+ . . .+aN\omega N\} . We define the proximity
function of H by
mf,H(r) =
\int
S(r)
\mathrm{l}\mathrm{o}\mathrm{g}
\| f\| \| H\|
| (f,H)|
\sigma n -
\int
S(1)
\mathrm{l}\mathrm{o}\mathrm{g}
\| f\| \| H\|
| (f,H)|
\sigma n,
where \| H\| =
\biggl( \sum N
i=0
| ai| 2
\biggr) 1/2
.
Let \varphi be a nonzero meromorphic function on \BbbC n, which are occasionally regarded as a mero-
morphic mapping into \BbbP 1(\BbbC ). The proximity function of \varphi is defined by
m(r, \varphi ) :=
\int
S(r)
\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x} (| \varphi | , 1)\sigma n.
As usual, by the notation “\| P ” we mean the assertion P holds for all r \in [0,\infty ) excluding a
Borel subset E of the interval [0,\infty ) with
\int
E
dr < \infty .
2.4. Some lemmas. The following results play essential roles in Nevanlinna theory (see [9,
12, 13]).
Theorem 2.1 (first main theorem). Let f : \BbbC n \rightarrow \BbbP N (\BbbC ) be a linearly nondegenerate mero-
morphic mapping and H be a hyperplane in \BbbP N (\BbbC ). Then
N(f,H)(r) +mf,H(r) = T (r, f), r > 1.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
418 PHAM HOANG HA, SI DUC QUANG
Theorem 2.2 (second main theorem). Let f : \BbbC n \rightarrow \BbbP N (\BbbC ) be a linearly nondegenerate mero-
morphic mapping and H1, . . . ,Hq be hyperplanes in general position in \BbbP N (\BbbC ). Then
\bigm\| \bigm\| (q - N - 1)T (r, f) \leq
q\sum
i=1
N
(N)
(f,Hi)
(r) + o(T (r, f)).
Lemma 2.1 (see [14]). Let f : \BbbC n \rightarrow \BbbP N (\BbbC ) be a linearly nondegenerate meromorphic map-
ping. Let H1, H2, . . . ,Hq be q hyperplanes in \BbbP N (\BbbC ) located in general position. Assume that
k \geq N - 1. Then\bigm\| \bigm\| \bigm\| \bigm\| \biggl( q - N - 1 - Nq
k + 1
\biggr)
T (r, f) \leq
q\sum
j=1
\biggl(
1 - N
k + 1
\biggr)
N
(N)
(f,Hj),\leq k(r) + o(T (r, f)) .
Lemma 2.2 (lemma on logarithmic derivative). Let f be a nonzero meromorphic function on
\BbbC n. Then \bigm\| \bigm\| \bigm\| \bigm\| m\biggl( r, \scrD \alpha (f)
f
\biggr)
= O
\bigl(
\mathrm{l}\mathrm{o}\mathrm{g}+ T (r, f)
\bigr)
, \alpha \in \BbbZ n
+.
Denote by \scrM \ast
n the Abelian multiplicative group of all nonzero meromorphic functions on \BbbC n .
Then the multiplicative group \scrM \ast
n/\BbbC \ast is a torsion free Abelian group.
Definition 2.1. Let G be a torsion free Abelian group and A = (a1, a2, . . . , aq) be a q-tuple
of elements ai in G. Let q \geq r > s > 1. We say that the q-tuple A has the property (Pr,s) if
any r elements al(1), . . . , al(r) in A satisfy the condition that for any given i1, . . . , is, 1 \leq i1 < . . .
. . . < is \leq r, there exist j1, . . . , js, 1 \leq j1 < . . . < js \leq r, with \{ i1, . . . , is\} \not = \{ j1, . . . , js\} such
that al(i1) . . . al(is) = al(j1) . . . al(js).
Proposition 2.1 (see [4]). Let G be a torsion free Abelian group and A = (a1, . . . , aq) be a
q-tuple of elements ai in G. If A has the property (Pr,s) for some r, s with q \geq r > s > 1, then
there exist i1, . . . , iq - r+2 with 1 \leq i1 < . . . < iq - r+2 \leq q such that ai1 = ai2 = . . . = aiq - r+2 .
Lemma 2.3. Let f be a linearly nondegenerate meromorphic mappings of \BbbC n into \BbbP N (\BbbC ). Let
H be a hyperplanes of \BbbP N (\BbbC ), d be a positive integer and k is a positive integer or +\infty with
k \geq d, then
N (d)(r, \nu 0(f,H),\leq k) \geq
k + 1
k + 1 - d
N (d)(r, \nu (f,H)) -
d
k + 1 - d
T (r, f).
Proof. We have
N (d)(r, \nu (f,H),\leq k) = N (d)(r, \nu (f,H)) - N (d)(r, \nu (f,H),>k) \geq
\geq N (d)(r, \nu (f,H)) -
d
k + 1
N(r, \nu 0(f,H),>k) =
= N (d)(r, \nu (f,H)) -
d
k + 1
\bigl(
N(r, \nu (f,H)) - N(r, \nu (f,H),\leq k)
\bigr)
\geq
\geq N (d)(r, \nu (f,H)) -
d
k + 1
T (r, f) +
d
k + 1
N (d)(r, \nu (f,H),\leq k).
This implies that
N (d)(r, \nu (f,H),\leq k) \geq
k + 1
k + 1 - d
N (d)(r, \nu (f,H)) -
d
k + 1 - d
T (r, f).
The lemma is proved.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
UNICITY THEOREMS WITH TRUNCATED MULTIPLICITIES OF MEROMORPHIC MAPPINGS . . . 419
3. Proof of Theorem 1.1. Suppose that f1 \not \equiv f2 . We set T (r) = T (r, f1) + T (r, f2).
For two index i, j \in \{ 1, . . . , 2N + 2\} , we set
Pij = (f1, Hi)(f2, Hj) - (f1, Hj)(f2, Hi).
By Jensen’s formula, we easily see that
NPij (r) \leq
\int
S(r)
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| (f1, Hi)(f2, Hj) - (f1, Hj)(f2, Hi)
\bigm| \bigm| \sigma n \leq
\leq
\int
S(r)
\mathrm{l}\mathrm{o}\mathrm{g} \| f1\| \sigma n +
\int
S(r)
\mathrm{l}\mathrm{o}\mathrm{g} \| f2\| \sigma n +O(1) = T (r) +O(1). (3.1)
Claim 3.1. Assume that Pij \not \equiv 0. Then we have
\sum
v=i,j
N
\bigl(
r,\mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Hv),\leq k, \nu (f2,Hv),\leq k
\bigr\} \bigr)
+
2N+2\sum
v=1
v \not =i,j
N
(1)
(fs,Hv),\leq k(r) \leq T (r) +O(1), s = 1, 2.
