On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity
We prove some uniqueness theorems for algebraically nondegenerate holomorphic curves sharing hyper-surfaces ignoring multiplicity.
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| Дата: | 2011 |
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2011
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508700136964096 |
|---|---|
| author | Phuong, Ha Tran Пхуонг, Ха Тран |
| author_facet | Phuong, Ha Tran Пхуонг, Ха Тран |
| author_sort | Phuong, Ha Tran |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:34:55Z |
| description | We prove some uniqueness theorems for algebraically nondegenerate holomorphic curves sharing hyper-surfaces ignoring multiplicity. |
| first_indexed | 2026-03-24T02:29:22Z |
| format | Article |
| fulltext |
UDC 517.5
Ha Tran Phuong (Thai Nguyen Univ. Education, Vietnam)
ON UNIQUENESS THEOREMS FOR HOLOMORPHIC CURVES
SHARING HYPERSURFACES
WITHOUT COUNTING MULTIPLICITY∗∗∗∗
PRO TEOREMY {DYNOSTI DLQ HOLOMORFNYX KRYVYX,
WO ROZDILQGT| HIPERPLOWYNY
BEZ VRAXUVANNQ KRATNOSTI
We prove some uniqueness theorems for algebraically nondegenerate holomorphic curves sharing hyper-
surfaces ignoring multiplicity.
Dovedeno deqki teoremy [dynosti dlq alhebra]çno nevyrodΩenyx holomorfnyx kryvyx, wo roz-
dilqgt\ hiperplowyny bez vraxuvannq kratnosti.
1. Introduction. In 1926, R. Nevanlinna proved that two non-constant meromorphic
maps f, g : C → P C1( ) satisfying f a j
−1( ) = g a j
−1( ) , for a1 , … , a5 ∈ P C1( ) dis-
tinct, we must have f ≡ g. In 1975, H. Fujimoto [5] generalized Nevanlinna’s result to
the case of meromorphic mappings of Cm into P Cn ( ) . Since that time, this problem
has been studied intensively. In this paper we give some uniqueness results for algeb-
raically nondegenerate holomorphic curves sharing sufficiently many nonlinear hyper-
surfaces in a projective space. To state our results, we first introduce some notations.
Let f : C → P Cn ( ) be a holomorphic map, let D be a hypersurface in P Cn ( ) of
degree d and Q be the homogeneous polynomial of degree d in n + 1 variables
with coefficients in C defining D, we define
E D z Q f zf ( ) : ( )= ∈ ={ }C � 0 ignoring multiplicity ,
E D z m Q f z z mf Q f( ) : ( , ) ( ) ( )= ∈ × = ={ }C N � �0 and ord ,
where ordγ ( )z is the order of holomorphic functions γ at z. Let D = D Dq1, ,…{ }
be a collection of hypersurfaces, we define
E E Df f
D
( ) : ( )D
D
=
∈
∪ and E E Df f
D
( ) : ( )D
D
=
∈
∪ .
In 1975, H. Fujimoto [5] proved the following theorem.
Theorem A. Let H = H H n1 3 2, ,…{ }+ be a collection of 3n + 2 hyperplanes in
general position in P Cn ( ) , and f, g : Cm → P Cn ( ) be meromorphic maps such that
f Hm( )C ⊄ and g Hm( )C ⊄ for any H ∈H . If
E H E Hf j g j( ) ( )= for any H j ∈H
then f ≡ g.
By Theorem A, the linearly nondegenerate meromorphic maps are uniquely deter-
mined by 3n + 2 hyperplanes in general position. In the last years, many uniqueness
theorems for holomorphic maps with hyperplanes have been established. For the case
hypersurfaces, in 2008, by using the second main theorem with ramification of An-
Phuong [1], H. T. Phuong [7], M. Dulock and M. Ru [4] proved some uniqueness
∗
Financial support provided to the author by Vietnam’s National Foundation for Science and
Technology Development (NAFOSTED).
© HA TRAN PHUONG, 2011
556 ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4
ON UNIQUENESS THEOREMS FOR HOLOMORPHIC … 557
theorems for algebraically nondegenerate holomorphic curves. Recently, G. Dethloff
and T. V. Tan [3] proved one uniqueness theorem for meromorphic maps in the case
moving hypersurfaces. Our contribution is to give some unicity results for algedraical-
ly nondegenerate holomorphic curves sharing sufficiently many hypersurfaces in gene-
ral position for Veronese embedding.
