MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF EDLICATION —————————–
VANGTY NOULORVANG
SOME THEORMS ON UNIQUENESS AND
FINITENESS OF MEROMORPHIC MAPPINGS
Specialized: Geometry and Topology Code: 9.46.10.05
SUMMARY OF THESIS DOCTOR MATHEMATICS
Hanoi, 01-2021
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The thesis is completed at: Hanoi University of Education
NScience instructor: Assoc. Prof. PHAM DUC THOAN
Assoc. Prof. PHAM HOANG HA
Rewier 1: Prof. Ha Huy Khoai
Rewier 2: Prof. Ta Thi Hoai An
Rewier 3: Prof. Tran Van Tan
The thesis was defended at the Thesis-level Thesis Judging Coun-
cil meeting at .......................
Thesis can be found at: - Thesis can be found at
-Library of Hanoi University of Educa-
tion
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INTRODUCTION
1. Rationale
Value distribution theory was built by the famous mathematician R. Nevanlinna in the 20 of the last century. Since its inception, this theory has attracted many great mathematicians around the world to study. Many remarkable results and great applications of this theory in different mathematical disciplines have been discovered. The basic content of the value distribution theory is to establish the second main theorem, nhichis about the relationship between the zero counting function and the increase of the charateristic functions. This theorem has many applications in studying the problem of uniqueness, finiteness, algebraic dependence, defect relations as well as the distribution of values of meromorphic mappings.
In order to establish a second main theorem for the meromor- phic mapping from Cm into the piojective space Pn(C), they based on the Logarithmic Derivative Lemma and the property of Wron- skian’s determinant (c-Casorati and p-Casorati) and replaced the Logarithmic Derivative Lemma by a similar lemma, which is called the c-differences or c-differences lemma for a zero-order meromor- phic map or for a meromorphic mapping of hyper-order less than 1 respectively. From there, they were able to study the uniqueness of these such as the general Picard’s theorem. This second main theo- rem is called the second main theorem p-differences or c-differences with targets. Using these approaches, in 2016, T. B. Cao and R. Ko- rhonen have established the second main theorem p-differences for the meromorphic mapping from Cm into the the piojective Pn(C) intersecting the superspattice at the hyperplanes in subgeral posi- tion.
In the one-dimensional case, since R. Halburd and R. Korhonen
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have given the lemma c-differences the secmd main theorem c- differences for the polymorphism function with super order less than 1, unique theorem Picard’s theorem similar to R. Nevanlinna’s 5 point theorem is well studied. There are many interesting results in this direction. For example, in 2009, J. Heittokangas et al. proved that if the f (z) meromorphic function f (z) has finite order shares 3 distinct values that count multiples with the f (z + c), then f is a periodic function with period c, that is, f (z) = f (z + c) for every z ∈ C. This Picard theorem is improved by these authors for the case of sharing two multiples and one non-multiples.
In early 2016, K. S. Charak, R. J. Korhonen and G. Kumar gave counterexamples to show that there is no unique theorem for the case that 1 value sharing counts multiplicities and two shared values without multiplicities. Note that, in R. Nevanlinna’s 5 point theorem, 5 shared values don’t need Count multiplicities. A question arises is there a Picard theorem in the case in the case the number of shared values that are not multiplied is 4. The authors have tried to answer the above question and they have obtained results for a meromorphic map of hypcr-order less than 1 sharing 4 values under one defect condition.
In 2018, W. Lin, X. Lin and A. Wu hnd a counter-example shoued that the result is no longer correct when the multiples of shared val- ues are interrupted. From there, they posed the problem of studying the uniqueness of Picard’s theorem when values are truncated mul- tiplicities. One of the goals when studying the unigcueness problem is to reduce the number of shared values. Accordingly, we pose re- search problems and improve the results of W. Lin, X. Lin and A. Wu.
The problem of the algebraic dependence of the meromorphic
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mappings from Cm to Pn(C) is studied in the paper of S. Ji and so far have announced many results. Some of the best recent results are of Z. Chen and Q. Yan, S. D. Quang, S. D. Quang and L. N. Quynh. Note that, by studying the algebraic dependance of the 3 meromorphic maps, having an inverse image intersecting the 2n + 2 hyperplance in general position, it helped S. D. Quang obtained the finiteness of that class of meromorphic mappings.
However, as noted above, reducing the number of hyperplance in the results is one of the important goals in value distribution theory. Therefore, we set out the purpose of studying the finiteness of the above polymorphic maps with the number of hyperplance less than 2n + 2 through the algebraic dependence of 3 meromorphic mappings.
From the above reasons, we choose the thesis “Some theorems on uniqueness and finiteness of meromorphic mappings" to study in depth the unique problems of the meromorphic mappings and their shifts, as well as the finite problems for the meromorphic mappings.
2. Objectives of research
The first purpose of the thesis is to give and prove some uniaueness theorems of the meromorphic functions f (z) on the complex plane which has hyperorder plane less than 1 share a part of the values with its f (z + c).
Following that, the dissertation establishes some second main the- orems and a some uniquenss Picard theorems for the meromorphic mapping from Cm into projective space Pn(C) intersecting interact with hypersurfaces.