In fact, we will prove the above inequality for s = 1. We see that for each
z \in \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} \nu (f1,Hv),\leq k, v = i, j, one has
\nu Pij (z) \geq \mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Hv),\leq k(z), \nu (f2,Hv),\leq k(z)
\bigr\}
.
Also, for each z \in
\bigcup
v \not =i,j
1\leq v\leq 2N+2
\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} \nu (f1,Hv),\leq k, since f(z) = g(z) we get Pij(z) = 0 and then
\nu Pij (z) \geq
2N+2\sum
v \not =i,j
v=1
N
(1)
(f1,Hv),\leq k(z).
Therefore, we obtain
\nu Pij (z) \geq \mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Hv),\leq k(z), \nu (f2,Hv),\leq k(z)
\bigr\}
+
2N+2\sum
v \not =i,j
v=1
N
(1)
(f1,Hv),\leq k(z),
for all z \in \BbbC n outside an analytic set of codimension two. By integrating both sides of the above
inequality, we have
\sum
v=i,j
N
\bigl(
r,\mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Hv),\leq k, \nu (f2,Hv),\leq k
\bigr\} \bigr)
+
2N+2\sum
v=1
v \not =i,j
N
(1)
(f1,Hv)
(r) \leq NPij (r).
Combining this inequality and (3.1), we obtain
\sum
v=i,j
N
\bigl(
r,\mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Hv),\leq k, \nu (f2,Hv),\leq k
\bigr\} \bigr)
+
2N+2\sum
v=1
v \not =i,j
N
(1)
(f1,Hv)
(r) \leq T (r) +O(1).
This proves the claim.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
420 PHAM HOANG HA, SI DUC QUANG
For each i \in \{ 1, . . . , 2N + 2\} , we define the divisor \mu i as follows:
\mu i(z) =
\left\{ 1 if \nu (f1,Hi)(z) \not = \nu (f2,Hi)(z),
0 for otherwise.
For each z \in \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} (\nu (f1,Hi)) \cup \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} (\nu (f1,Hj)), we easily see that
if z \in \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} (\nu (f1,Hi),\leq k) \cup \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} (\nu (f1,Hj),\leq k), then by the assumption (a) of the theorem, for
each s = 1, 2, we have
\mu i(z) \leq \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu (f1,Hi),\leq k(z), \nu (f2,Hi),\leq k(z)\} -
- \nu
(N)
(f1,Hi),\leq k(z) - \nu
(N)
(f2,Hi),\leq k(z) +N\nu
(1)
(fs,Hi),\leq k(z);
otherwise z \in \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} (\nu (f1,Hi),>k) \cup \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} (\nu (f1,Hj),>k), then
\mu i(z) \leq \nu
(1)
(f1,Hi),>k(z) + \nu
(1)
(f2,Hi),>k(z).
Therefore, we have
\mu i \leq \mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Hi),\leq k, \nu (f2,Hi),\leq k
\bigr\}
- \nu
(N)
(f1,Hi),\leq k -
- \nu
(N)
(f2,Hi),\leq k +N\nu
(1)
(fs,Hi),\leq k + \nu
(1)
(f1,Hi),>k + \nu
(1)
(f2,Hi),>k
outside an analytic set of codimension two. This yields that
N(r, \mu i) \leq N
\bigl(
r,\mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Hi),\leq k, \nu (f2,Hi),\leq k
\bigr\} \bigr)
- N
(N)
(f1,Hi),\leq k(r) - N
(N)
(f2,Hi),\leq k(r)+
+NN
(1)
(fs,Hi),\leq k(r) +N
(1)
(f1,Hi),>k(r) +N
(1)
(f2,Hi),>k(r). (3.2)
Claim 3.2:
2N+2\sum
i=1
N(r, \mu i) \leq
(N + 1)(N + 8)
2(k + 1 - N)
.
Indeed, by changing indices if necessary, we may assume that
(f1, H1)
(f2, H1)
\equiv (f1, H2)
(f2, H2)
\equiv . . . \equiv (f1, Hk1)
(f2, Hk1)\underbrace{} \underbrace{}
group 1
\not \equiv (f1, Hk1+1)
(f2, Hk1+1)
\equiv . . . \equiv (f1, Hk2)
(f2, Hk2)\underbrace{} \underbrace{}
group 2
\not \equiv
\not \equiv (f1, Hk2+1)
(f2, Hk2+1)
\equiv . . . \equiv (f1, Hk3)
(f2, Hk3)\underbrace{} \underbrace{}
group 3
\not \equiv . . . \not \equiv
(f1, Hks - 1+1)
(f2, Hks - 1+1)
\equiv . . . \equiv (f1, Hks)
(f2, Hks)\underbrace{} \underbrace{}
group s
,
where ks = 2N + 2.
For each 1 \leq i \leq 2N + 2, we set
\tau (i) =
\left\{ i+N + 1 if i \leq N + 1,
i - N - 1 if i > N + 1.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
UNICITY THEOREMS WITH TRUNCATED MULTIPLICITIES OF MEROMORPHIC MAPPINGS . . . 421
Since f1 \not \equiv f2, the number of elements of every group is at most N . Hence,
(f1, Hi)
(f2, Hi)
and
(f1, H\tau (i))
(f2, H\tau (i))
belong to distinct groups. This means that P\tau (i)i \not \equiv 0, 1 \leq i \leq 2N +2. From Claim 3.1,
we have
T (r) \geq
\sum
v=i,\tau (i)
N
\bigl(
r,\mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Hv),\leq k, \nu (f2,Hv),\leq k
\bigr\} \bigr)
+
2N+2\sum
v=1
v \not =i,\tau (i)
N
(1)
(fs,Hv),\leq k(r) +O(1), s = 1, 2.
Summing-up of both sides of the above inequality over all pairs (i, \tau (i)), by (3.2) and Lemma 2.3,
for each s = 1, 2 we get
(N + 1)T (r) \geq
2N+2\sum
v=1
N
\bigl(
r,\mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Hv),\leq k, \nu (f2,Hv),\leq k
\bigr\} \bigr)
+
+N
2N+2\sum
v=1
N
(1)
(fs,Hv),\leq k(r) +O(1) \geq
\geq
2N+2\sum
v=1
\left( N(r, \mu i) +
\sum
t=1,2
\Bigl(
N
(N)
(ft,Hv),\leq k(r) - N
(1)
(f1,Hv),>k(r)
\Bigr) \right) =
=
2N+2\sum
v=1
\left( N(r, \mu i) +
\sum
t=1,2
\biggl(
N
(N)
(ft,Hv),\leq k(r) -
1
N
N
(N)
(f1,Hv),>k(r)
\biggr) \right) =
=
2N+2\sum
v=1
\left( N(r, \mu i) +
\sum
t=1,2
\biggl(
N + 1
N
N
(N)
(ft,Hv),\leq k(r) -
1
N
N
(N)
(f1,Hv)
(r)
\biggr) \right) =
=
2N+2\sum
v=1
\left( N(r, \mu i) +
\sum
t=1,2
\biggl(
N + 1
N
\biggl(
k + 1
k + 1 - N
N
(N)
(ft,Hv)
(r) - N
k + 1 - N
T (r, ft)
\biggr)
-
- 1
N
N
(N)
(f1,Hv)
(r)
\biggr) \right) \geq
\geq
2N+2\sum
v=1
\left( N(r, \mu i) +
\sum
t=1,2
1
N
\biggl(
(N + 1)(k + 1)
k + 1 - N
- 1
\biggr)
N
(N)
(ft,Hv)
(r) - N
k + 1 - N
T (r)
\right) \geq
\geq (N + 1)
\biggl(
1
N
\biggl(
(N + 1)(k + 1)
k + 1 - N
- 1
\biggr)
- 2N
k + 1 - N
\biggr)
T (r) +
2N+2\sum
v=1
N(r, \mu i).
