Tt tiếng anh một số định lí cơ bản thứ hai và sự phụ thuộc đại số của ánh xạ phân hình vào không gian xạ ảnh phức với mục tiêu di động

Tt tiếng anh một số định lí cơ bản thứ hai và sự phụ thuộc đại số của ánh xạ phân hình vào không gian xạ ảnh phức với mục tiêu di động Tt tiếng anh một số định lí cơ bản thứ hai và sự phụ thuộc đại số của ánh xạ phân hình vào không gian xạ ảnh phức với mục tiêu di động Tt tiếng anh một số định lí cơ bản thứ hai và sự phụ thuộc đại số của ánh xạ phân hình vào không gian xạ ảnh phức với mục tiêu di động Tt tiếng anh một số định lí cơ bản thứ hai và sự phụ thuộc đại số của ánh xạ phân hình vào không gian xạ ảnh phức với mục tiêu di động Tt tiếng anh một số định lí cơ bản thứ hai và sự phụ thuộc đại số của ánh xạ phân hình vào không gian xạ ảnh phức với mục tiêu di động

MINISTRY OF EDUCATION AND TRAINING

HANOI NATIONAL UNIVERSITY OF EDUCATION

This dissertation has been written at Hanoi National University of Education.

Supervisors: Prof Dr Si Duc Quang and Asso Prof Dr Pham Duc Thoan.

Referee 1: Prof Dr Sc Ha Huy Khoai, Thang Long University.

Referee 2: Prof Dr Tran Van Tan, Hanoi National University of Education.

Referee 3: Asso Prof Dr Nguyen Thac Dung, University of Science, Vietnam National versity, Ha Noi.

1 Rationale

In 1925, R Nevanlinna started researching about the value ditribution of morphic functions on C and created a new theory known as value distributiontheory or Nevanlinna theory In this theory, there are two main theorems: theFirst Main Theorem and the Second Main Theorem With these theorems, heproved two famous results about the uniqueness of meromorphic functions known

mero-as Nevalinna’s five and four values theorems Up to now, there are many researchworks to extend these results One of expansions is case where the targets aresmall functions or pairs of small functions One of the best recent publications is

of P Li and Y Zhang They proved that, two non-constant meromorphic tions if have they have the same inverse images, with multiplicities, of three pairs

func-of small functions and have the same inverse images, regardless func-of multiplicity, func-ofother pair of small functions, then linked by a quasi-M¨obius transformation Thisresult does not refer to the case where multiplicities truncated by a level and donot consider the case where all zeros of functions, with multiplicities at a level

do not need to be counted If these cases are solved, it will be further improvedresults for this direction of research

In Nevanlinna theory, the study, improvement and introduction of new forms

of Second Main Theorem is always the main issue of interest to many authors.The Second Main Theorem for meromorphic mappings of many complex variables

is first given by H Cartan in 1933 After that, this result was improved by W.Stoll, M Ru, M Shirosaki, D D Thai and S D Quang Recently, S D Quangintroduced some Second Main Theorems for meromorphic mappings of Cm into

Pn(C) intersecting moving hyperplanes with counting function truncated to level

n or the case where each counting function has a different weight Therefore,

an interesting question is whether it is possible to combine both generalizationdirections to obtain Second Main Theorems for meromorphic mappings of Cm

into Pn(C) and moving hyperplanes where the counting functions are truncatedmultiplicity and have different weights, optimization and more applications.