Finally, the dissertation studies the finiteness through establishing the theorem of algebraic dependence of 3 meromorphic mappings from Cm into the projective space Pn(C) intersecting with 2n + 1
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hyperplanes in general position.
3. Object and scope of research
The research object of the dissertation is some uniqueness Picard theorems and the problem of algebraic dependence as well as the finiteness of meromorphic mappings.
The thesis is studied in the scope of Nevanlinna theory for mero-
morphic mapping.
4. Methodology
To solve the problems posed in the dissertation, we use the methods of value distribution theory and complex geometry. Besides using traditional techniques, we come up with new techniques to achieve the goals set out in the thesis.
5. Scientific and practical significances
The dissertation contributes to deepening the results on the uniqueness and finiteness of the meromorphic functions or the mero- morphic mappings. Besides enriching these problems, the disserta- tion also gives new results for the algebraic dependence of 3 mero- morphic mappings in the projective space with few hyperplanes.
The dissertation is a useful reference for students, graduate stu-
dents and postgraduate students in this direction.
6. Structure
The structure of the dissertation consists of four main chapters. Overview Chapter devoted to analyzing some research results related to the content of the topic of domestic and foreign authors. The remaining three chapters present preparation knowledge as well as detailed evidence for the new results of the topic.
Chapter I. Overview.
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Chapter II. The uniqueness for the meromorphic function that
has hyper degree less than 1
Chapter III. The uniqueness for zero ordcr meromorphic map-
ping
Chapter IV.The algebraic degneracy and finiteness of meromor-
phic mappings. The dissertation is based on three published articles.
7. Place of writing the dissertation
Hanoi National University of Education.
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Chapter 1: Overview
I. The uniqueness for the meromorphic function that has hyper order less than 1
Finding the conditions for the meromorphic function f (z) on the complex plane coincides with its shift f (z + c) has been vigorously studied in recent years. Since the work of R. Halburd and R. Korho- nen, there have been many interesting uniqueness theorems similar to Nevanlinna’s 5 point theorem. For example, in 2009, J. Heit- tokangas and colleagues considered this problem for a meromorphic function f (z) with a finite order on complex plane sharing 3 CM values with its shift f (z + c). The results are then improved for the case of sharing two CM values and one IM value by these authors.
In 2016, K. S. Charak, R. Korhonen and G. Kumar gave an example to show that the case of sharing one CM value and two IM values (and thus three IM values) is not happening in general case.
The concept of partial sharing of values of a meromorphic function with hyper order less than 1 was introduced by K. S. Charak, R. Korhonen and G. Kumar. They have a uniquess theorem for a meromorphic function with hyper order less than 1 that shares partialhy four values the IM with its shift under the following defect condition.
Theorem A. Let f be a nonconstant meromorphic function of hyper-order γ(f ) < 1 and c ∈ C \ {0}. Let a1, a2, a3, a4 ∈ ˆS(f ) be four distinct periodic functions with period c. If δ(a, f ) > 0 for some a ∈ ˆS(f ) and
E(aj, f (z)) ⊆ E(aj, f (z + c)), j = 1, 2, 3, 4
then f (z) = f (z + c) for all z ∈ C.
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In 2018, W. Lin, X. Lin and A. Wu [11] obtained a counterexam- ple which showed that Theorem A does not hold when the condition "partially shared values E(aj, f (z)) ⊆ E(aj, f (z + c)), j = 1, 2" is replaced by the condition "truncated partially shared values E≤k(aj, f (c)) ⊆ E≤k(aj, f (z + c)), j = 1, 2 " with a positive in- teger k, even if f (z) and f (z + c) share a3, a4 CM. Then, they introduced the following results under a reduced deficiency assump- tion Θ(0, f ) + Θ(∞, f ) > 2 k+1. An example was also given to show that this condi-tion is necessary and sharp.
Theorem B. Let f be a nonconstant meromorphic function of hyper-order γ(f ) < 1 and c ∈ C \ {0}. Let k1, k2 be two positive integers, and let a1, a2 ∈ S(f )\{0}, a3, a4 ∈ ˆS(f ) be four distinct periodic functions with period c such that f (z) and f (z + c) share a3, a4 CM and
k+1, where k := min{k1, k2}, then
E≤kj(aj, f (z)) ⊆ E≤kj(aj, f (z + c)), j = 1, 2.
If Θ(0, f ) + Θ(∞, f ) > 2 f (z) = f (z + c) for all z ∈ C.
Theorem C. Let f be a nonconstant meromorphic function of hyper-order γ(f ) < 1, Θ(∞, f ) = 1 and c ∈ C \ {0}. Let a1, a2, a3 ∈ S(f ) be three distinct periodic functions with period c such that f (z) and f (z + c) share a3 CM and
E≤k(aj, f (z)) ⊆ E≤k(aj, f (z + c)), j = 1, 2.
If k ≥ 2 then f (z) = f (z + c) for all z ∈ C.
As an application of Theorems B and C, the above authors gave the sufficient conditions for periodicity of meromorphic functions as follows.