Thus,
2N+2\sum
i=1
N(r, \mu i) \leq
N2 - 1
k + 1 - N
T (r).
This proves the claim.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
422 PHAM HOANG HA, SI DUC QUANG
Claim 3.3. For i, j \in \{ 1, . . . , 2N + 2\} such that Pij \not \equiv 0, we have
\bigm\| \bigm\| \bigm\| T (r) (1)
\geq NPij (r)
(2)
\geq
\sum
v=i,j
N
\bigl(
r,\mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Hv),\leq k, \nu (f2,Hv),\leq k
\bigr\} \bigr)
+
2N+2\sum
v=1
v \not =i,j
N
(1)
(fs,Hv),\leq k(r)
(3)
\geq
(3)
\geq
\sum
v=i,j
\biggl(
N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f2,Hv),\leq k(r) - NN
(1)
(fs,Hv),\leq k(r)
\biggr)
+
2N+2\sum
v=1
v \not =i,j
N
(1)
(fs,Hv),\leq k(r)
(4)
\geq
(4)
\geq
\biggl(
1 - N(N + 1)
k + 1 - N
\biggr)
T (r) +O(1), s = 1, 2.
Indeed, inequalities (1) and (2) are clear. The third inequality follows from the inequality
\mathrm{m}\mathrm{i}\mathrm{n}\{ a, b\} \geq \mathrm{m}\mathrm{i}\mathrm{n}\{ a, n\} - \mathrm{m}\mathrm{i}\mathrm{n}\{ b, n\} - 1 for two integers a and b. We will prove the last inequality.
Without loss of generality, we may assume that i = 1 and j = \tau (1). Then we obtain
\sum
v=1,\tau (1)
\biggl(
N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f2,Hv),\leq k(r) - NN
(1)
(fs,Hv),\leq k(r)
\biggr)
+
+
2N+2\sum
v=1
v \not =i,j
N
(1)
(fs,Hv),\leq k(r) - T (r) \geq
\geq
N+1\sum
t=1
\Biggl( \sum
v=t,\tau (t)
\biggl(
N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f2,Hv),\leq k(r) - NN
(1)
(fs,Hv),\leq k(r)
\biggr)
+
+
2N+2\sum
v=1
v \not =i,j
N
(1)
(fs,Hv),\leq k(r) - T (r)
\Biggr)
\geq
\geq
2N+2\sum
t=1
\biggl(
N
(N)
(f1,Ht),\leq k(r) +N
(N)
(f2,Ht),\leq k(r)
\biggr)
- (N + 1)T (r) \geq
\geq
\biggl(
(N + 1)(k + 1) - N(2N + 2)
k + 1 - N
\biggr)
T (r) - (N + 1)T (r) =
- N(N + 1)
k + 1 - N
T (r).
This proves the last inequality of the claim.
Assume that Hi = \{ ai0\omega 0 + . . . + aiN\omega N = 0\} . We set hi =
(f1, Hi)
(f2, Hi)
, 1 \leq i \leq 2N + 2.
Then
hi
hj
=
(f1, Hi)(f2, Hj)
(f1, Hj)(f2, Hi)
does not depend on representations of f1 and f2, respectively. Since\sum N
k=0
aikf1k - hi
\sum N
k=0
aikf2k = 0, 1 \leq i \leq 2N + 2, it implies that \mathrm{d}\mathrm{e}\mathrm{t}(ai0, . . . , aiN , ai0hi, . . .
. . . , aiNhi; 1 \leq i \leq 2N + 2) = 0.
For each subset I \subset \{ 1, 2, . . . , 2N + 2\} , put hI =
\prod
i\in I
hi . Denote by \scrI the set of all
combinations I = (i1, . . . , iN+1) with 1 \leq i1 < . . . < iN+1 \leq 2N + 2.
For each I = (i1, . . . , iN+1) \in \scrI , define
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
UNICITY THEOREMS WITH TRUNCATED MULTIPLICITIES OF MEROMORPHIC MAPPINGS . . . 423
AI = ( - 1)
(N+1)(N+2)
2
+i1+...+iN+1 \mathrm{d}\mathrm{e}\mathrm{t}(airl; 1 \leq r \leq N + 1, 0 \leq l \leq N)\times
\times \mathrm{d}\mathrm{e}\mathrm{t}(ajsl; 1 \leq s \leq N + 1, 0 \leq l \leq N),
where J = (j1, . . . , jN+1) \in \scrI such that I \cup J = \{ 1, 2, . . . , 2N + 2\} .
Then
\sum
I\in \scrI
AIhI = 0.
Take I0 \in \scrI . Then
AI0hI0 = -
\sum
I\in \scrI ,I \not =I0
AIhI , i.e., hI0 = -
\sum
I\in \scrI ,I \not =I0
AI
AI0
hI .
Remark that for each I \in \scrI , then
AI
AI0
\not \equiv 0.
Denote by t the minimal number satisfying the following: There exist t elements I1, . . . , It \in
\in \scrI \setminus \{ I0\} and t nonzero constants bi \in \BbbC such that hI0 =
\sum t
i=1
bihIi . It is easy to see that
t \leq
\biggl(
2N + 2
N + 1
\biggr)
- 1.
Since hI0 \not \equiv 0 and by the minimality of t, it follows that the family \{ hI1 , . . . , hIt\} is linearly
independent over \BbbC .
