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MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION NGUYEN VAN AN SOME SECOND MAIN THEOREMS AND ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS INTO THE COMPLEX PROJECTIVE SPACES WITH MOVING TARGETS Major: Geometry and Topology Code: 9.46.01.05 SUMMARY OF DOCTORAL THESIS IN MATHEMATICS Ha Noi - 2024 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 Uni- versity, Ha Noi INTRODUCTION 1 Rationale In 1925, R Nevanlinna started researching about the value ditribution of mero- morphic functions on C and created a new theory known as value distribution theory or Nevanlinna theory In this theory, there are two main theorems: the First Main Theorem and the Second Main Theorem With these theorems, he proved two famous results about the uniqueness of meromorphic functions known as Nevalinna’s five and four values theorems Up to now, there are many research works to extend these results One of expansions is case where the targets are small 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 func- tions if have they have the same inverse images, with multiplicities, of three pairs of small functions and have the same inverse images, regardless of multiplicity, of other pair of small functions, then linked by a quasi-M¨obius transformation This result does not refer to the case where multiplicities truncated by a level and do not 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 improved results 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 Quang introduced 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 generalization directions to obtain Second Main Theorems for meromorphic mappings of Cm 1 into Pn(C) and moving hyperplanes where the counting functions are truncated multiplicity and have different weights, optimization and more applications In another direction, to generalize the results of H Cartan, W Stoll and other mathematicians 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 moving targets without using the lemma on logarithmic derivative and the notion of gen- eralized Wronskian of meromorphic mappings However, this result is much weaker than recent results of S D Quang Therefore, a question arises here whether it is possible 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 parabolic manifolds into Pn(C) with moving hyperplanes and the counting functions are truncated multiplicity to level n, also extended the results of S D Quang and proven simplification An important application of the Second Main Theorem is the study of the algebraic dependence of meromorphic mappings into Pn(C) through their inverse assumptions of moving hyperplanes The first result on algebraic dependence of meromorphic 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 these results is quite large From here, it also opens up the problem of improving the theorems on algebraic dependence of meromorphic mappings such that the num- ber of moving targets participating is reduced as well as consider the more general source space is parabolic manifold From the above reasons, we choose the topic "Some Second main theo- rems and algebraic dependences of meromorphic mappings into the complex projective spaces with moving targets ", to construct new forms of the Second Main Theorems with truncated counting functions for meromor- phic mappings optimize the known theorems, and apply the results to study the properties of the mappings 2 Purpose of research The first aim of the thesis is to generalize generalize the results of the theorems about two meromorphic functions that share four pairs of small functions The 2 second aim of