Theorem D. Assume that f and g are two nonconstant mero- morphic function with Θ(∞, f ) = Θ(∞, g) = 1, where f has a
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nonzero periodic c ∈ C \ {0} with hyper-order γ(f ) < 1. Let k1, k2 be two positive integers, a1, a2, a3 ∈ S(f ) be three distinct periodic functions with period c such that f and g share a3 CM and
E≤k(aj, f ) ⊆ E≤k(aj, g), j = 1, 2. Then g is a function with periodic T , where T ∈ {c, 2c}, that is g(z) = g(z + T ) for all z ∈ C.
In this thesis the first aim is to generalize and improve Theorems B and C by reducing the number of shared values. The second aim is to give some uniqueness theorems in this direction as well as some of their applications. Namely, we will prove the following results.
Theorem 2.2.1. Let f be a meromorphic function of hyper- order γ(f ) < 1 and let c ∈ C \ {0}. Let a1, a2, a3 ∈ ˆS(f ) be three distinct periodic functions with period c and let k be a positive integer. Assume that f (z) and f (z + c) share partially a1; a2 CM, i.e.,
E(a1, f (z)) ⊆ E(a1, f (z + c)), E(a2, f (z)) ⊆ E(a2, f (z + c))
and
2a3−(a1+a2) } then,
E≤k(a3, f (z)) ⊆ E≤k(a3, f (z + c)). k+1 for some a ∈ ˆS(f ) \ {a3, a3(a1+a2)−2a1a2
If Θ(a, f ) > 2 f (z) = f (z + c) for all z ∈ C.
Corollary 2.2.2. Let f be a nonconstant meromorphic function of hyper-order γ(f ) < 1, Θ(∞, f ) = 1 and c ∈ C \ {0}. Let a1, a2 ∈ S(f ) be two distinct periodic functions with period c such that f (z) and f (z + c) share partially a1 CM, i.e.,
E(a1, f (z)) ⊆ E(a1, f (z + c))
and
E≤k(a2, f (z)) ⊆ E≤k(a2, f (z + c)).
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If k ≥ 2, then f (z) = f (z + c) for all z ∈ C.
In the case k = ∞, we have the following theorem.
Theorem 2.2.3. Let f be a meromorphic function of hyper- order γ(f ) < 1 and let c ∈ C \ {0}. Let a1, a2, a3 ∈ ˆS(f ) be three distinct periodic functions with period c. Assume that f (z) and f (z + c) share partially a1, a2 CM and share partially a3 IM, i.e.,
E(a1, f (z)) ⊆ E(a1, f (z + c)) E(a2, f (z)) ⊆ E(a2, f (z + c))
and
E(a3, f (z)) ⊆ E(a3, f (z + c)). If Θ(a, f ) > 0 for some a ∈ ˆS(f ) \ {a3}, then f (z) = f (z + c) for all z ∈ C.
Omitting the deficiency assumption, we will have the following
results.
Theorem 2.2.4. Let f be a meromorphic function of hyper- order γ(f ) < 1 and let c ∈ C \ {0}. Let k, l be two positive integers and let a1, a2, a3, a4 ∈ ˆS(f ) be four distinct periodic functions with period c. Assume that f (z) and f (z + c) share partially a1, a2 CM and
E≤k(a3, f (z)) ⊆ E≤k(a3, f (z+c)), E≤l(a4, f (z)) ⊆ E≤l(a4, f (z+c)).
= for all z ∈ C. Moreover, the latter occurs only when
. = −a3−a1 a3−a2
Then the following statements hold: (i) If kl > min{k, l} + 2, then f (z) = f (z + c) or f (z)−a1 f (z)−a2 − f (z+c)−a1 f (z+c)−a2 a4−a1 a4−a2 (ii) If max{k, l} = ∞, then f (z) = f (z + c) for all z ∈ C.
Using the idea in the proof of Theorem D, we get a similar result which is considered an application of Theorem 2.2.1 and Corollary 2.2.2.
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Theorem 2.3.1. Assume that f and g are two nonconstant meromorphic func-tions with Θ(∞, f ) = Θ(∞, g) = 1, where f has a nonzero periodic c ∈ C \ {0} with hyper-order γ(f ) < 1. Let k be a positive integer and let a1, a2 ∈ S(f ) be two distinct periodic functions with period c such that f and g share partially a1 CM and
E≤k(a2, f ) ⊆ E≤k(a2, g).
If k ≥ 2, then g g is a function with periodic c, that is g(z) = g(z + c) for all z ∈ C.
By the same argument as in the proof of Theorem 2.3.1, we also
get a result in this form from applying Theorem 2.2.4.
Theorem 2.3.2 Assume that f and g are two nonconstant meromorphic func-tions, where f has a nonzero periodic c ∈ C\{0} with hyper-order γ(f ) < 1. Let k, l be two positive integers and let a1, a2 ∈ S(f ) \ {0} be two distinct periodic functions with period c such that
E(0, f ) ⊆ E(0, g), E(∞, f ) ⊆ E(∞, g)
and
E≤k(a1, f ) ⊆ E≤k(a1, g), E≤l(a2, f ) ⊆ E≤l(a2, g).
Then the following statements hold:
(i) If kl > min{k, l} + 2, then g is a function with periodic T , where T ∈ {c, 2c}, that is g(z) = g(z + T ) for all z ∈ C.