Now let Dj be a hypersurface of degree d in P Cn ( ) , which is defined by a ho-
mogeneous polynomial Qj of degree d. Then
Q z zj n( , , )0 … = a z zk
i
n
i
k
n
k kn
d
0
0
0 …
=
∑ ,
where ik0 + … + ikn = d for k = 0, 1, … , nd and nd =
n d
n
+
– 1. We denote by
a = ( , , )a and0 … the vector associated with Dj (or with Qj ).
Let D = D Dq1, ,…{ } be a collection of arbitrary hypersurfaces and Qj be the
homogeneous polynomial in C z zn0, ,… of degree d j defining Dj for j =
= 1, … , q. We define the minimal index of degrees of D by
δD : min , ,= …{ }d dq1 .
Let mD be the least common multiple of the d j for j = 1, … , q and denote
nD =
n m
n
+
D
– 1.
For j = 1, … , q, we set Qj
∗ = Qj
m d jD /
and let a j
∗ be the vector associated with Qj
∗ .
The collection D is said to be in general position for Veronese embedding if q > nD
and for any distinct i1 , … , inD +1 ∈ 1, ,…{ }q , the vectors ai1
∗ ,…, a inD +
∗
1
are linearly
independent.
Recall that the collection D = D Dq1, ,…{ } is said to be in N-subgeneral position
if q > N and for any distinct i1 , … , iN+1 ∈ 1, ,…{ }q
Di
k
N
k
=
+
= ∅
1
1
∩ ,
where N be a positive integer such that N ≥ n. It is seen that, for hyperplanes, general
position for Veronese embedding is equivalent to the usual concept of hyperplanes in
general position (namely n-subgeneral position). For hypersurfaces, general position
for Veronese embedding implies nD -subgeneral position.
The following results we obtained in this paper.
Theorem 1.1. Let f and g be algebraically nondegenerate holomorphic curves
from C into P Cn ( ) . Let D = D Dq1, ,…{ } be a collection of q ≥ nD + 2 +
+ 2 2nD D/δ hypersurfaces in general position for Veronese embedding in P Cn ( )
such that f z( ) = g z( ) for all z ∈ E f ( )D ∪ Eg ( )D . Then f ≡ g.
Theorem 1.2. Let f and g be algebraically nondegenerate holomorphic curves
from C into P Cn ( ) . Let D = D Dq1, ,…{ } be a collection of q ≥ nD + 2 +
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4
558 HA TRAN PHUONG
+ 2nD D/δ hypersurfaces in general position for Veronese embedding in P Cn ( )
such that
(a) f z( ) = g z( ) for all z ∈ E f ( )D ∪ Eg ( )D ,
(b) E Df i( ) ∩ E Df j( ) = ∅ and E Dg i( ) ∩ E Dg j( ) = ∅ for all i ≠ j ∈
∈ 1, ,…{ }q .
Then f ≡ g.
Theorems 1.1 and 1.2 are uniqueness theorems for algebraically nondegenerate me-
romorphic curves in the case hypersurface, they have shown the sufficient conditions
for two algebraically nondegenerate meromorphic curves being equivalent. Note that,
if n = 1, hypersurfaces are distinct points, Theorem 1.2 becomes Nevanlinna’s five
theorem. And in Theorem 1.2 too, if mD = 1, δD = 1, nD = n and the number of
hyperplanes is 3n + 2 as in Fujimoto’s result. Furthermore, the number of hypersurfa-
ces in our results is smaller that in Dulock – Ru’s result.
2. Preliminaries from Nevanlinna – Cartan theory. In this section, we introduce
some notations in Nevanlinna – Cartan theory and recall some results, which are
necessary for proofs of our main results.
Let f : C → P Cn ( ) be a holomorphic map and f = ( , , )f fn0 … be a reduced rep-
resentative of f. The Nevanlinna – Cartan characteristic function T rf ( ) is defined by
T rf ( ) =
1
2 0
2
π
θθ
π
log f re di( )∫ ,
where f z( ) = max ( )f z0{ , … , f zn ( ) } . The above definition is independent, up
to an additive constant, of the choice of the reduced representation of f.