Assume that t \geq 2. Consider the meromorphic mapping h : \BbbC n \rightarrow \BbbP t - 1(\BbbC ) with a reduced
representation h = (dhI1 : . . . : dhIt), where d is meromorphic on \BbbC n . We see that if z is a zero
or pole of some dhIj , then it must be zero or pole of some hi, i.e., \mu i(z) = 1. Then by the second
main theorem, we have\bigm\| \bigm\| \bigm\| T (r, h) \leq t\sum
i=1
N
(t - 1)
dhIi
(r) +N
(t - 1)
dhI0
(r) + o(T (r, h)) \leq
\leq (t - 1)
\Biggl(
t\sum
i=1
N
(1)
dhIi
(r) +N
(1)
dhI0
(r)
\Biggr)
+ o(T (r, h)) \leq
\leq (t - 1)(t+ 1)
2N+2\sum
i=1
N(r, \mu i) + o(T (r, h)) + o(T (r)) \leq
\leq (t - 1)(t+ 1)(N2 - 1)
k + 1 - N
T (r) + o(T (r, h)) + o(T (r)).
This yields that \| T (r, h) = o(T (r)).
On the other hand, we get
3T (r, h) \geq N
(1)
hI1
hI2
- 1
(r) +N
(1)
hI2
hI0
- 1
(r) +N
(1)
hI0
hI1
- 1
(r) +O(1).
Since
hI
hJ
= 1 on the set
\bigcup
j\in ((I\cup J)\setminus (I\cap J))c Ej , where Ej = \{ z \in \BbbC n : \nu (fs,Hj),\leq k(z) > 0\} and
\bigl(
(I1 \cup I2) \setminus (I1 \cap I2)
\bigr) c \cup \bigl( (I2 \cup I0) \setminus (I2 \cap I0)
\bigr) c \cup \bigl( (I0 \cup I1) \setminus (I0 \cap I1)
\bigr) c
=
= \{ 1, . . . , 2N + 2\} , s = 1, 2,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
424 PHAM HOANG HA, SI DUC QUANG
it implies that
N
(1)
hI1
hI2
- 1
(r) +N
(1)
hI2
hI0
- 1
(r) +N
(1)
hI0
hI1
- 1
(r) \geq
2N+2\sum
i=1
N
(1)
(fs,Hi),\leq k(r) \geq
1
N
2N+2\sum
i=1
N
(N)
(fs,Hi),\leq k(r).
Hence, for each s = 1, 2, we obtain\bigm\| \bigm\| \bigm\| 3T (r, h) \geq 1
N
2N+2\sum
i=1
N
(N)
(fs,Hi),\leq k(r) \geq
\geq 1
N(k + 1 - N)
\biggl(
(N + 1)(k + 1) - N(2N + 2)
\biggr)
T (r, fs)+
+o(T (r)) =
(N + 1)(k - 2N + 1)
N(k + 1 - N)
T (r, fs) + o(T (r)).
Therefore, we have
6(t - 1)(t+ 1)(N2 - 1)
k + 1 - N
T (r) \geq (N + 1)(k - 2N + 1)
N(k + 1 - N)
T (r) + o(T (r)).
This implies that
k \leq 6(t - 1)(t+ 1)(N2 - N) + 2N - 1 \leq 6(m - 2)m(N2 - N) + 2N - 1.
This is a contradiction.
Thus, t = 1. Then
hI0
hI1
= \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t} \not = 0. Hence, for each I \in \scrI , there is J \in \scrI \setminus \{ I\}
such that
hI
hJ
= \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t} \not = 0. Consider the free Abelian subgroup generated by the family
\{ [h1], . . . , [h2N+2]\} of the torsion free Abelian group \scrM \ast
n/\BbbC \ast . Then the family \{ [h1], . . . , [h2N+2]\}
has the property P2N+2,N+1 . It implies that there exist 2N + 2 - 2N = 2 elements, without loss of
generality we may assume that they are [h1], [h2], such that [h1] = [h2]. Then
h1
h2
= \tau \in \BbbC \ast .
Suppose that \tau \not = 1. Since for each z \in
\bigcup 2N+2
i=3 \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} \nu (f1,Hi),\leq k \setminus
\bigcup
i=1,2(f
- 1
i (H1) \cup f - 1
i (H2))
we have
h1(z)
h2(z)
= 1, the set
\bigcup 2N+2
i=3 \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} \nu (f1,Hi),\leq k is a subset of
\bigcup
i=1,2(f
- 1
i (H1) \cup f - 1
i (H2)),
and, hence, it is a subset of
\bigcup
i=1,2
\bigl(
\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} \nu (fi,H1),>k \cup \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} \nu (fi,H2),>k
\bigr)
. By Lemma 2.1, we have\bigm\| \bigm\| \bigm\| \bigm\| (N - 1)(k + 1) - 2N2
k + 1 - N
(T (r, f1) + T (r, f2)) \leq
\leq
\sum
s=1,2
2N+2\sum
i=3
N
(N)
(fs,Hi),\leq k(r) + o(T (r)) \leq
\leq
\sum
s=1,2
N
\sum
i=1,2
N
(1)
(fs,Hi),>k + o(T (r)) \leq
\leq
\sum
s=1,2
N
k + 1
\sum
i=1,2
N(fs,Hi),>k + o(T (r)) \leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
UNICITY THEOREMS WITH TRUNCATED MULTIPLICITIES OF MEROMORPHIC MAPPINGS . . . 425
\leq 2N
k + 1
T (r) + o(T (r)).
Letting r - \rightarrow +\infty , we get
(N - 1)(k + 1) - 2N2
k + 1 - N
\leq 2N
k + 1
. This is a contradiction. Thus, \tau = 1,
i.e., h1 = h2 .
Now we consider
P1\tau (1) = P1(N+2) = (f1, H1)(f2, HN+2) - (f2, H1)(f1, HN+2) =
=
(f1, H1)
(f1, H2)
\biggl(
(f1, H2)
(f1, HN+2)
- (f2, H2)
(f2, HN+2)
\biggr)
\not \equiv 0.
For a point z \not \in \cup \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} \nu (f1,H2),>k \cup \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} \nu (f1,HN+2),>k \cup \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} \nu (f2,HN+2),>k, we see that
if z \in \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} \nu (f1,Hv),\leq k for some v \not = 1, N + 1, then P1(N+2)(z) = 0, i.e.,
\nu P1(N+2)
(z) \geq
\sum
v \not =1
\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,Hv),\leq k\} ,
if z \in \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} \nu (fs,H1),\leq k, then z is a zero of P1(N+2) with multiplicity at least \nu (f1,H1),\leq k(z) +
+\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,H1),\leq k\} , hence,
\nu P1(N+2)
(z) \geq \mathrm{m}\mathrm{i}\mathrm{n}\{ N, \nu (f1,H1),\leq k\} (z) + \mathrm{m}\mathrm{i}\mathrm{n}\{ N, \nu (f2,H1),\leq k(z)\} -
- (N - 1)\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,H1),\leq k(z)\} ,
if z \in \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p} \nu (fs,HN+2),\leq k, then z is a zero of P1(N+2) with multiplicity at least
\nu (f1,HN+2),\leq k(z) + \mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,HN+2),\leq k\} , hence,
\nu P1(N+2)
(z) \geq \mathrm{m}\mathrm{i}\mathrm{n}\{ N, \nu (f1,HN+2),\leq k(z)\} +
+\mathrm{m}\mathrm{i}\mathrm{n}\{ N, \nu (f2,HN+2),\leq k(z)\} - N \mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,H1),\leq k(z)\} .