(ii) If max{k, l} = ∞, then g is a function with periodic c, that is g(z) = g(z + c) for all z ∈ C.
II. The uniqueness for zero order meromorphic mapping
In recent years, the Second Main Theorem for meromorphic mappings intersecting hypersurfaces has been investigated by many
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authors such as T. V. Tan and V. V. Truong, M. Ru, S. D. Quang and other authors. For example, in 2004, M. Ru proved a second main theorem for the nondegenerate mappings mappings to Pn(C) intersecting with the hyperface family in general position, this is a breakthrough result. In 2017, S. D. Quang obtaineda second main theorem for the general case of meromorphic mappings intersecting by hypersurfaces ingenrdl position using the Chow weights.
q. Denote by ˜f = (f0 :
· · ·
p (cid:88)
For the purpose of studying the uniqueness or the Picard theorem of the meromorphic maps from Cm into Pn(C) having order 0 intersecting hypersurfaces, we have studied and gave some results for the distribution the q-differences value of the complex-variable meromorphic mapping intersecting with hypersurfaces located in subgcneral position based on the ideas of M. Ru and S. D. Quang. Theorem 3.2.1. Let q = (q1, . . . , qm) ∈ Cm with qj (cid:54)= 0 for all 1 ≤ j ≤ m and let f : Cm → Pn(C) be a meromorphic mapping with zero-order. Assume that f is algebraically nondegenerate over the field φ0 : fn) a reduced (local) representation of f . Let Qj be hypersurfaces of degree dj (1 ≤ j ≤ p) located in N -subgeneral position in Pn(C). Let d be the least common multiple of all dj. Then there exists a large positive u which is divisible by d, such that
i=1
(q − (N − n + 1)(n + 1)) Tf (r) ≤ NQi( ˜f )(r) 1 di
− NCq(f I1,...,f IM )(r) N − n + 1 un+1 (n+1)! + O(un)
+ o (Tf (r))
on a set of logarithmic density 1, where Ij = (ij0, . . . , ijn), (cid:32)
(cid:33) . |Ij| = ij0 + · · · + ijn = u and M = u + d u
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We have the following result similar to the Nochka-Cartan theorem
p (cid:88)
with truncated multiplicities. Theorem 3.2.3. Let q = (q1, . . . , qm) ∈ Cm with qj (cid:54)= 0 for all 1 ≤ j ≤ m and let f : Cm → Pn(C) be a meromorphic mapping with zero-order. Assume that f is algebraically nondegenerate q. Denote by ˜f = (f0 : · · · : fn) a reduced(local) over the field φ0 representation of f. Let Qj be hypersurfaces of degree dj (1 ≤ j ≤ p), located in N -subgeneral position in Pn(C). Let d be the least common multiple of all dj. Then, for every (cid:15) > 0, we have
j=1
(r) + o (Tf (r)) (p − (N − n + 1)(n + 1) − (cid:15)) Tf (r) ≤ ¯N [M0,q] Qj( ˜f ) 1 dj
on a set of logarithmic density 1, where M0 = 4(ed(N − n + 1)(n + 1)2I((cid:15)−1))n − 1.
Here, by the notation I(x) we denote the smallest integer number
which is not smaller than the real number x. Theorem 3.3.1. Let q = (q1, . . . , qm) ∈ Cm with qj (cid:54)= 0, 1 for all j ∈ {1, . . . , m} and let f : Cm → Pn(C) be a zero- order meromorphic mapping. Let Q1, . . . , Qp be hypersurfaces in Pn(C), located in N -subgeneral position with common degree d. (cid:32) (cid:33)
Put M = − 1. Assume that f is forward invariant over n + d n
Qj with respect to the rescaling τq(z) = qz. If p ≥ M + 2N − n + 1 then the image of f is contained in a linear subspace over the (cid:20)
q of dimension ≤
p [ p−N −1 M −n+1]+1
field φ0 (cid:21) . M − n − 1 +
In the case of hyperplanes in subgeneral position in Pn(C), we have M = n. Moreover, when |qi| (cid:54)= 1 for all i ∈ {1, . . . , m}, then f (z) = f (qz) implies that f must be a constant mapping. Immediately, we have the following corollary.
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Corollary 3.3.5. Let f be a zero-order meromorphic mapping of Cm into Pn(C), and let τq(z) = qz, where q = (q1, . . . , qm) ∈ Cm with qj (cid:54)= 0 for all j ∈ {1, . . . , m}. Assume that τq((f, Hj)−1) ⊂ (f, Hj)−1 (counting multiplicity) hold for distinct hyperplanes {Hj}p j=1 in N -subgeneral position in Pn(C). If p > 2N then f (qz) = f (z). In particular, if |qi| (cid:54)= 1 for all i ∈ {1, . . . , m}, then f is constant.