Let D be a hypersurface in P Cn ( ) of degree d. Let Q be the homogeneous po-
lynomial of degree d defining D. The proximity function of f is defined by
m r Df ( , ) =
1
2 0
2
π
θ
π θ
θ
log∫
( )
( )
f re
Q f re
d
i d
i�
.
Let n r Df ( , ) be the number of zeros of Q f� in the disk z ≤ r, counting multi-
plicity. For any positive integer k, let n r D kf ( , , )≤ be the number of zeros having
multiplicity ≤ k of Q f� in the disk z ≤ r, counting multiplicity and let
n r D kf ( , , )> be the number of zeros having multiplicity > k of Q f� in the disk
z ≤ r, counting multiplicity. The integrated counting functions are defined by
N r Df ( , ) =
n t D n D
t
dt
f f
r ( , ) ( , )−
∫
0
0
+ n D rf ( , ) log0 ,
N r Df k, ( , )≤ =
n t D k n D k
t
dt
f f
r ( , , ) ( , , )≤ − ≤
∫
0
0
+ n D k rf ( , , ) log0 ≤ ,
N r Df k, ( , )> =
n t D k n D k
t
dt
f f
r ( , , ) ( , , )> − >
∫
0
0
+ n D k rf ( , , ) log0 > .
For any positive integers ∆, k, let n r Df
∆ ( , ) be the number of zeros of Q f� in the
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4
ON UNIQUENESS THEOREMS FOR HOLOMORPHIC … 559
disk z ≤ r, where any zero is counted with multiplicity if its multiplicity is less than
or equal to ∆, and ∆ times otherwise. Let n r D kf
∆ ( , , )≤ (resp. n r D kf
∆ ( , , )> ) be
the number of zeros having multiplicity ≤ k (resp. > k) of Q f� in the disk z ≤ r,
where any zero is counted if its multiplicity is less than or equal to ∆, and ∆ times
otherwise, too. The integrates truncated counting functions are defined by
N r Df
∆ ( , ) =
n t D n D
t
dt
f f
r ∆ ∆( , ) ( , )−
∫
0
0
+ n D rf
∆ ( , ) log0 ,
N r Df k
∆
, ( , )≤ =
n t D k n D k
t
dt
f f
r ∆ ∆( , , ) ( , , )≤ − ≤
∫
0
0
+ n D k rf
∆ ( , , ) log0 ≤ ,
N r Df k
∆
, ( , )> =
n t D k n D k
t
dt
f f
r ∆ ∆( , , ) ( , , )> − >
∫
0
0
+ n D k rf
∆ ( , , ) log0 > .
Next, we will recall the following lemma which show properties of integrated coun-
ting functions. The proof can be found in [7].
Lemma 2.1. With the above notations we have
1) N r Df ( , ) = N r Df k, ( , )≤ + N r Df k, ( , )> ,
2) N r Df
∆ ( , ) = N r Df k, ( , )≤
∆ + N r Df k, ( , )>
∆ ,
3) N r Df
∆ ( , ) ≤ N r Df ( , ) ,
4) N r Df
1 ( , ) ≤ N r Df
∆ ( , ) ≤ ∆N r Df
1 ( , ) ,
5) N r Df k, ( , )≤
1 ≤ N r Df k, ( , )≤
∆ ≤ ∆ N r Df k, ( , )≤
1 ,
6) N r Df k, ( , )>
1 ≤ N r Df k, ( , )>
∆ ≤ ∆ N r Df k, ( , )>
1 ,
7)
1
1k
N r Df k+ ≤, ( , )∆ + N r Df k, ( , )>
∆ ≤
∆
k
N r Df+ 1
( , ) .
First main theorem. Let f : C → P Cn ( ) be a holomorphic map, and D be a hy-
persurface in P Cn ( ) of degree d. If f D( )C ⊄ , then for every real number r with
0 < r < ∞,
m r Df ( , ) + N r Df ( , ) = dT rf ( ) + O( )1 ,
where O( )1 is a constant independent of r.