the thesis is to improve the Second Main Theorems with weighted counting functions The last aim of the thesis is to improve the Second Main Theorems for meromorphic mappings from parabolic manifolds into complex pro- jective space n-dimensional Applying the obtained results, the thesis gives some results for the problem of algebraic dependence of meromorphic mappings in some cases 3 Object and scope of research The research object of the thesis is meromorphic mappings from parabolic manifolds and Cm into Pn(C) Research scope in Value Distribution Theory 4 Research Methods We based on the research methods, the traditional techniques of Complex Geometry and Value Distribution Theory, and we added new techniques to solve the problems posed in the thesis 5 Scientific and practical significances The thesis contributes to to rich and profound understanding of value distribu- tion of meromorphic mappings as well as the relationship between these mappings under the condition of the inverse set of the targets The thesis is also one of the references for bachelor, master and PhD students in this direction of research 6 Structures of thesis Together with the Introduction; Conclusion and recommendations; Works re- lated to the thesis; References, the thesis includes three chapters: Chapter 1 Two meromorphic functions on the complex plane having the same inverse images of four pairs of small functions Chapter 2 Second Main Theorems for meromorphic mappings of Cm inter- secting moving hyperplanes with weighted counting functions and its applications Chapter 3 The Second Main Theorem for meromorphic mappings intersect- ing moving hyperplanes on parabolic manifolds and its applications The thesis is based on three articles published in international scientific journals (SCIE) 7 The place where the thesis is carried out This dissertation has been written at Hanoi National University of Education 3 OVERVIEW In the thesis, we research three main problems Problem 1: Studying the dependency relationship between two mero- morphic functions on C having the same inverse images of four pairs of small functions 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) More general, h and g is said to be having the same inverse images IM (or CM ) of pair (a, b) if h − a and g − b having the same zeros ignore multiples (or counting multiples) The study of the uniqueness of two meromorphic functions having the same inverse 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 the same inverse images of targets and would replace targets are values by small functions This brings us to the problem of studying the relationship by quasi- Mo¨bius transformations between two meromorphic functions that share pairs of small functions IM (or CM ) outside a certain set, or say they have the same inverse images IM ∗ (or CM ∗) Definition 1 We say that two meromorphic functions h and g share (a, b) with multiplicities truncated to level n, or share (a, b) CMn∗ in another word, if min{n, νh0−a(z)} = min{n, νg0−b(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 quasi- Mo¨bius transformation of g if there exist small (with respect to g) functions 4 αi (1 ≤ i ≤ 4) with α1α4 − α2α3̸ ≡ 0 such that h = α1g + α2 In particular, the α3g + α4 function h is called a Mo¨bius transformation of g if all functions αi (1 ≤ i ≤ 4) are constants An interesting question arises here: “Are there any quasi-Mo¨bius transformation between h and g if they share some pairs of small functions IM ∗ or CM ∗?” In 2003, P C Hu, P Li and C C Yang improved Nevanlinna’s four values theorem to: if two meromorphic functions have the same inverse images CM ∗ 