III. The finiteness of meromorphic mappings
The problem of the algebraic dependence of the complex sereral- variable meromorphic mapping on the complex project space for fixed targets was first studied by S. Ji and W. Stoll. After that, their results were developed by many authors such as H. Fujimoto, Z. Chen and Q. Yan, S. D. Quang, S. D. Quang and L. N. Quynh. More specifically, H. Fujimoto introduced the degeneracy theorem for n + 2 meromorphic mappings sharing 2n + 2 hypnplanes with truncated multiplicities to level n(n+1) 2 +n. Recently, S. D. Quang has obtained the algebraic degenerate theorem for three meromorphic maps and used it to give results on the finiteness of meromorphic mapping sharing 2n + 2 hyperplane located in genwral position without multiplicities.
In 2019, S. D. Quang proved the following theorem, in which he did not need to count all zeros with multiples greater than a certain value.
2n+2 (cid:88)
Theorem E Let H1, . . . , H2n+2 be hyperplanes in general position in Pn(C). Assume that
j=1
< . n + 1 n(3n + 1) 1 kj + 1
j=1 , 1), we
Then for three mappings f 1, f 2, f 3 ∈ F(f, {Hj, kj}2n+2
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have f 1 ∧ f 2 ∧ f 3 ≡ 0 on Cm.
j=1, n) and let {Hi}q
In 2015, S. D. Quang and L. N. Quynh found a sucient condition for the algebraic dependence of three meromorphic maps sharing less than 2n + 2 hyperplanes in general position as follows.
√ Theorem F Let f 1, f 2, f 3 ∈ F(f, {Hj}q i=1 be a family of q hyperplanes of Pn(C) in general position. If q > 2n + 5 + then one of the following assertions 28n2 + 20n + 1 4 holds:
3
3]+1
such that (i) There exist (cid:2) q
3]+1
)
3]+1
, = · · · = = ) (f u, Hi2) (f v, Hi2) (f u, Hi1) (f v, Hi1) (cid:3) + 1 hyperplanes Hi1, . . . , Hi[ q (f u, Hi[ q (f v, Hi[ q
(ii) f 1 ∧ f 2 ∧ f 3 ≡ 0 on Cm.
Clearly, to obtain the assertion (ii) in Theorem B, they need to assume that (i) does not occur. The question is whether we can ignore this condition for q < 2n + 2? The first aim of this section is to give a positive answer for the above question. For our purpose, we will rearrange hyperplanes into suitable groups and use the technique “rearranging counting functions” due to D. D. Thai and S. D. Quang as well as consider a new auxiliary function. These help us obtain a complete theorem for the algebraic dependence of three meromorphic sharing 2n + 1 hyperplanes in general position as follows.
Theorem 4.2.1. Let H1, . . . , H2n+1 be hyperplanes in general position in Pn(C) (n ≥ 5). Let f 1, f 2, f 3 : Cm → Pn(C) be
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j=1 , n). If
2n+1 (cid:88)
meromorphic mappings which belong to F(f, {Hj, kj}2n+1
i=1
< , n − 4 2n(2n + 1) 1 ki + 1
then f 1 ∧ f 2 ∧ f 3 ≡ 0 on Cm.
We would like to emphasize that Theorem A plays an essential role in S. D. Quang ’s proof for finiteness of meromorphic mappings. Here is his result.
Theorem G Let f be a linearly nondegenerate meromorphic mappings of Cm into Pn(C). Let H1, . . . , H2n+2 be 2n + 2 hyper- planes of Pn(C) in general position and k1, . . . , kn+2 be positive integers or +∞. Assume that
2n+2 (cid:88)
(cid:27)
i=1
, , . < min (cid:26) n + 1 3n2 + n 5n − 9 24n + 12 n2 − 1 10n2 + 8n 1 ki + 1
i=1 , 1) ≤ 2.
Then (cid:93)F(f, {Hi, ki}2n+2
The following question arises naturally at this moment: By using Quang’s techniques in [7] and by Theorem 1.0.11, do we obtain a result on finiteness for meromorphic mappings sharing 2n + 1 hyperplanes in general position with truncated multiplicities to level n? The second aim of this section is also to give a positive answer for this question. Namely, we have the following
2n+1 (cid:88)
Theorem 4.3.1. Let f be a linearly nondegenerate meromor- phic mappings of Cm into Pn(C). Let H1, . . . , H2n+1 be 2n + 1 hyperplanes of Pn(C) in general position and let k1, . . . , kn+1 be positive integers or +∞ such that
i=1
< . n − 4 2n(2n + 1) 1 ki + 1
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i=1 , n) ≤ 2.
If n ≥ 8 then (cid:93)F(f, {Hi, ki}2n+1
Chapter 2: The uniqueness of the zero rder meromorphic mapping
Chapter 2 is written based on the article [1] in the published works related to the thesis.