In 1933, H. Cartan [2] proved the following theorem.
Theorem 2.1. Let f : C → P Cn ( ) be a linearly nondegenerate holomorphic map,
and let H j , 1 ≤ j ≤ q, be hyperplanes in P Cn ( ) in general position. Then
q n T rf− + −( )( ) ( )1 ε ≤ N r Hf
n
j
j
q
( , )
=
∑
1
+ S rf ( ) ,
where S rf ( ) = o T rf ( )( ) and inequality holds for all large r outside a set of finite
Lebesgue measure.
3. Proofs of Theorems 1.1 and 1.2. To prove our theorems we need the following
lemma.
Lemma 3.1. Let f : C → P Cn ( ) be an algebraically nondegenerate holomorphic
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4
560 HA TRAN PHUONG
map and D1 , … , Dq be hypersurfaces in P Cn ( ) of degree d j , j = 1, … , q, such
that the collection D = D Dq1, ,…{ } is in general position for Veronese embedding
in P Cn ( ) . Then for every ε > 0,
q n T rf− − −( )D 1 ε ( ) ≤
1
1 d
N r D
j
f
n
j
j
q
D ( , )
=
∑ + S rf ( ) ,
where inequality holds for all large r outside a set of finite Lebesgue measure.
Proof. Let f = ( , , )f fn0 … be a reduced representative of f, where f0 , … , fn
are entire functions on C without common zeros, and Qj , j ≤ 1 ≤ q , be the
homogeneous polynomials in C[ , , ]z zn0 … of degree d j defining Dj . Of course
we may assume that q ≥ nD + 1.
We first claim that it suffices to prove the lemma in the case that all of the d j are
equal to mD . Indeed, if we have the lemma in that case, then we know that for ε > 0
as in statement of the lemma that
q n T rf− − −( )D 1 ε ( ) ≤
1
1 m
N r Qf
n
j
m d
j
q
j
D
D D( , )
/
=
∑ + S rf ( ) .
Note that if z ∈C is the zero of Q fj � with multiplicity a then z is the zero of
Q fj
m d jD / � with multiplicity a
m
d j
D . This implies that
N r Qf
n
j
m d jD D,
/( ) ≤
m
d
N r Q
j
f
n
j
D D ( , ) .
Hence
q n T rf− − −( )D 1 ε ( ) ≤
1
1 m
N r Qf
n
j
m d
j
q
j
D
D D( , )
/
=
∑ + S rf ( ) ≤
≤
1
1 d
N r Q
j
f
n
j
j
q
D ( , )
=
∑ + S rf ( ) .
Therefore, without loss of generality, we can assume that D1 , … , Dq have a same
degree of mD .
We recall the lexicographic ordering on the n-tuples of natural numbers: let J =
= j1{ , … , jn} , I = i1{ , … , in} ∈ Nn , J < I if and only if for some b ∈ 1{ , … , n}
we have j il l= for l < b and j ib b< . With the n-tuples I = i1{ , … , in} of non-
negative integers, we denote σ( ) :I i jj
= ∑ .
Let ( : : )z zn0 … be a homogeneous coordinates in P Cn ( ) and let
I In0, ,…{ }D
be a set of (n + 1)-tuples such that σ( )I j = mD , j = 0, … , nD , and
I Ii j< for i < j ∈ ∈ 0, ,…{ }nD . For z = ( : : )z zn0 … ∈ P Cn ( ) , we write z I as
z zi
n
in
0
0 … where I = i in0, ,…{ } ∈ I In0, ,…{ }D
. Then we may write the set of mo-
nomials of degree mD as z zI In0 , ,…{ }D .
Denote
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4
ON UNIQUENESS THEOREMS FOR HOLOMORPHIC … 561
�
Dm : P Cn ( ) → P CnD ( )
be the Veronese embedding of degree mD . Let ( : : )w wn0 …
D
be a homogeneous
coordinate in P CnD ( ) . Then �d is given by
�
Dm ( )z = w wm0( ) : : ( )z z…
D( ) , where w j
I j( )z z= , j = 0, … , nD .