In another direction, to generalize the results of H Cartan, W Stoll and othermathematicians have studied the replacement Cm by parabolic manifolds Re-cently, by using a technique of Y Liu from Schmidt’s subspace problem of Dio-phantine approximation, Q Yan established Second Main Theorem for movingtargets without using the lemma on logarithmic derivative and the notion of gen-eralized Wronskian of meromorphic mappings However, this result is much weakerthan recent results of S D Quang Therefore, a question arises here whether it ispossible to combine the technique of Q Yan and method of S D Quang to estab-lish the Second Main Theorem for meromorphic mappings class from parabolicmanifolds into Pn(C) with moving hyperplanes and the counting functions aretruncated multiplicity to level n, also extended the results of S D Quang andproven simplification

An important application of the Second Main Theorem is the study of thealgebraic dependence of meromorphic mappings into Pn(C) through their inverseassumptions of moving hyperplanes The first result on algebraic dependence ofmeromorphic mappings in this direction is given by M Ru in 2001 After that,

M Ru’s result improved by the authors P D Thoan, P V Duc and S D Quang,but can see that the number of moving hyperplanes in the assumption of theseresults is quite large From here, it also opens up the problem of improving thetheorems on algebraic dependence of meromorphic mappings such that the num-ber of moving targets participating is reduced as well as consider the more generalsource space is parabolic manifold

From the above reasons, we choose the topic "Some Second main rems and algebraic dependences of meromorphic mappings into thecomplex projective spaces with moving targets ", to construct new forms

theo-of the Second Main Theorems with truncated counting functions for phic mappings optimize the known theorems, and apply the results to study theproperties of the mappings

meromor-2 Purpose of research

The first aim of the thesis is to generalize generalize the results of the theoremsabout two meromorphic functions that share four pairs of small functions The

second aim of the thesis is to improve the Second Main Theorems with weightedcounting functions The last aim of the thesis is to improve the Second MainTheorems for meromorphic mappings from parabolic manifolds into complex pro-jective space n-dimensional Applying the obtained results, the thesis gives someresults for the problem of algebraic dependence of meromorphic mappings in somecases.

3 Object and scope of research

The research object of the thesis is meromorphic mappings from parabolicmanifolds and Cm into Pn(C) Research scope in Value Distribution Theory

4 Research Methods

We based on the research methods, the traditional techniques of ComplexGeometry and Value Distribution Theory, and we added new techniques to solvethe problems posed in the thesis

5 Scientific and practical significances

The thesis contributes to to rich and profound understanding of value tion of meromorphic mappings as well as the relationship between these mappingsunder the condition of the inverse set of the targets The thesis is also one of thereferences for bachelor, master and PhD students in this direction of research

7 The place where the thesis is carried out

This dissertation has been written at Hanoi National University of Education

In the thesis, we research three main problems

Problem 1: Studying the dependency relationship between two morphic functions on C having the same inverse images of four pairs ofsmall functions

mero-Let two meromorphic functions h and g Let values a, b ∈ C Two functions h

and g is said to be having the same inverse images IM (or CM) of a if h − a

and g − a having the same zeros ignore multiples (or counting multiples) Moregeneral, h and g is said to be having the same inverse images IM (or CM) ofpair (a, b) if h − a and g − b having the same zeros ignore multiples (or countingmultiples)

The study of the uniqueness of two meromorphic functions having the sameinverse images of distinct values or pairs of values of interest to many mathemati-cians

The next interesting question is whether we can weaken the conditions of thesame inverse images of targets and would replace targets are values by smallfunctions This brings us to the problem of studying the relationship by quasi-M¨obius transformations between two meromorphic functions that share pairs ofsmall functions IM (or CM) outside a certain set, or say they have the sameinverse images IM∗ (or CM∗)

Definition 1 We say that two meromorphic functions h and g share (a, b) withmultiplicities truncated to level n, or share (a, b) CMn∗ in another word, if

min{n, νh−a0 (z)} = min{n, νg−b0 (z)}

for all z ∈ C outside a discrete set of counting function equal to S(r, h) + S(r, g)

We will say that h and g share (a, b) IM∗ if n = 1 and say that h and g share

(a, b) CM∗ if n = ∞

Definition 2 We say that the meromorphic function h is said to be a M¨obius transformation of g if there exist small (with respect to g) functions

quasi-αi (1 ≤ i ≤ 4) with α1α4 − α2α3 ̸≡ 0 such that h = α1g + α2

of four pairs of small functions then linked by a quasi-M¨obius transformation

In addition, in 2014 S D Quang and L N Quynh considered the case wherethe number of pairs of functions is more than 5 but the condition of sharing theinverse images is made weaker One of the best results available at present wasgiven by P Li and C C Yang as follows