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 where the number of pairs of functions is more than 5 but the condition of sharing the inverse images is made weaker One of the best results available at present was given 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; ai̸ = aj, bi̸ = bj ∀i̸ = 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-Mo¨bius transformation of g In this thesis, we have improved the above result with counting function trun- cated to a level and showing the relationship between the two mesomorphic func- tions is as follows Theorem 1.3.6 Let f and g be non-constant meromorphic functions and ai, bi (i = 1, , 4; ai̸ = aj, bi̸ = bj ∀i̸ = j) be small functions (with respect to f and g) If f and g share the pair (a1, b1) IM ∗ and share three pairs (ai, bi), (i = 2, 3, 4) CM4∗ then f is a quasi-Mo¨bius transformation of g Moreover there is a permutation (i1, i2, i3, i4) of {1, 2, 3, 4} such that f − ai1 · ai3 − ai2 = g − bi1 · bi3 − bi2 or f − ai1 · ai3 − ai2 = g − bi1 · bi4 − bi2 f − ai2 ai3 − ai1 g − bi2 bi3 − bi1 f − ai2 ai3 − ai1 g − bi2 bi4 − bi1 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 Let f and g be non-constant meromorphic functions and ai, bi (i = 1, , 4; ai̸ = aj, bi̸ = bj ∀i̸ = j) be small functions with respect to f and g such 5 that min{νf0−ai,≤k(z), 4} = min{νg0−bi,≤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 f − ai1 · ai3 − ai2 = g − bi1 · bi3 − bi2 or f − ai1 · ai3 − ai2 = g − bi1 · bi4 − bi2 f − ai2 ai3 − ai1 g − bi2 bi3 − bi1 f − ai2 ai3 − ai1 g − bi2 bi4 − bi1 Problem 1 is solved in chapter 1 of the thesis Problem 2: Studying the Second Main Theorem and algebraic depen- dency of meromorphic mappings from Cm into Pn(C) with the condition of inverse images of moving targets In 1926, R Nevanlinna established Second Main Theorem for meromorphic functions on C and complex values with counting functions are truncated multi- plicity to level 1 After that, in 1933, H Cartan extended this result to the class of meromorphic functions from Cm into Pn(C) with moving hyperplanes in general position 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 not have the truncation level for the counting functions of the inverse image of moving hyperplanes In some recent years, this theory have been studied very intesively with many results established M Ru proved the first version of Second Main Theorem with the counting functions are truncated multiplicity (to level n) for nondegenerate holomorphic mappings of C into Pn(C) intersecting moving hyperplanes And then, this result was 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}i=1 q (q ≥ 2n + 1) be q meromorphic mappings of Cm into Pn(C)∗ in general position such that (f, ai)̸ ≡ 0 (1 ≤ i ≤ q) Then we have q q Tf (r) ≤ 2n + 1 [n] N(f,ai)(r) + o(Tf (r)) + O(max Tai(r)) 1≤i≤q i=1 6 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 E dr < ∞ In 2016, S D Quang improved these results to the following Theorem C Let f : Cm → Pn(C) be a meromorphic mapping Let {ai}i=1 q (q ≥ 2n − k + 2) be meromorphic mappings of Cm into Pn(C)∗ in general position such that (f, ai)̸ ≡ 0 (1 ≤ i ≤ q), where rank R{ai}(f ) = k + 1 Then the following assertion holds: q q [k] (a) Tf (r) ≤ N(fi,a)(r) + o(Tf (r)) + O(max Tai(r)), 2n − k + 2 1≤i≤q i=1 q q − (n + 2k − 1) (b) Tf (r) ≤ [k] n+k+1 N(fi,a)(r) + o(Tf (r)) + O(max Tai(r)) 1≤i≤q i=1 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}i=1 q (q ≥ 2n − k + 2) be meromorphic mappings of Cm into Pn(C)∗ in general position such that (f, ai)̸ ≡ 0 (1 ≤ i ≤ q), where rank R{ai}(f ) = k + 1 Then we have q − (n − k) q (a) n+2 Tf (r) ≤ [k] N(f,ai)(r) + o(Tf (r)) + O(max Tai(r)), 1≤i≤q i=1 q − 2(n − k) q (b) k(k + 2) Tf (r) ≤ [1] N(f,ai)(r) + o(Tf (r)) + O(max Tai(r)) 1≤i≤q i=1 In another direction, in 2016, S D Quang initially introduced the second main theorem with weighted counting functions He has generalized partially the above results (the assertion (a) of Theorem A) to the case where each counting function has a different weight His result is stated as follows Theorem E Let f : Cm → Pn(C) be a meromorphic mapping Let {ai}i=1 q (q ≥ 2n − k + 2) be meromorphic mappings of Cm into Pn(C)∗ in general position such that (f, ai)̸ ≡ 0 (1 ≤ i ≤ q), where k + 1 = rank R{ai}(f ) Let λ1, , λq be q positive number with (2n − k + 2) max1≤i≤q λi ≤ i=1 q λi Then the following assertions hold: || i=1 q λi q [k] Tf (r) ≤ λiN(f,ai)(r) + o(Tf (r)) + O(max Tai(r)) 2n − k + 2 1≤i≤q i=1 In thesis, we prove a complete generalization of these above results in this direction 7 Theorem 2.2.2 Let f : Cm → Pn(C) be a meromorphic mapping Let {aj}j=1 q (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≤q λj ≤ j=1 q λj Then for every positive number j=1 q λj η ∈ [max1≤j≤q λj, 2n − k + 2], we have j=1 q λj − (n − k)η Tf (r) ≤ q [k] λjN(f,aj)(r) + o(Tf (r)) + O( max Taj (r)) n+2 j=1 1≤j≤q We see that, this theorem is a generalization of the above theorems through the following specific cases 1) Letting λ1 = · · · = λq = 1 and η = 1, from Theorem 2.2.2, we get assertion a) of Theorem D j=1 q λj 2) Letting η = , from Theorem 2.2.2, we get Theorem E 2n − k + 2 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 meromorphic mappings Let aj : Cm → Pn(C)∗ (1 ≤ j ≤ q) be slowly moving hyperplanes 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 l be a positive integer with 2 ≤ l ≤ λ Assume that for each z ∈ Aj (1 ≤ j ≤ q) and for any dλn2(2n + 1) 1 ≤ i1 < · · · < il < q, fi1(z) ∧ · · · ∧ fil(z) = 0 If q > 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 is reduced 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) In the above results, all λ−l+1 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 of the mappings fi and the hyperplanes aj with the mutiplicity not exceed a certain number kj < +∞ and have been obtained more general results L N Quynh give 8 for 1 ≤ i < j ≤ q If q ≥ 2n + 1, then q q 2n + 1Tf (r, s0) ≤ [n] Nf,gj (r, s0) + O( max Tgj (r, s0)) + o(Tf (r, s0)), j=1 1≤j≤q where “ ≤ ” means that the inequality holds for all r ∈ [s0, +∞] except for a finite Borel measure subset Here, our first purpose is to improve Theorem G and extend Theorem D to the case of meromorphic mappings from an admissible parabolic manifold into 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 generalized Wronskian of meromorphic mappings, which are very hard to be established in the 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 meromorphic maps located in general position with q ≥ 2n − k + 2, where rank R{gj}(f ) = k + 1 Assume that (f, gj)̸ ≡ 0 for all 1 ≤ j ≤ q and dim(Supp ν0(f,gi) ∩ Supp ν0(f,gj)) ≤ m − 2 for 1 ≤ i < j ≤ q Then the following assertions hold: q − (n − k) q (a) n+2 Tf (r, s0) ≤ [k] N(f,gi)(r, s0) + o(Tf (r, s0)) + O(max Tgi(r, s0)), 1≤i≤q i=1 q − 2(n − k) q (b) k(k + 2) Tf (r, s0) ≤ [1] N(f,gi)(r, s0)+o(Tf (r, s0))+O(max Tgi(r, s0)) 1≤i≤q i=1 Theorem 3.2.2(a) is an improvement of Theorem G Also Theorem 5 is a general- ization of Theorem D For the last aim of this problem, we will apply Theorem 3.2.2 to prove an al- gebraic dependence theorem for meromorphic mappings from parabolic manifolds sharing moving hyperplanes of Pn(C) in general position regardless of multiplicity Namely, we will prove the following 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 