Now we set
F = ( : : )F Fn0 …
D
= �
Dm f� ,
then Fj
I j= f , j = 0, … , nD . Then F is a holomorphic map from C to P CnD ( )
and F = ( , , )F Fn0 …
D
is a reduced representative of F. We know from the assump-
tion that f is algebraically nondegenerate hence F is linearly nondegenerate.
For any hypersurface Dj ∈ ( , , )D Dq1 … , let a j = ( , , )a aj jn0 …
D
be the vec-
tor associated with Dj , we set
L j = a wj0 0 + … + a win nD D
.
Then L j is a linear form in P CnD ( ) . Let H j be a hyperplane in P CnD ( ) , which
is made by L j , we say that the hyperplane H j associated with Dj . Hence for the
collection of hypersurfaces ( , , )D Dq1 … in P Cn ( ) , we have the collection of hy-
perplanes ( , , )H Hq1 … of associate hyperplanes in P CnD ( ) . By the assumption
that (D1 , … , Dq ) is in general position for Veronese embedding in P Cn ( ) , we have
that (H1 , … , Hq ) is in general position in P CnD ( ) .
By the definition of F we have for any j = 1, … , q,
a Fj j k
k
n
a F. : .=
=
∑
0
D
= Dj ( )f .
So
N r Df j( , ) = N r HF j( , ) and N r Df
n
j
D ( , ) = N r HF
n
j
D ( , ) . (3.1)
Furthermore by the first main theorem,
T rF ( ) = m T rfD ( ) + O( )1 , S rF ( ) = S rf ( ) . (3.2)
With the assumption of Lemma 3.1, applying Theorem 2.1 to holomorphic map
F : C → P CnD ( ) and the hyperplanes H j , j = 1, … , q, we have
( ) ( )q n T rF− − −D 1 ε ≤ N r HF
n
j
q
j
D
=
∑
1
( , ) + S rF ( ) (3.3)
holds for all large r outside a set of finite Lebesgue measure.
Combining with (3.1), (3.2) and (3.3) together, we have
( ) ( )q n m T rf− − −D D1 ε ≤ N r Df
n
j
q
j
D
=
∑
1
( , ) + S rf ( ) .
This concludes the proof of the lemma.
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4
562 HA TRAN PHUONG
Proof of Theorem 1.1. Assume for the sake contradiction that f � g. Then there
are two numbers, α, β ∈ 0, ,…{ }n , α ≠ β such that f gα β � f gβ α . Let k be a
sufficiently large positive integer, which will be chosen later, and ε be a real number
such that 0 < ε < 1. For any Dj ∈D , for all large r outside a set of finite Lebesgue
measure, by the first main theorem we have
N r Df
n
j
D ( , ) = N r Df k
n
j, ( , )≤
D + N r Df k
n
j, ( , )>
D =
=
k
k
N r Df k
n
j+ ≤1 , ( , )D +
1
1k
N r Df k
n
j+ ≤, ( , )D + N r Df k
n
j, ( , )>
D ≤
≤
k
k
N r Df k
n
j+ ≤1 , ( , )D +
n
k
N r Df k j
D
+ ≤
1
1
, ( , ) + n N r Df k jD , ( , )>
1 ≤
≤
k
k
N r Df k
n
j+ ≤1 , ( , )D +
n
k
N r Df k j
D
+ ≤
1
, ( , ) +
n
k
N r Df k j
D
+ >
1
, ( , ) ≤
≤
k
k
N r Df k
n
j+ ≤1 , ( , )D +
n
k
N r Df j
D
+ 1
( , ) ≤
≤
k
k
N r Df k
n
j+ ≤1 , ( , )D +
d n
k
T r
j
f
D
+ 1
( ) + O( )1 ,
where d j is degree of Dj . So
1
d
N r D
j
f
n
j
D ( , ) ≤
k
d k
N r D
j
f k
n
j
( )
( , ),+ ≤1
D +
n
k
T rf
D
+ 1
( ) + O( )1 .