2.1 Some preparation knowledge
q (cid:88)
Lemma 2.1.3. Let f be a nonconstant meromorphic function on C. Let a1, a2, . . . , aq (q ≥ 3) be q distinct small meromorphic functions of f on C. Then the following holds
i=1
N (cid:0)r, (q − 2)T (r, f ) ≤ (cid:1) + S(r, f ). 1 f − ai
Lemma 2.1.4. Let f be a nonconstant meromorphic function and c ∈ C. If f is of finite order, then
(cid:19)
m(cid:0)r, (cid:1) = O T (r, f ) (cid:18)log r r f (z + c) f (z)
for all r outside of a subset E zero logarithmic density. If the hyper-order γ(f ) of f is less than one, then for each (cid:15) > 0, we have (cid:19)
m(cid:0)r, (cid:1) = o f (z + c) f (z) (cid:18) T (r, f ) r1−γ(f )−(cid:15)
for all r outside of a subset finite logarithmic measure. Lemma 2.1.5. Let T : R+ → R+ be a non-decreasing continuous function, and let s ∈ (0, +∞) such that hyper-order of T is strictly less than one, i.e.,
r→∞
< 1, γ = lim sup log+ log+ T (r) log r
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Then (cid:19)
, T (r + s) = T (r) + o (cid:18) T (r) r1−γ−(cid:15)
where (cid:15) > 0 and r → ∞ outside a subset of finite logarithmic measure.
For each meromorphic function f , we denote fc(z) = f (z + c) and
q (cid:88)
∆cf := fc − f. Lemma 2.1.6. Let c ∈ C and let f be a meromorphic function of hyper-order γ(f ) < 1 such that ∆cf (cid:54)≡ 0. Let q ≥ 2 and a1(z), . . . , aq(z) be distinct meromorphic periodic small functions of f with period c. Then
k=1
m(cid:0)r, m(r, f ) + (cid:1) ≤ 2T (r, f ) − Npair(r, f ) + S1(r, f ), 1 f − ak
where
(cid:1). Npair(r, f ) = 2N (r, f ) − N (r, ∆cf ) + N (cid:0)r, 1 ∆cf
2.2 The uniqueness for the meromorphic function with hyper order less than 1
In this section, we give and prove the Theorems 1.0.1, 1.0.3, 1.0.4.
2.3 Periodic property of the meromorphic functions with hyper order less than 1
In this section, we give some necessary conditions for a meromor- phic function less than 1 to be periodic. This is an application of the Theorem 1.0.1 and corollary 1.0.2. Specifically, we prove the Theorem 1.0.5, 1.0.6.
Chapter 3: The uniqueness problem for the meromorphic mapping with zero order
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Chapter 3 is written based on the article [2] in the published works
0
related to the thesis.
k+1 (cid:92)
3.1. Some preparation knowledge Lemma 3.1.1. Let q ∈ Cm \ {0}. For a positive integer M , set Iα = {(i0, . . . , in) ∈ Nn+1 : i0 + · · · + in = α}, Ij ∈ Iα for all j ∈ {1, . . . , M }. Then the meromorphic mapping f = (f0 : · · · : fn) : Cm → Pn(C) with zero-order satisfies Cq(f ) = (cid:0)f I1, . . . , f IM (cid:1) ≡ 0 if and only if the functions f0, . . . , fn are Cq alge-braically dependent over the field φ0 q. Lemma 3.1.2. Let Q1, . . . , Qk+1 be hypersurfaces in Pn(C) of the same degree d such that
i=1 Then there exist n hypersurfaces P2, . . . , Pn+1 of the forms
k−n+t (cid:88)
Qi = ∅.
j=2
i=1 Pi = ∅.
Pt = ctjQj, ctj ∈ C, t = 2, . . . , n + 1,
such that (cid:84)n+1 Lemma 3.1.3. Let {Qi}i∈R be a family of hypersurfaces in Pn(C) of the common degree d, and let f be a meromorphic mapping of Cm into Pn(C). Assume that (cid:84) i∈R Qi = ∅. Then there exist positive constants α and β such that
|Qi( ˜f )| ≤ β|| ˜f ||d. α|| ˜f ||d ≤ max i∈R
The following lemma is called the q-difference logarithmic deriva- tive lemma, which is analogous to the logarithmic derivative lemma.
Lemma 3.1.4. Let f be a non-constant zero-order meromorphic mapping of Cm into C and q = (q1, . . . , qm) ∈ Cm with qj (cid:54)= 0 for
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all j, then (cid:18) (cid:19)
m r, = o(T (r, f (z))) f (qz) f (z)
on a set of logarithmic density 1.
3.2 The Second main theorem for q-differences
In this section, we will give and prove Theorem 1.0.7, 1.0.8.
3.3 Picard’s theorem
In this section, we will give and prove Theorems 1.0.9 and Corol-
lary 1.0.10.
To prove Theorem 1.0.9, we need the following main lemma.
j}j∈R ∩ span{Q∗
j is a homogeneous polynomial defining Qj.