nonconstant meromorphic 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 maps located in general position and Tgj (r, s0) = o(max1≤t≤λ Tft(r, s0)) for r → ∞, 1 ≤ 10 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 = j=1 q Aj Assume that 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 the mappings from Cm into Pn(C) Namely, for d = 1 from Theorem 3.3.1 we give the 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, our result in Theorem 3.3.1 is an improvement of the result of Q Yan because we used a Second main theorem where has a better upper bound of the characteristic function Problem 3 is solved in chapter 3 of the thesis 11 Chapter 1 TWO MEROMORPHIC FUNCTIONS ON THE COMPLEX PLANE HAVING THE SAME INVERSE IMAGES OF FOUR PAIRS OF SMALL FUNCTIONS in this chapter, we study the pairs of meromorphic functions on complex plane having the same inverse images (IM ∗ or CM ∗) of four pairs of small functions and consider the inverse images that having multiplicities truncated by a certain level With certain assumptions we show that those pairs of functions difference by one Mo¨bius transformation on C Our results in this chapter are Theorem 1.3.6 and Theorem 1.3.7 This chapter is based on the article no [1] in Works related to the thesis 1.1 Some definition and results from Nevanlinna theory for mero- morphic functions In this section, we present again the definitions of proximity function, charac- teristic function of a meromorphic function Then, we present the known results that need to be used in the proof of the main theorems as Lemma on Logarith- mic Derivatives, First and Second Main Theorems for meromorphic functions on complex plane with small functions In 2004, K Yamanoi proved the Second Main Theorem for meromorphic func- tions with small functions where the counting functions are truncated multiplicity to level 1 12 Theorem 1.1.7 Let f be a non-constant meromorphic function on C and a1, , aq (q ≥ 3) be q distinct small (with respect to f ) meromorphic functions on C Then, for each ϵ > 0, the following holds q (q − 2 − ϵ)T (r, f ) ≤ N [1](r, νf−ai 0 ) + S(r, f ) i=1 In 2008,D D Thai and S D Quang gave a Second Main Theorem for mero- morphic mappings from Cm into Pn(C) with moving hyperplanes Theorem 1.1.8 Let f be a non-constant meromorphic function on C Let a1, , aq (q ≥ 3) be q distinct small (with respect to f ) meromorphic functions on C Then the following holds q q || T (r, f ) ≤ [1] 0 3 N (r, νf−ai) + o(T (r, f )) i=1 1.2 Nevanlinna theory for holomorphic curves In this section, we present again some results of Nevanlinna theory for holomorphic curves from C into Pn(C) Theorem 1.2.3 (First Main Theorem) Let f be a holomorphic curve from C into Pn(C) Let H is a hyperplane of Pn(C) such that f (C)̸ ⊂ H Then, we have Tf (r) = Nf (r, H) + mf (r, H) + O(1) Theorem 1.2.6 (Second Main Theorem of H Cartan) Let f : C → Pn(C) be a linearly nondegenerate holomorphic function and let {Hi}i=1 q (q ≥ n + 2) be a set of q hyperplanes in Pn(C) in general position Then q (q − n − 1)Tf (r) ≤ Nf[n](r, Hi) + S(r, f ) i=1 1.3 Two meromorphic functions having the same inverse images of four pairs small functions In this section, we prove the lemmas and propositions about the link between two meromorphic functions that share some pairs of small functions and use these results to prove Theorem 1.3.6 13 Lemma 1.3.2 Let f be a nonconstant meromorphic function and a be a small function (with respect to f ) Then for each positive integer k (k may be +∞) we have N [1](r, νf−a 0 ) ≤ k N≤k [1] (r, νf−a 0 ) + 1 T (r, f ) + S(r, f ) k+1 k+1 Lemma 1.3.3 Let f and g be