Therefore, this implies that
\nu P1(N+2)
\geq
\sum
v=1,N+2
\bigl(
\mathrm{m}\mathrm{i}\mathrm{n}\{ N, \nu (f1,Hv),\leq k\} +\mathrm{m}\mathrm{i}\mathrm{n}\{ N, \nu (f1,Hv),\leq k\} - N \mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,Hv),\leq k\}
\bigr)
+
+
2N+2\sum
v \not =1,N+2
v=1
\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,Hv),\leq k\} +\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,H1),\leq k\} -
- (N + 1)
\bigl(
\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,H2),>k\} +\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,HN+2),>k\} +\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f2,HN+2),>k\}
\bigr)
.
By integrating both sides of the above inequality, we obtain
T (r) \geq NP1(N+2)
(r) \geq
\sum
v=1,N+2
\Bigl(
N
(N)
(f1,Hv),\leq k(z) +N
(N)
(f2,Hv),\leq k(z) - NN
(1)
(f1,Hv),\leq k(z)
\Bigr)
+
+
2N+2\sum
v \not =1,N+2
v=1
N
(1)
(f1,Hv),\leq k(r) +N
(1)
(f1,H1),\leq k(r) -
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
426 PHAM HOANG HA, SI DUC QUANG
- (N + 1)
\Bigl(
N
(1)
(f1,H2),>k(r) +N
(1)
(f1,HN+2),>k(r) +N
(1)
(f2,HN+2),>k(r)
\Bigr) (by Claim 3.3(4))
\geq
(by Claim 3.3(4))
\geq
\biggl(
1 - N(N + 1)
k + 1 - N
\biggr)
T (r) +N
(1)
(f1,H1),\leq k(r) -
- N + 1
k + 1
\bigl(
T (r, f1) + T (r)
\bigr)
+ o(T (r)).
Similarly, we have
T (r) \geq
\biggl(
1 - N(N + 1)
k + 1 - N
\biggr)
T (r) +N
(1)
(f1,H1),\leq k(r) -
N + 1
k + 1
\bigl(
T (r, f2) + T (r)
\bigr)
+ o(T (r)).
Therefore,
T (r) \geq
\biggl(
1 - N(N + 1)
k + 1 - N
\biggr)
T (r) +N
(1)
(f1,H1),\leq k(r) -
3(N + 1)
2(k + 1)
T (r) + o(T (r)).
Thus, \bigm\| \bigm\| \bigm\| N (1)
(f1,H1),\leq k(r) \leq
\biggl(
N(N + 1)
k + 1 - N
+
3(N + 1)
2(k + 1)
\biggr)
T (r) + o(T (r)). (3.3)
On the other hand, since f1 \not = f2, for each i \not = 1 there exists an index j such that P1j \not \equiv 0 and
Pij \not \equiv 0. Therefore, by Claim 3.3, we easily see that
N(N + 1)
k + 1 - N
T (r) \geq
\sum
v=i,j
N
\bigl(
r,\mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Hv),\leq k, \nu (f2,Hv),\leq k
\bigr\} \bigr)
+
2N+2\sum
v=1
v \not =i,j
N
(1)
(f1,Hv),\leq k(r) -
-
\left( \sum
v=1,j
\Bigl(
N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f2,Hv),\leq k(r) - NN
(1)
(f1,Hv),\leq k(r)
\Bigr)
+
2N+2\sum
v=1
v \not =i,j
N
(1)
(f1,Hv),\leq k(r)
\right) \geq
\geq N
\bigl(
r,\mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Hi),\leq k, \nu (f2,Hi),\leq k
\bigr\} \bigr)
+N
(1)
(f1,Hi),\leq k(r) -
-
\Bigl(
N
(N)
(f1,H1),\leq k(r) +N
(N)
(f2,H1),\leq k(r) - (N + 1)N
(1)
(f1,H1),\leq k(r)
\Bigr)
\geq
\geq 2N
(1)
(f1,Hi),\leq k(r) - (N - 1)N
(1)
(f1,H1),\leq k(r).
This implies that\bigm\| \bigm\| \bigm\| N (1)
(f1,Hi),\leq k(r) \leq
N(N + 1)
2(k + 1 - N)
T (r) +
N - 1
2
N
(1)
(f1,H1),\leq k(r) \leq
\leq
\biggl(
N2(N + 1)
2(k + 1 - N)
+
3(N2 - 1)
4(k + 1)
\biggr)
T (r) + o(T (r)). (3.4)
Now applying Lemma 2.1 and using (3.3) and (3.4), we have
(N + 1)(k + 1) - N(2N + 2)
k + 1 - N
T (r, f1) \leq
2N+1\sum
i=1
N
(N)
(f,Hi),\leq k(r) + o(T (r)) \leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
UNICITY THEOREMS WITH TRUNCATED MULTIPLICITIES OF MEROMORPHIC MAPPINGS . . . 427
\leq
2N+1\sum
i=1
NN
(1)
(f,Hi),\leq k(r) + o(T (r)) \leq
\leq
\biggl(
N(2N + 1)
\biggl(
N2(N + 1)
2(k + 1 - N)
+
3(N2 - 1)
4(k + 1)
\biggr)
+
+
\biggl(
N(N + 1)
k + 1 - N
+
3(N + 1)
2(k + 1)
\biggr) \biggr)
T (r) + o(T (r)) \leq
\leq N(2N + 1)
N2(N + 1)
k + 1 - N
T (r) + o(T (r)).
Similarly, we obtain
(N + 1)(k + 1) - N(2N + 2)
k + 1 - N
T (r, f2) \leq N(2N + 1)
N2(N + 1)
k + 1 - N
T (r) + o(T (r)).
Then
(N + 1)(k + 1) - N(2N + 2)
k + 1 - N
T (r) \leq 2N(2N + 1)
N2(N + 1)
k + 1 - N
T (r) + o(T (r)).
Letting r - \rightarrow +\infty , we get
(N + 1)(k + 1) - N(2N + 2)
k + 1 - N
\leq 2N(2N + 1)
N2(N + 1)
k + 1 - N
.
This implies that
k \leq 4N4 + 2N3 + 2N - 1 \leq 6N4 + 2N - 1 \leq
\leq 6(2N+1 - 2)2N+1 + 2N - 1 \leq 6(m - 2)m+ 2N - 1.
This is a contradiction. Hence, f1 \equiv f2 .