It implies that
1
1 d
N r D
j
f
n
j
q
j
D
=
∑ ( , ) ≤
k
k d
N r D
jj
q
f k
n
j+ =
≤∑
1
1
1
, ( , )D +
qn
k
T rf
D
+ 1
( ) + O( )1 . (3.4)
From Lemma 3.1, the inequality (3.4) becomes
( – ) ( )q n T rf− −D 1 ε ≤
k
k
N r Df k
n
j
q
j+ ≤
=
∑
1 1
, ( , )D +
qn
k
T rf
D
+ 1
( ) + S rf ( ) ≤
≤
k
k
N r Df k
n
j
q
jδD
D
( )
( , ),+ ≤
=
∑
1 1
+
qn
k
T rf
D
+ 1
( ) + S rf ( ) .
This is equivalent to
q
qn
k
n T rf−
+
− −
D
D
1
1 – ( )ε ≤
k
k
N r Df k
n
j
q
jδD
D
( )
( , ),+ ≤
=
∑
1 1
+ S rf ( ) .
So
q k n n k T rf( ) ( ) ( ) ( )+ − − + + +( )1 1 1D D ε ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4
ON UNIQUENESS THEOREMS FOR HOLOMORPHIC … 563
≤
k
N r Df k
n
j
q
jδD
D
, ( , )≤
=
∑
1
+ S rf ( ) ≤
≤
n k
N r Df k
j
q
j
D
Dδ
, ( , )≤
=
∑ 1
1
+ S rf ( ) . (3.5)
Assume that z0 ∈C is a zero of D fj � with multiplicity less than or equal to k,
then z E f0 ∈ ( )D ∪ Eg ( )D . This implies that g z( )0 = f z( )0 , so
f z
f z
α
β
( )
( )
0
0
=
=
g z
g z
α
β
( )
( )
0
0
, namely z0 is a zero of the function
f
f
α
β
–
g
g
α
β
. Note that by the hypo-
thesis D is in general position for Veronese embedding then there exist at most nD
hypersurfaces Dj in D such that D f zj � ( )0 = 0. This implies that
N r Df k
j
q
j, ( , )≤
=
∑ 1
1
≤ n N rf f g gD α β α β/ / ( )− .
Therefore, by properties of counting function, (3.5) becomes
q k n n k T rf( ) ( ) ( ) ( )+ − − + + +( )1 1 1D D ε ≤
≤
n k
N r S rf f g g f
D
D
2
δ α β α β/ / ( ) ( )− + ≤
n k
T r T rf g
D
D
2
δ
( ) ( )+( ) + S rf ( ) . (3.6)
Similarly for the holomorphic map g, we have
q k n n k T rg( ) ( ) ( ) ( )+ − − + + +( )1 1 1D D ε ≤
≤
n k
T r T rf g
D
D
2
δ
( ) ( )+( ) + S rg ( ) . (3.7)
Adding the inequalities (3.6) and (3.7), we have
q k n n k T r T rf g( ) ( ) ( ) ( ) ( )+ − − + + +( ) +( )1 1 1D D ε ≤
≤
2 2n k
T r T rf g
D
Dδ
( ) ( )+( ) + S rf ( ) + S rr ( ) .
This concludes that
q k n( )+ −1 D – ( ) ( )n kD + + +1 1ε –
2 2n kD
Dδ
≤
S r S r
T r T r
f g
f g
( ) ( )
( ) ( )
+
+
holds for all large r. Let r → ∞, we have
q k n( )+ −1 D – ( ) ( )n kD + + +1 1ε –
2 2n kD
Dδ
≤ 0.
This is equivalent to
k q n nδ ε δD D D D− + + −( )( )1 2 2 + q qn n− − + +( )D D D( )1 ε δ ≤ 0. (3.8)
If we take
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4
564 HA TRAN PHUONG
k >
( )
( )
qn q n
n n
D D D
D D D D
− + + +
− + + −
1
1 2 2
ε δ
δ ε δ
,
then since the hypothesis that q ≥ nD + 2 +
2 2nD
Dδ
, we have a contradiction. Hence
f g f gi j j i≡ for any i ≠ j ∈ 0, ,…{ }n , namely f g≡ . This is the conclusion of the
proof of Theorem 1.1.