(cid:33) Lemma 3.3.2. Let Q1, . . . , Qp be hypersurfaces of the common degree d in Pn(C), located in N -subgeneral position. Put M = (cid:32) (cid:105) + N + 1. If p ≥ M + 2N − n + 1 − 1 and let ˜p = (cid:104) p−N −1 M −n+1 n + d n
· · · :
then exists a subset U ⊂ {1, . . . , p} with |U | ≥ ˜p satisfying the condition: (∗) for every subset R ⊂ U with |R| ≤ ˜p − N − 1, we i }i∈R∗ (cid:54)= {0}, where R∗ = U \ R and have span{Q∗ Q∗ Lemma 3.3.3. Let f : Cm → Pn(C) be a meromorphic mapping with a reduced representation ˜f = (f0 : fn) and let q = (q1, . . . , qm) ∈ Cm with qj (cid:54)= 0 for all j. Assume that σ(f ) = 0 and all zeros of f0, . . . , fnare forward invariant with respect to the (cid:33) (cid:32)
− 1. If rescaling τq(z) = qz. Let d ∈ N∗ and put M = n + d n
q for all i, j ∈ {0, . . . , M } such that Ii (cid:54)= Ij, then f0, . . . , fn are algebraically nondegenerate over the field φ0 q. Lemma 3.3.4. Let f = (f0 : · · · : fn) be a meromorphic mapping of Cm to Pn(C) such that σ(f ) = 0 and all q = (q1, . . . , qm) ∈ Cm
for each d, (cid:54)∈ φ0 f Ii f Ij
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with qj (cid:54)= 0, 1 for all j. Assume that all zeros of f0, . . . , fn are forward invariant with respect to the rescaling τq(z) = qz. Let (cid:32) (cid:33)
d ∈ N∗, put M = − 1. Let S1 ∪ · · · ∪ Sl be a portion of n + d n
{0, . . . , M } formed in such a way that i and j are in the same
q. If f I0 + · · · + f IM = 0 then
∈ φ0 f Ii f Ij
f Ij = 0 for all k ∈ {1, . . . , l}.
class Sk if and only if (cid:80) j∈Sk
Chapter 4: The algebraic degeneracy and finiteness of meromorphic mappings
Chapter 4 is written based on the article [3] in the published works
related to the dissertation.
In this chapter, we prove a theorem of algebraic degeneracy of three meromorphic mappings from Cm to Pn(C) intersecting 2n + 1 hyperplane in general position with truncated to leveln. That said, the intersections with a multiple greater than actually a certain value will not need to be counted. As an application we have a theorem on the finiteness of these meromorphic mappings.
4.1 Some basic properties and additional results in Nevanlinna theory Theorem 4.1.1. [The first main theorem]. Let f : Cm → Pn(C) be a linearly nondegenerate meromorphic mapping and H be a hyperplane in Pn(C). Then
N(f,H)(r) + mf,H(r) = Tf (r), r > 1.
Theorem 4.1.2. [The second main theorem]. Let f : Cm → Pn(C) be a linearly nondegenerate meromorphic mapping and
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(f,Hi)(r) + o(Tf (r)).
i=1
H1, . . . , Hq be hyperplanes in general position in Pn(C). Then q (cid:88) N [n] ||(q − n − 1)Tf (r) ≤
F , 1
G, 1
H
(cid:1) = 0 for Lemma 4.1.3. If Φα(F, G, H) = 0 and Φα (cid:0) 1
all α with |α| ≤ 1, then one of the following assertions holds:
F are all constants.
H and H
G, G
i=1, 1).
(i) F = G, G = H or H = F . (ii) F
Theorem 4.1.4. Let f 1, f 2, f 3 be three mappings in F(f, {Hi, ki}q Suppose that there exist s, t, l ∈ {1, . . . , q} such that
P := (cid:54)≡ 0.
(f 1, Hs) (f 1, Ht) (f 1, Hl) (f 2, Hs) (f 2, Ht) (f 2, Hl) (f 3, Hs) (f 3, Ht) (f 3, Hl) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Then we have
(f 1,Hi),≤ki
i=s,t,l q (cid:88)
(cid:88) (r)) T (r) ≥ NP (r) ≥ {ν(f u,Hi),≤ki}) − N [1] (N (r, min 1≤u≤3
(f 1,Hi),≤ki
i=1
u=1 Tf u(r). 1
i=1, n)
n , then every g ∈ F(f, {Hi, ki}q
ki+1 < n−1
N [1] (r), + 2
where T (r) = (cid:80)3 Lemma 4.1.5. If (cid:80)2n+1 i=1 is linearly nondegenerate and
||Tg(r) = O(Tf (r)) và ||Tf (r) = O(Tg(r)).
4.2 The algerbraic degeneracy of thrre meromorphic mappings sharing 2n + 1 hypreplanes
In this section, we prove Theorem 1.0.11. However, we need the
following Lemma.
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Lemma 4.2.2. Let q, N be two integers satisfying q ≥ 2N + 2, N ≥ 2 and q be even. Let {a1, a2, . . . , aq} be a family of vectors in a 3-dimensional vector space such that rank{aj}j∈R = 2 for any subset R ⊂ Q = {1, . . . , q} with cardinality (cid:93)R = N + 1. Then there exists a partition (cid:83)q/2 j=1 Ij of {1, . . . , q} satisfying (cid:93)Ij = 2 and rank{ai}i∈Ij = 2 for all j = 1, . . . , q/2. 4.3 The finitenees of meromorphic mappings sharing 2n + 1 hypreplanes
In this section, we prove Theorem 1.0.12. However, we need the
following Lemma.