Theorem 1.1 is proved.
4. Proof of Theorem 1.2. Assume that f1, f2 have reduce representation fi = (fi0 : . . . : fiN ),
i = 1, 2. We will use the same notations which are introduced in the proof of Theorem 1.1. Define
I = I(f1) \cup I(f2) \cup 1\leq t<s\leq q
\bigl\{
z \in \BbbC n | \nu (f1,Ht),\leq k(z)\nu (f1,Hs),\leq k(z) > 0
\bigr\}
.
Then I is an analytic set of codimension 2 or empty set. For each i \in \{ 1, . . . , q\} , we set
Ni(r) = N
(N)
(f1,Hi),\leq k(r) +N
(N)
(f1,Hi),\leq k(r) - (N + d)N
(1)
(f1,Hi),\leq k(r).
For each permutation I = (i1, . . . , iq) of \{ 1, . . . , q\} , we define TI the set of all r \in [1,+\infty ) such
that
Ni1(r) \geq Ni2(r) \geq . . . \geq Niq(r).
Then we see that
\bigcup
I TI = [1,+\infty ). Therefore, there exists a permutation, for instance it is I =
= (1, . . . , q) such that
\int
TI
dr = +\infty .
We also remark that with the assumptions of (a) or (b) or (c), one always has k \geq N.
We first prove the assertion (a).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
428 PHAM HOANG HA, SI DUC QUANG
Claim 4.1. If Pij \not \equiv 0, then the following assertion holds:
T (r) \geq
\sum
v=i,j
\Bigl(
N
(N)
(f,Hv),\leq k(r)+N
(N)
(f,Hv),\leq k(r) - (N+d)N
(1)
(f,Hv),\leq k(r)
\Bigr)
+
q\sum
v=1
dN
(1)
(f,Hv),\leq k(r)+o(T (r)).
Indeed, we fix a point z \not \in I satisfying \nu (f1,Ht),\leq k(z) > 0, t \not = i, j. Suppose that f1l(z)f2l(z) =
= 0, 0 \leq l \leq N. Then f2l(z) = 0, 0 \leq l \leq N. This means that z \in I(f2). This is impossible.
Hence, there exists an index l such that f1l(z)f2l(z) \not = 0. This implies that
\scrD \alpha
\biggl(
Pij
f1lf2l
\biggr)
(z) = 0 \forall | \alpha | < d.
Hence, \nu Pij (z) \geq d.
Now we fix a point z \not \in I satisfying \nu (f1,Ht),\leq k(z) > 0, t = i, j, for instance t = i. Then we
easily have the following:
\nu Pij (z) \geq \mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\nu (f1,Ht),\leq k(z), \nu (f2,Ht),\leq k(z)
\bigr\}
\geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
N, \nu (f1,Ht),\leq k(z)
\bigr\}
+\mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
N, \nu (f2,Ht),\leq k(z)
\bigr\}
- N \mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
1, \nu (f1,Ht),\leq k(z)
\bigr\}
.
Therefore, we get
\nu Pij
\sum
v=i,j
\bigl(
\mathrm{m}\mathrm{i}\mathrm{n}\{ N, \nu (f1,Ht),\leq k\} +\mathrm{m}\mathrm{i}\mathrm{n}\{ N, \nu (f2,Ht),\leq k\} -
- (N + d)\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,Ht),\leq k\}
\bigr)
+ d
q\sum
v=1
\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,Hv),\leq k\}
outside an analytic set of codimension two. This implies that
NPij \geq
\sum
v=i,j
\Bigl(
N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f2,Hv),\leq k(r) - (N + d)N
(1)
(f1,Hv),\leq k(r)
\Bigr)
+ d
q\sum
v=1
N
(1)
(f1,Hv),\leq k(r).
By Jensen’s formula, it is also clear that
T (r) \geq NPij (r) + o(T (r)).
This proves the claim.
Now we suppose that there exists an index i \in
\biggl\{
1, . . . ,
\biggl[
q
2
\biggr]
+1
\biggr\}
such that P1i \not \equiv 0. For r \in TI ,
we obtain \bigm\| \bigm\| \bigm\| T (r) \geq \sum
v=1,i
\Bigl(
N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f2,Hv),\leq k(r) -
- (N + d)N
(1)
(f1,Hv),\leq k(r)
\Bigr)
+ d
q\sum
v=1
N
(1)
(f1,Hv),\leq k(r) \geq
\geq 2
[q/2]
2[q/2]\sum
v=1
\Bigl(
N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f2,Hv),\leq k(r) -
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
UNICITY THEOREMS WITH TRUNCATED MULTIPLICITIES OF MEROMORPHIC MAPPINGS . . . 429
- (N + d)N
(1)
(f1,Hv),\leq k(r)
\Bigr)
+ d
q\sum
v=1
N
(1)
(f1,Hv),\leq k(r) \geq
\geq 2
q
q\sum
v=1
\Bigl(
N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f2,Hv),\leq k(r) -
- (N + d)N
(1)
(f1,Hv),\leq k(r)
\Bigr)
+ d
q\sum
v=1
N
(1)
(f1,Hv),\leq k(r) =
=
2
q
q\sum
v=1
\Bigl(
N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f2,Hv),\leq k(r)
\Bigr)
+
+
\biggl(
d - 2(N + d)
q
\biggr) q\sum
v=1
N
(1)
(f1,Hv),\leq k(r) \geq
\geq
\biggl(
2
q
+
d
2N
- N + d
Nq
\biggr) q\sum
v=1
(N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f2,Hv),\leq k(r)) \geq
\geq
\biggl(
2
q
+
d
2N
- N + d
Nq
\biggr) \biggl(
(q - N - 1)(k + 1) - Nq
k + 1 - N
\biggr)
T (r) + o(T (r)) =
=
dq + 2N - 2d
2Nq
(q - N - 1)k - Nq + q - N - 1
k + 1 - N
T (r) + o(T (r)).
Letting r - \rightarrow +\infty (r \in TI), we get
1 \geq dq + 2N - 2d
2Nq
(q - N - 1)k - Nq + q - N - 1
k + 1 - N
.
Thus,
k(dq(q - N - 3) - 2(N + 1)(N - d)) \leq d(N - 1)(q - 2)q + (qd+ 2N - 2d)(N + 1).
By the assumption of the theorem we see that dq(q - N - 3) - 2(N + 1)(N - d) > 0, and then the
above inequality yields that
k \leq d(N - 1)(q - 2)q + (qd+ 2N - 2d)(N + 1)
dq(q - N - 3) - 2(N + 1)(N - d)
.
This is a contradiction. Thus, there does not exist the index i such that P1i \not \equiv 0 with i \leq [q/2] + 1.
Therefore, we have
(f1, H1)
(f2, H1)
= . . . =
(f1, H[q/2]+1)
(f2, h[q/2]+1)
.