two meromorphic functions on C Let {ai}3i=1 and {bi}3i=1 be two sets of small (with respect to f ) meromorphic functions on C with ai̸ = aj and bi̸ = bj for all 1 ≤ i < j ≤ 3 Assume that min{1, νf−ai,≤k 0 (z)} = min{1, νg−bi,≤k 0 (z)} (1 ≤ i ≤ 3) for all z outside a discrete subset S of counting function equal to S(r, f ) If k ≥ 3 then || T (r, f ) = O(T (r, g)) and || T (r, g) = O(T (r, f )) In particular, S(r, f ) = S(r, g) Lemma 1.3.4 Let f and g be non-constant meromorphic functions and ai, bi (i = 1, 2, 3, 4) (ai̸ = aj, bi̸ = bj, i̸ = j) be small functions (with respect to f and g) such that min{νf0−ai,≤k(z), 1} = min{νg0−bi,≤k(z), 1} (1 ≤ i ≤ 4) for all z outside a discrete subset S of counting function equal to S(r, f ) + S(r, g) Assume that f is a quasi-Mo¨bius transformation of g If k ≥ 3 then there is a permutation (i1, i2, i3, i4) of (1, 2, 3, 4) such that f − ai1 · ai3 − ai2 = g − bi1 · bi3 − bi2 or f − ai1 · ai3 − ai2 = g − bi1 · bi4 − bi2 f − ai2 ai3 − ai1 g − bi2 bi3 − bi1 f − ai2 ai3 − ai1 g − bi2 bi4 − bi1 The following proposition gives an evaluation for the divisor counting function of two meromorphic functions and the pairs of small functions with respect to them Proposition 1.3.5 Let F and G be non-constant meromorphic functions and Ai, Bi (i = 1, 2, 3) (Ai̸ = Aj, Bi̸ = Bj, i̸ = j) be small functions (with respect to F and G) Assume that F is not a quasi-Mo¨bius transformation of G Then for every positive integer n we have the following inequality N (r, ν) ≤ N [1](r, |νF −A1 0 − νG−B1 0 |) + N [1](r, |νF −A2 0 − νG−B2 0 |) + S(r), where S(r) = o(T (r, F ) + T (r, G)) outside a finite Borel measure set of [1, +∞) and ν is the divisor defined by ν(z) = max {0 , min{ν 0 − A 3 (z ), ν 0G− B3 (z )} − 1} F 14 We prove the first main result about the link between two meromorphic func- tions that have same inverse images of three pairs CM4∗ and another pair IM ∗ Theorem 1.3.6 Let f and g be non-constant meromorphic functions and ai, bi (i = 1, 2, 3, 4) (ai̸ = aj, bi̸ = bj, i̸ = j) be small functions (with respect to f and g) If f and g share the pair (a1, b1) IM ∗ and share three pairs (ai, bi), (i = 2, 3, 4) CM4∗ then f is a quasi-Mo¨bius transformation of g Moreover there is a permutation (i1, i2, i3, i4) of {1, 2, 3, 4} such that f − ai1 · ai3 − ai2 = g − bi1 · bi3 − bi2 or f − ai1 · ai3 − ai2 = g − bi1 · bi4 − bi2 f − ai2 ai3 − ai1 g − bi2 bi3 − bi1 f − ai2 ai3 − ai1 g − bi2 bi4 − bi1 In the next part, we prove a theorem about the link between two meromorphic functions that share four pairs of small functions with multiplicities truncated to level 4, where all zeros with multiplicities at least k > 865 are omitted Theorem 1.3.7 Let f and g be non-constant meromorphic functions and ai, bi (i = 1, 2, 3, 4) (ai̸ = aj, bi̸ = bj, i̸ = j) be small functions with respect to f and g such that min{νf0−ai,≤k(z), 4} = min{νg0−bi,≤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 f − ai1 · ai3 − ai2 = g − bi1 · bi3 − bi2 or f − ai1 · ai3 − ai2 = g − bi1 · bi4 − bi2 f − ai2 ai3 − ai1 g − bi2 bi3 − bi1 f − ai2 ai3 − ai1 g − bi2 bi4 − bi1 15 Chapter 2 SECOND MAIN THEOREMS FOR MEROMORPHIC MAPPINGS ON Cm INTERSECTING MOVING HYPERPLANES WITH WEIGHTED COUNTING FUNCTIONS AND ITS APPLICATIONS In this chapter, we prove a Second main theorem for meromorphic mappings from Cm into Pn(C) with moving targets where the counting functions are truncated multiplicity and have different weights Based on this theorem, we prove the al- gebraic dependence theorem of a family of meromorphic mappings from Cm into Pn(C) Chapter 2 is written based on articles [2] in Works related to the thesis 2.1 