Lemma 4.3.2. With the assumption of Theorem 1.0.12, let h and g be two elements of the family F(f, {Hi, ki}2n+1 i=1 , n). If there exists a constant λ and two indices i, j such that
i=1 , n),
= λ (h, Hi) (h, Hj) (g, Hi) (g, Hj)
then λ = 1. Lemma 4.3.3. Let f 1, f 2, f 3 be three elements of F(f, {Hi, ki}2n+1 where ki (1 ≤ i ≤ 2n + 1) are positive integers or +∞. Suppose that f 1 ∧ f 2 ∧ f 3 ≡ 0 and Vi ∼ Vj for some distinct indices i and j. Then f 1, f 2, f 3 are not distinct.
ij := Φα(F ij Φα
1 , F ij
2 , F ij
3 ) ≡ 0
m with |α| = 1. Then for every
Lemma 4.3.4. With the assumption of Theorem 1.0.12, let f 1, f 2, f 3 be three maps in F(f, {Hi, ki}2n+1 i=1 , n). Suppose that f 1, f 2, f 3 are distinct and there are two indices i, j ∈ {1, 2, . . . , 2n+ 1} (i (cid:54)= j) such that Vi (cid:54)∼= Vj and
for every α = (α1, . . . , αm) ∈ Z+ t ∈ {1, . . . , 2n + 1} \ {i}, the following assertions hold:
it ≡ 0 for all |α ≤ 1|,
(i) Φα
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3 are distinct and
1 , F ti (cid:88)
(ii) if Vi ∼= Vt, then F ti
2 , F ti N [1]
(f,Hs),≤ks
(f,Ht),≤kt
(f,Hi),≤ki
N [1] (r) ≥ (r) − N [1] (r)
s(cid:54)=i,t 3 (cid:88)
(f u,Hs),>ks
u=1
s=i,t
m with |α| = 1 such that Φα
ij ≡ 0. Then we
(cid:88) N [1] −2 (r) + o(T (r)).
Lemma 4.3.5. With the assumption of Theorem 1.0.12, let f 1, f 2, f 3 be three maps in F(f, {Hi, ki}2n+1 i=1 , n). Suppose that f 1, f 2, f 3 are distinct and there are two indices i, j ∈ {1, 2, . . . , 2n+ 1} (i (cid:54)= j) and α ∈ Z+ have
3 (cid:88)
3 (cid:88)
(f,Ht),≤kt
(f u,Hi),≤ki
(f u,Hj),≤kj
u=1
u=1
t=1,t(cid:54)=i,j
(cid:88) N [n] N [n] N [1] T (r) ≥ (r) + (r) + 2 (r)
(f,Hi),≤ki
(f,Hj),≤kj
(r) − (n + 1)N 1 (r) + N (r, νj)
3 (cid:88)
(cid:19) − (2n + 1)N 1 (cid:18)
(f u,Hj),>kj
(f u,Hi),>ki
u=1 + o(T (r)),
(cid:1)N [1] (cid:1)N [1] − + (cid:0)1 + (cid:0)1 + n − 1 3 2n − 2 3
where νj := {z : ki ≥ ν(f u,Hi)(z) = ν(f t,Hi)(z)} for each permuta- tion (u, v, t) of (1, 2, 3).
Conclusions and recommendations
Conclusions The thesis researches some unigueness problems such that, algebraic dependence and finiteness of the meromorphic map- pings. The dissertation has achieved some of the following results: • Prove some uniqueness theorem for the meromorphic function with hyper order less than 1 in the complex plane. • Prove the second main theorem for the meromorphic mapping from
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Cn into the project space Pn(C) of zero order intersecting the hy- peraces in subgeneral position. Applies to wide mining of Picard’s unique theorem for meromorphic mappings intersecting hypersur- faces. • Prove the theorem of the algebraic dependence of three meromor- phic maps from Cn into Pn(C) intersecting the 2n + 1 hyperplanes in general position. Application gives a theorem on the finiteness of those mappings.
Recommendations for the next research
In the process of researching the issues of the dissertation, we think
about some next research directions as follows. • In the dissertation, we have proved the second main theorem and the Picard-’s theorem for the meromorphic mapping of zero order from Cm into Pn(C) intersecting superfacial families. In the near future, we will study how to come up with unique theorems for the mapping of this taxonomy with a family of hyperfaces where the number of participating hyperfaces is smaller. • We continue to work on the Picard-type uniqueness theorems of the meromorphic functions of zero order or the meromorphic func- tions with hyper oeder less than one in the complex plane. • We do research to try to generalize the theorem of algebraic depen- dence and finiteness for them meromorphic maps on more general manifolds, such as Kahler manifolds. We also study these theorems when they join as hyperaces or hyperplanes, but are considered un- der more general terms of multiplicities and meromorphic .
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The published works related to the dissertation.
[1] N. Vangty and P. D. Thoan, On partial value sharing results of meromorphic functions with their shifts and its applications, Bull. Korean Math., 57 (2020), No. 5, 1083-1094.
[2] P. D. Thoan, N. H. Nam and N. Vangty, q-differences the- orems for meromorphic maps of several complex variables intersecting hypersurfaces, Asian-European J. Math., Vol. 14, No. 3 (2021) 2150040 (21 pages).
[3] P. D. Thoan and N. Vangty, Algebraic dependences and finitenness of meromorphic mappings sharing 2n + 1 hy- perplanes with truncated multiplicities, Kodai Math. J., 43 (2020), 504-523.