The assertion (a) is proved.
We prove the assertion (b). As the first part, we use the same notations.
Claim 4.2. If Pij \not \equiv 0, then the following assertion holds:
T (r) \geq
\Bigl(
N
(N)
(f1,Hi),\leq k(r) +N
(N)
(f2,Hi),\leq k(r) - (N + d)N
(1)
(f1,Hi),\leq k(r)
\Bigr)
+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
430 PHAM HOANG HA, SI DUC QUANG
+
q\sum
v=1
dN
(1)
(f,Hv),\leq k(r) + o(T (r)).
As in the proof of the first assertion, we fix z \not \in I satisfying \nu (f1,Ht),\leq k(z) > 0, t \not = i. Then there
exists an index l such that f1l(z)f2l(z) \not = 0. This implies that
\scrD \alpha
\biggl(
Pij
f1lf2l
\biggr)
(z) = 0 \forall | \alpha | < d.
Hence, \nu Pij (z) \geq d.
For a point z \not \in I satisfying \nu (f1,Hi),\leq k(z) > 0, we also get
\nu Pij (z) \geq \mathrm{m}\mathrm{i}\mathrm{n}\{ N, \nu (f1,Hi),\leq k(z)\} +\mathrm{m}\mathrm{i}\mathrm{n}\{ N, \nu (f2,Hi),\leq k(z)\} - N \mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,Hi),\leq k(z)\} .
Therefore, we have
\nu Pij \geq \mathrm{m}\mathrm{i}\mathrm{n}\{ N, \nu (f1,Hi),\leq k\} +\mathrm{m}\mathrm{i}\mathrm{n}\{ N, \nu (f2,Hi),\leq k\} -
- (N + d)\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,Hi),\leq k\} + d
q\sum
v=1
\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,Hv),\leq k\}
outside an analytic set of codimension two. This implies that
NPij \geq N
(N)
(f1,H1),\leq k(r) +N
(N)
(f2,H1),\leq k(r) - (N + d)N
(1)
(f1,H1),\leq k(r) + d
q\sum
v=1
N
(1)
(f1,Hv),\leq k(r).
By Jensen’s formula, we get
T (r) \geq NPij (r) + o(T (r)).
This proves the claim.
Now we suppose that f1 \not = f2, then there exists an index i \not = 1 such that P1i \not \equiv 0. For r \in TI ,
we obtain\bigm\| \bigm\| \bigm\| T (r) \geq \Bigl( N (N)
(f1,H1),\leq k(r) +N
(N)
(f2,H1),\leq k(r) - (N + d)N
(1)
(f1,H1),\leq k(r)
\Bigr)
+ d
q\sum
v=1
N
(1)
(f1,Hv),\leq k(r) \geq
\geq 1
q
q\sum
v=1
\Bigl(
N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f2,Hv),\leq k(r) - (N + d)N
(1)
(f1,Hv),\leq k(r)
\Bigr)
+ d
q\sum
v=1
N
(1)
(f1,Hv),\leq k(r) =
=
1
q
q\sum
v=1
\Bigl(
N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f2,Hv),\leq k(r)
\Bigr)
+
\biggl(
d - N + d
q
\biggr) q\sum
v=1
N
(1)
(f1,Hv),\leq k(r) \geq
\geq
\biggl(
1
q
+
d
2N
- N + d
2Nq
\biggr) q\sum
v=1
(N
(N)
(f1,Hv),\leq k(r) +N
(N)
(f1,Hv),\leq k(r)) \geq
\geq
\biggl(
1
q
+
d
2N
- N + d
2Nq
\biggr) \biggl(
(q - N - 1)(k + 1) - Nq
k + 1 - N
\biggr)
T (r) + o(T (r)) =
=
dq +N - d
2Nq
(q - N - 1)k - Nq + q - N - 1
k + 1 - N
T (r) + o(T (r)).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
UNICITY THEOREMS WITH TRUNCATED MULTIPLICITIES OF MEROMORPHIC MAPPINGS . . . 431
Letting r - \rightarrow +\infty (r \in TI), we have
1 \geq dq +N - d
2Nq
(q - N - 1)k - Nq + q - N - 1
k + 1 - N
.
Thus,
k(dq2 - (N + 1)(d+ 1)q - N2 + 1) \leq q(N - 1)(dq - N - 1) + (dq +N - 1)(N + 1).
By the assumption of the theorem, we see that dq2 - (N + 1)(d+ 1)q - N2 + 1 > 0, and then the
above inequality yields that
k \leq q(N - 1)(dq - N - 1) + (dq +N - 1)(N + 1)
dq2 - (N + 1)(d+ 1)q - N2 + 1
.
This is a contradiction. Thus, f1 = f2 . The assertion (b) is proved.
We prove the assertion (c). Suppose that f1 \not = f2, then there exist two index i, j \in \{ 0, . . . , N\}
such that P = f1if2j - f1jf2i \not \equiv 0.
We fix a point z \not \in I with \nu (f1,Hv)(z),\leq k > 0. Then there exists an index l such that f1l(z)f2l(z) \not =
\not = 0. This implies that
\scrD \alpha
\biggl(
P
f1lf2l
\biggr)
(z) = 0 \forall | \alpha | < N.
Hence, \nu P (z) \geq d. Therefore, we have
\nu P \geq d
q\sum
v=1
\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu (f1,Hv),\leq k\}
outside an analytic set of codimension two. By Lemma 2.3, this implies that
NPij \geq d
q\sum
v=1
N
(1)
(f1,Hv),\leq k(r) \geq
d
N
q\sum
v=1
N
(N)
(f1,Hv),\leq k(r) \geq
d
N
(q - N - 1)(k + 1) - Nq
k + 1 - N
T (r, f1).
By Jensen’s formula and the above inequality, we have
T (r) \geq NPij (r) + o(T (r)) \geq d
N
(q - N - 1)(k + 1) - Nq
k + 1 - N
T (r, f1) + o(T (r)).
Similarly, we get
T (r) \geq d
N
(q - N - 1)(k + 1) - Nq
k + 1 - N
T (r, f2) + o(T (r)).
Therefore,
T (r) \geq d
2N
(q - N - 1)(k + 1) - Nq
k + 1 - N
T (r) + o(T (r)).
Letting r - \rightarrow +\infty (r \in TI), we get
1 \geq d
2N
(q - N - 1)(k + 1) - Nq
k + 1 - N
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
432 PHAM HOANG HA, SI DUC QUANG
Thus,
k(d(q - N - 1) - 2N) \leq d(Nq - q +N + 1) - 2N2 + 2N.
By the assumption of the theorem, we see that d(q - N - 1) - 2N > 0, and then the above inequality
yields that
k \leq d(Nq - q +N + 1) - 2N2 + 2N
d(q - N - 1) - 2N
.