Theorem A Let f and g be non-constant meromorphic functions and ai, bi (i =

1, , 4; ai ̸= aj, bi ̸= bj ∀i ̸= j) be small functions with respect to f and g If

f and g share three pairs (ai, bi), (i = 1, 2, 3) CM∗, and share the fourth pair

(a4, b4) IM∗, then f is a quasi-M¨obius transformation of g

In this thesis, we have improved the above result with counting function cated to a level and showing the relationship between the two mesomorphic func-tions is as follows

trun-Theorem 1.3.6

Let f andg be non-constant meromorphic functions and ai, bi (i = 1, , 4; ai ̸=

aj, bi ̸= bj ∀i ̸= j) be small functions (with respect to f and g) If f and g sharethe pair (a1, b1) IM∗ and share three pairs (ai, bi), (i = 2, 3, 4) CM4∗ then f is aquasi-M¨obius transformation of g Moreover there is a permutation (i1, i2, i3, i4)

In the next theorem, we will consider the case where all zeros of functions f −ai

with multiplicities at least k > 865 do not need to be counted

Theorem 1.3.7 Letf andg be non-constant meromorphic functions andai, bi (i =

1, , 4; ai ̸= aj, bi ̸= bj ∀i ̸= j) be small functions with respect to f and g such

min{νf −a0

i ,≤k(z), 4} = min{νg−b0

i ,≤k(z), 4} (1 ≤ i ≤ 4)

for all z outside a discrete set S of counting function equal to S(r, f ) + S(r, g)

If k > 865 then there is a permutation (i1, i2, i3, i4) of {1, 2, 3, 4} such that

Problem 1 is solved in chapter 1 of the thesis

Problem 2: Studying the Second Main Theorem and algebraic dency of meromorphic mappings from Cm into Pn(C) with the condition

depen-of inverse images depen-of moving targets

In 1926, R Nevanlinna established Second Main Theorem for meromorphicfunctions on C and complex values with counting functions are truncated multi-plicity to level1 After that, in 1933, H Cartan extended this result to the class ofmeromorphic functions from Cm into Pn(C) with moving hyperplanes in generalposition and the counting functions are truncated multiplicity to level n

Theory of the Second Main Theorem for meromorphic functions from Cm into

Pn(C) with moving hyperplanes was started studied by W Stoll, M Ru and M.Shirosaki in 1990’s In that time, almost all given second main theorems do nothave the truncation level for the counting functions of the inverse image of movinghyperplanes In some recent years, this theory have been studied very intesivelywith many results established

M Ru proved the first version of Second Main Theorem with the countingfunctions are truncated multiplicity (to level n) for nondegenerate holomorphicmappings of C into Pn(C) intersecting moving hyperplanes And then, this resultwas extended by D D Thai and S D Quang for meromorphic mappings from

Cm into Pn(C)

Theorem B Let f : Cm → Pn(C) be a meromorphic mapping and let {ai}qi=1(q ≥ 2n + 1) be q meromorphic mappings of Cm into Pn(C)∗ in general positionsuch that (f, ai) ̸≡ 0 (1 ≤ i ≤ q) Then we have

Here, by the notation “|| P ” we mean the assertion P holds for all r ∈ [0, ∞)

outside a Borel subset E of the interval [0, ∞) with REdr < ∞ In 2016, S D.Quang improved these results to the following

Theorem C Let f : Cm → Pn(C) be a meromorphic mapping Let {ai}qi=1(q ≥ 2n − k + 2) be meromorphic mappings of Cm into Pn(C)∗ in general positionsuch that (f, ai) ̸≡ 0 (1 ≤ i ≤ q), where rankR{ai}(f ) = k + 1 Then the followingassertion holds:

(a)

q2n − k + 2Tf(r) ≤

Recently, S D Quang made further improvements his result to the following

Theorem D Let f : Cm → Pn(C) be a meromorphic mapping Let {ai}qi=1(q ≥ 2n − k + 2) be meromorphic mappings of Cm into Pn(C)∗ in general positionsuch that (f, ai) ̸≡ 0 (1 ≤ i ≤ q), where rankR{ai}(f ) = k + 1 Then we have(a)

Theorem E Let f : Cm → Pn(C) be a meromorphic mapping Let {ai}qi=1 (q ≥2n − k + 2) be meromorphic mappings of Cm into Pn(C)∗ in general position suchthat (f, ai) ̸≡ 0 (1 ≤ i ≤ q), where k + 1 = rankR{ai}(f ) Let λ1, , λq be

q positive number with (2n − k + 2) max1≤i≤qλi ≤ Pq

i=1λi Then the followingassertions hold:

Theorem 2.2.2 Let f : Cm → Pn(C) be a meromorphic mapping Let {aj}qj=1(q ≥ 2n − k + 2) be meromorphic mappings of Cm into Pn(C)∗ in general po-sition such that (f, aj) ̸≡ 0 (1 ≤ j ≤ q) and λ1, , λq be q positive num-bers with (2n − k + 2) max1≤j≤qλj ≤ Pq

j=1λj Then for every positive number

η ∈ [max1≤j≤qλj,

Pq j=1λj2n − k + 2], we have

In the last of Problem 2, we used Theorem 2.2.2 to study algebraic dependence

of meromorphic mappings sharing moving hyperplanes regardless of multiplicities

In 2001, M Ru proved the following theorem

Theorem F Let f1, · · · , fλ : Cm → Pn(C) (λ ≥ 2) be nonconstant meromorphicmappings Let aj : Cm → Pn(C)∗ (1 ≤ j ≤ q) be slowly moving hyperplanes ingeneral position Assume that(fi, aj) ̸≡ 0 and (f1, aj)−1{0} = · · · = (fλ, aj)−1{0}

for each 1 ≤ i ≤ λ, 1 ≤ j ≤ q Denote Aj = (f1, aj)−1({0}) Let l be a positiveinteger with 2 ≤ l ≤ λ Assume that for each z ∈ Aj (1 ≤ j ≤ q) and for any

1 ≤ i1 < · · · < il < q, fi1(z) ∧ · · · ∧ fil(z) = 0 If q > dλn

2(2n + 1)

λ − l + 1 , then

f1, · · · , fλ are algebraically dependent over C, i.e., f1 ∧ · · · ∧ fλ ≡ 0 on Cm

After that, the result of M Ru has been improved and extended by P D.Thoan - P V Duc and S D Quang when the number of moving hyperplanes isreduced Namely, for d = 1, the authors show that the condition for the num-ber of hyperplanes is q > λn(2n + 1) − (n − 1)(λ − 1)

λ − l + 1 In the above results, all

intersecting points of the mappings and the moving hyperplanes are considered.And then, H H Giang and L N Quynh consider only the intersecting points ofthe mappings fi and the hyperplanes aj with the mutiplicity not exceed a certainnumber kj < +∞ and have been obtained more general results L N Quynh give

a conditions to f1, · · · , fλ are algebraically dependent over C is

Theorem 2.3.2 Let f1, · · · , fλ : Cm → Pn(C) (λ ≥ 2) be nonconstant morphic mappings Let aj : Cm → Pn(C)∗ (1 ≤ j ≤ q) be slowly moving hyper-planes in general position Assume that (fi, aj) ̸≡ 0 and (f1, aj)−1{0} = · · · =(fλ, aj)−1{0} for each 1 ≤ i ≤ λ, 1 ≤ j ≤ q Denote Aj = (f1, aj)−1({0}).Let l1, , lq be q positive integers with 2 ≤ lj ≤ λ Assume that for each

mero-z ∈ Aj (1 ≤ j ≤ q) and for any 1 ≤ i1 < · · · < ilj < q, fi1(z) ∧ · · · ∧ filj(z) = 0.If

q > dλk(2n − k + 2) − dλ(k − 1) +

Pq j=1lj

then f1 ∧ · · · ∧ fλ ≡ 0 on Cm

We see that, this result is more general than Quynh’s result and improve theresults of P D Thoan, P V Duc and S D Quang