Nevanlinna theory for meromorphic mappings on Cm into pro- jective spaces and hyperplanes To prepare for the proof of the two main theorems of this chapter in the follow- ing section, we recall the basic functions in Nevanlinna theory for meromorphic mappings from Cm into Pn(C) Then we introduce First main theorem for those mappings with moving targets Theorem 2.1.5 Let f be a meromorphic mapping from Cm into Pn(C) and a be a meromorphic mapping from Cm into Pn(C)∗ such that (f, a)̸ ≡ 0 Then Tf (r) + Ta(r) = mf,a(r) + N (r, ν(f,a)) + O(1) (r > 1) 16 Theorem 2.1.6 Let M be a connected complex manifold of dimension m Let A be a pure (m − 1)-dimensional analytic subset of M Let V be a complex vector space of dimension n + 1 > 1 Let p and k be integers with 1 ≤ p ≤ k ≤ n + 1 Let fi : M → P (V ), 1 ≤ i ≤ k, be meromorphic mappings Assume that f1, , fk are in p-special position on A Then we have µf1∧···∧fk ≥ (k − p + 1)νA Here, by µf1∧···∧fλ we denote the divisor associated to f1 ∧ · · · ∧ fλ Theorem 2.1.7 Let fi : Cm → Pn(C), 1 ≤ i ≤ k be meromorphic mappings located in general position Assume that 1 ≤ k ≤ n Then Nµf1∧···∧fλ (r) + m(r, f1 ∧ · · · ∧ fλ) ≤ Tfi(r) + O(1) 1≤i≤λ 2.2 Second Main Theorem for meromorphic mappings on Cm inter- secting moving hyperplanes with weighted counting functions In this section, we quote the lemma of S D Quang and use this lemma to prove Second Main Theorem for meromorphic mappings and moving targets with weighted counting functions Lemma 2.2.1 Let f : Cm → Pn(C) be a meromorphic mapping Let {aj}j=1 2n−k+2 be meromorphic mappings of Cm into Pn(C)∗ in general position such that (f, aj)̸ ≡ 0 (1 ≤ j ≤ 2n − k + 2), where rank R{aj}(f ) = k + 1 Then there exists a subset J ⊂ {1, , 2n − k + 2} with |J| = n + 2 satisfying || Tf (r) ≤ [k] N(f,aj)(r) + o(Tf (r)) + O( max Taj (r)) 1≤j≤2n−k+2 j∈J One of our main results in this chapter is the improvement of Second main theorem for meromorphic mappings from Cm into Pn(C) where the characteristic function is bounded above by the sum of counting functions have different weights Theorem 2.2.2 Let f : Cm → Pn(C) be a meromorphic mapping Let {aj}j=1 q (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≤q λj ≤ j=1 q λj Then for every positive number 17 j=1 q λj η ∈ [max1≤j≤q λj, 2n − k + 2], we have j=1 q λj − (n − k)η Tf (r) ≤ q [k] λjN(f,aj)(r) + o(Tf (r)) + O( max Taj (r)) n+2 j=1 1≤j≤q 2.3 Algebraic dependence of meromorphic mappings on Cm having the same inverse images of moving hyperplanes In this section, to consider the algebraic dependence of meromorphic mappings of Cm, we quote the following lemma of L N Quynh Lemma 2.3.1 Let hi : Cm → Pn(C) (1 ≤ i ≤ p ≤ n + 1) be meromorphic mappings with reduced representations hi := (hi0 : · · · : hin) Let ai : Cm → Pn(C) (1 ≤ i ≤ p ≤ n + 1) be moving hyperplanes with reduced representa- tions ai := (ai0 : · · · : ain) Put h˜i := ((hi, a1) : · · · : (hi, an=1)) Assume that a1, · · · , an+1 are located in general position such that (hi, aj)̸ ≡ 0 (1 ≤ i ≤ p, 1 ≤ j ≤ n + 1) Let S be a pure (n − 1)− dimensional analytic subset of Cm such that S̸ ⊂ (a1 ∧ · · · ∧ an+1)−1{0} Then h1 ∧ · · · ∧ hp ≡ 0 on S if and only if h˜1 ∧ · · · ∧ h˜p ≡ 0 on S By using Theorem 2.2.2, we prove the following theorem that shows the con- dition for algebraic dependence of meromorphic mappings from Cm into Pn(C) Theorem 2.3.2 Let f1, · · · , fλ : Cm → Pn(C) (λ ≥ 2) be nonconstant mero- 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 z ∈ Aj (1 ≤ j ≤ q) and for any 1 ≤ i1 < · · · < ilj < q, fi1(z) ∧ · · · ∧ filj (z) = 0 j=1 q lj If dλk(2n − k + 2) − dλ(k − 1) + , q> λ+1 then f1 ∧ · · · ∧ fλ ≡ 0 on Cm 18