This is a contradiction. Thus, f1 = f2 . The assertion (c) is proved.
Acknowledgements. This work is a part of the doctoral thesis of Pham Hoang Ha at Hanoi
National University of Education, written under a joint work with Si Duc Quang with some improve-
ments. Pham Hoang Ha would like to thank the university for the support he received.
References
1. Aihara Y. Finiteness theorem for meromorphic mappings // Osaka J. Math. – 1998. – 35. – P. 593 – 616.
2. Chen Z., Yan Q. Uniqueness theorem of meromorphic mappings from \BbbC n into \BbbP N (\BbbC ) sharing 2N + 3 hyperplanes
in \BbbP N (\BbbC ) regardless of multiplicities // Intern. J. Math. – 2009. – 20. – P. 717 – 726.
3. Dethloff G., Tan T. V. Uniqueness theorems for meromorphic mappings with few hyperplanes // Bull. Sci. Math. –
2009. – 133. – P. 501 – 514.
4. Fujimoto H. The uniqueness problem of meromorphic maps into the complex projective space // Nagoya Math. J. –
1975. – 58. – P. 1 – 23.
5. Fujimoto H. Uniqueness problem with truncated multiplicities in value distribution theory // Nagoya Math. J. –
1998. – 152. – P. 131 – 152.
6. Cao H. Z., Cao T. B. Uniqueness problem for meromorphic mappings in several complex variables with few
hyperplanes // Acta Math. Sinica (English Ser.). – 2015. – 31. – P. 1327 – 1338.
7. Ji S. Uniqueness problem without multiplicities in value distribution theory // Pacif. J. Math. – 1988. – 135. –
P. 323 – 348.
8. Nevanlinna R. Einige Eideutigkeitssäte in der Theorie der meromorphen Funktionen // Acta Math. – 1926. – 48. –
P. 367 – 391.
9. Noguchi J., Ochiai T. Introduction to geometric function theory in several complex variables // Transl. Math. Monogr. –
Providence, Rhode Island: Amer. Math. Soc., 1990. – 80.
10. Quang S. D. Unicity problem of meromorphic mappings sharing few hyperplanes // Ann. Polon. Math. – 2011. –
102. – P. 255 – 270.
11. Smiley L. Geometric conditions for unicity of holomorphic curves // Contemp. Math. – 1983. – 25. – P. 149 – 154.
12. Stoll W. Introduction to value distribution theory of meromorphic maps // Lect. Notes Math. – 1982. – 950. –
P. 210 – 359.
13. Stoll W. Value distribution theory for meromorphic maps // Aspects Math. E. – 1985. – 7.
14. Thai D. D., Quang S. D. Uniqueness problem with truncated multiplicities of meromorphic mappings in several
complex variables // Intern. J. Math. – 2006. – 17. – P. 1223 – 1257.
15. Pham H. H., Quang S. D., Do D. T. Unicity theorems with truncated multi-plicities of meromorphic mappings
in several complex variables sharing small identical sets for moving targets // Intern. J. Math. – 2010. – 21. –
P. 1095 – 1120.
16. Tu Z.-H. Uniqueness problem of meromorphic mappings in several complex variables for moving targets // Tohoku
Math. J. – 2002. – 54. – P. 567 – 579.
Received 12.10.14,
after revision — 02.02.19
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
|
| id | umjimathkievua-article-1448 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:35Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a8/dde294147ffbaff3c4d0fc44571b0ca8.pdf |
| spelling | umjimathkievua-article-14482019-12-05T08:55:13Z Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets Теореми єдиностi з обрiзаними кратностями для мероморфних вiдображень за кiлькома змiнними для невеликої кiлькостi об’єктiв Pham, Hoang Ha Si, Duc Quang Фам, Хоанг Га Сі, Дук Куанг The purpose of our paper is twofold. Our first aim is to prove a uniqueness theorem for meromorphic mappings of $C^n$ into $P^N(C)$ sharing $2N + 2$ hyperplanes in the general position with truncated multiplicities, where all common zeros with multiplicities more than a certain number do not need to be counted. Second, we consider the case of mappings sharing less than $2N + 2$ hyperplanes. These results are improvements of some recent results. Робота має двi основнi мети. По-перше, доведено теорему єдиностi для мероморфних вiдображень з $C^n$ в $P^N(C)$, що подiляють $2N+2$ гiперплощини загального положення з обрiзаними кратностями, де всi спiльнi нулi з кратностями, що перевищують деяке число, можна не враховувати. По-друге, розглянуто випадок, коли вiдображення подiляють менше, нiж $2N + 2$ гiперплощини. Отриманi результатi покращують деякi вiдомi новi результати. Institute of Mathematics, NAS of Ukraine 2019-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1448 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 3 (2019); 412-432 Український математичний журнал; Том 71 № 3 (2019); 412-432 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1448/432 Copyright (c) 2019 Pham Hoang Ha; Si Duc Quang |
| spellingShingle | Pham, Hoang Ha Si, Duc Quang Фам, Хоанг Га Сі, Дук Куанг Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets |
| title | Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets |
| title_alt | Теореми єдиностi з обрiзаними кратностями
для мероморфних вiдображень за кiлькома змiнними
для невеликої кiлькостi об’єктiв |
| title_full | Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets |
| title_fullStr | Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets |
| title_full_unstemmed | Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets |
| title_short | Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets |
| title_sort | unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1448 |
| work_keys_str_mv | AT phamhoangha unicitytheoremswithtruncatedmultiplicitiesofmeromorphicmappingsinseveralcomplexvariablesforfewfixedtargets AT siducquang unicitytheoremswithtruncatedmultiplicitiesofmeromorphicmappingsinseveralcomplexvariablesforfewfixedtargets AT famhoangga unicitytheoremswithtruncatedmultiplicitiesofmeromorphicmappingsinseveralcomplexvariablesforfewfixedtargets AT sídukkuang unicitytheoremswithtruncatedmultiplicitiesofmeromorphicmappingsinseveralcomplexvariablesforfewfixedtargets AT phamhoangha teoremiêdinostizobrizanimikratnostâmidlâmeromorfnihvidobraženʹzakilʹkomazminnimidlânevelikoíkilʹkostiobêktiv AT siducquang teoremiêdinostizobrizanimikratnostâmidlâmeromorfnihvidobraženʹzakilʹkomazminnimidlânevelikoíkilʹkostiobêktiv AT famhoangga teoremiêdinostizobrizanimikratnostâmidlâmeromorfnihvidobraženʹzakilʹkomazminnimidlânevelikoíkilʹkostiobêktiv AT sídukkuang teoremiêdinostizobrizanimikratnostâmidlâmeromorfnihvidobraženʹzakilʹkomazminnimidlânevelikoíkilʹkostiobêktiv |