Problem 2 is solved in chapter 2 of the thesis

Problem 3: Studying the Second Main Theorem and algebraic dency of meromorphic mappings from parabolic manifolds into Pn(C)

depen-In this problem, we establish a Second Main Theorem in the case where thesource space is a parabolic manifold Namely, we improve the theorem D above byreplacing Cm by an admissible parabolic manifold of dimension m Recently, byusing a technique from Diophantine approximation of Y Liu, Q Yan extendedthe Second Main Theorem to the case of meromorphic mappings on parabolicmanifolds as follows

Theorem G Let (M, τ ) be an admissible parabolic manifold of dimension m Let

s0 is a fixed positive number Let g1, , gq : M → Pn(C)∗ be meromorphic mapslocated in general position Let f : M → Pn(C) be a nonconstant meromorphicmapping such that Ricτ = o(Tf(r, s)) and logY (r) = o(Tf(r, s)) for r → ∞.Assume that (f, gj) ̸≡ 0 for 1 ≤ j ≤ q and dim(Supp ν(f,g0

i )∩Supp ν0

(f,g j )) ≤ m−2

Pn(C) We have found a way to combine the methods of S D Quang and Q Yan

to avoid using the lemma on logarithmic derivative and the notion of generalizedWronskian of meromorphic mappings, which are very hard to be established inthe case of parabolic manifolds Our result is stated as follows

Theorem 3.2.2 Let (M, τ ) be an admissible parabolic manifold of dimension m

with a majorant function Y (r) Let s0 is a fixed positive number Let f : M →

Pn(C) be a nonconstant meromorphic map such that Ricτ = o(Tf(r, s0)) and

logY (r) = o(Tf(r, s0)) for r → ∞ Let g1, , gq : M → Pn(C)∗ be meromorphicmaps located in general position with q ≥ 2n − k + 2, where rankR{gj}(f ) = k + 1.Assume that (f, gj) ̸≡ 0 for all 1 ≤ j ≤ q and dim(Supp ν(f,g0

al-Theorem 3.3.1 Let (M, τ ) be an admissible parabolic manifold of dimension

m with a majorant function Y (r) Let f1, , fλ : M → Pn(C) be nonconstantmeromorphic maps with Ricτ(r, s) = o(Tft(r, s0)) and logY (r) = o(Tft(r, s))

for r → ∞, 1 ≤ t ≤ λ Let g1, , gq : M → Pn(C)∗ be meromorphic mapslocated in general position and Tgj(r, s0) = o(max1≤t≤λTft(r, s0)) for r → ∞, 1 ≤

j ≤ q Assume that (ft, gj) ̸≡ 0 for 1 ≤ j ≤ q, 1 ≤ t ≤ λ Assume that

Aj = (f1, gj)−1(0) = (f2, gj)−1(0) = · · · = (fλ, gj)−1(0) for each j = 1, , q,and dim(Ai ∩ Aj) ≤ m − 2 for 1 ≤ i ≤ j ≤ q Define A = Sq

j=1Aj Assumethat f1, , fλ are in l− special position on A, where l is an integer with 2 ≤

l ≤ λ Then f1, , fλ are in special position, i.e., f1 ∧ · · · ∧ fλ ≡ 0 on M, if

q > n(n + 2)λ

λ − l + 1.

Our above result extends and improves the previous results for the case of themappings from Cm into Pn(C) Namely, for d = 1 from Theorem 3.3.1 we givethe previous results of M Ru and this result also improves the results of P D.Thoan, P V Duc and S D Quang We would also like to emphasize that, ourresult in Theorem 3.3.1 is an improvement of the result of Q Yan because weused a Second main theorem where has a better upper bound of the characteristicfunction

Problem 3 is solved in chapter 3 of the thesis