Proof of Theorem 1.2. Assume for the sake contradiction that f � g. Then there
are two numbers α, β ∈ 0, ,…{ }n , α ≠ β such that f gα β � f gβ α . Let k be a suf-
ficiently large positive integer, which will be chosen later, and ε be a real number such
that 0 < ε < 1. With the hypothesis in Theorem 1.2 and the proof of Theorem 1.1, we
have
q k n n k T rf( ) ( ) ( ) ( )+ − − + + +( )1 1 1D D ε ≤
≤
n k
N r Df k
j
q
j
D
Dδ
, ( , )≤
=
∑ 1
1
+ S rf ( ) . (3.9)
We know that, if z0 ∈C is a zero of D fj � with multiplicity less than or equal
to k, then z0 will be a zero of the function
f
f
α
β
–
g
g
α
β
. By the hypothesis we have
E Df i( ) ∩ E Df j( ) = ∅
for any pair i ≠ j ∈ 1, ,…{ }q . So if z0 is a zero of D fj � then z0 will be not a
zero of D fj � for all i ≠ j ∈ 1, ,…{ }q . Hence
N r Df k
j
q
j, ( , )≤
=
∑ 1
1
≤ N rf f g g( / ) ( / )( )α β α β− ≤ T rf ( ) + T rg ( ) + O( )1 .
Therefore, (3.9) becomes
q k n n k T rf( ) ( ) ( ) ( )+ − − + + +( )1 1 1D D ε ≤
≤
n k
T r T rf g
D
Dδ
( ) ( )+( ) + S rf ( ) . (3.10)
Similarly for the holomorphic map g, we have
q k n n k T rg( ) ( ) ( ) ( )+ − − + + +( )1 1 1D D ε ≤
≤
n k
T r T rf g
D
Dδ
( ) ( )+( ) + S rg ( ) . (3.11)
Since the inequalities (3.10) and (3.11), we have
q k n n k T r T rf g( ) ( ) ( ) ( ) ( )+ − − + + +( ) +( )1 1 1D D ε ≤
≤
2 n k
T r T rf g
D
Dδ
( ) ( )+( ) + S rf ( ) + S rg ( ) .
Hence
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4
ON UNIQUENESS THEOREMS FOR HOLOMORPHIC … 565
q k n n kδ δ εD D D D( ) ( ) ( )+ − − + + +1 1 1 – 2 n kD ≤
S r S r
T r T r
f g
f g
( ) ( )
( ) ( )
+
+
holds for all large r. Let r → ∞, we have
k q n nδ ε δD D D D− + + −( )( )1 2 + q n n− − + +( )D D D( )1 ε δ ≤ 0.
If we take
k >
( )
( )
qn q n
q n n
D D D
D D D D
− + + +
− + + −
1
1 2
ε δ
δ ε δ
,
then since the hypothesis that q ≥ nD + 2 +
2 nD
Dδ
, we have a contradiction. Hence
f g f gi j j i≡ for any i ≠ j ∈ 0, ,…{ }n , namely f g≡ . This is the conclusion of the
proof of Theorem 1.2.
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thematica (Cluj). – 1993. – 7. – P. 80 – 103.
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Received 28.10.10
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4
|
| id | umjimathkievua-article-2741 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:29:22Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/8d/f958df6c8c3d26c33089907e539b978d.pdf |
| spelling | umjimathkievua-article-27412020-03-18T19:34:55Z On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity Про теореми єдності для голоморфних кривих, що розділяють гіперплощини без врахування кратності Phuong, Ha Tran Пхуонг, Ха Тран We prove some uniqueness theorems for algebraically nondegenerate holomorphic curves sharing hyper-surfaces ignoring multiplicity. Доведено деякі теореми єдності для алгебраічих голоморфних кривих, що розділяють гіперплощини без врахування кратності. Institute of Mathematics, NAS of Ukraine 2011-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2741 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 4 (2011); 556-565 Український математичний журнал; Том 63 № 4 (2011); 556-565 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2741/2238 https://umj.imath.kiev.ua/index.php/umj/article/view/2741/2239 Copyright (c) 2011 Phuong Ha Tran |
| spellingShingle | Phuong, Ha Tran Пхуонг, Ха Тран On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity |
| title | On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity |
| title_alt | Про теореми єдності для голоморфних кривих, що розділяють гіперплощини без врахування кратності |
| title_full | On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity |
| title_fullStr | On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity |
| title_full_unstemmed | On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity |
| title_short | On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity |
| title_sort | on uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2741 |
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