một số định lý về tính duy nhất và tính hữu hạn của họ các ánh xạ phân hình TT

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 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 Council meeting at Thesis can be found at: - Thesis can be found at -Library of Hanoi University of Education INTRODUCTION 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 meromorphic mapping from Cm into the piojective space Pn(C), they based on the Logarithmic Derivative Lemma and the property of Wronskian’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 meromorphic map or for a meromorphic mapping of hyper-order less than respectively From there, they were able to study the uniqueness of these such as the general Picard’s theorem This second main theorem is called the second main theorem p-differences or c-differences with targets Using these approaches, in 2016, T B Cao and R Korhonen 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 position In the one-dimensional case, since R Halburd and R Korhonen have given the lemma c-differences the secmd main theorem cdifferences for the polymorphism function with super order less than 1, unique theorem Picard’s theorem similar to R Nevanlinna’s 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 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 value sharing counts multiplicities and two shared values without multiplicities Note that, in R Nevanlinna’s point theorem, 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 The authors have tried to answer the above question and they have obtained results for a meromorphic map of hypcr-order less than sharing 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 values are interrupted From there, they posed the problem of studying the uniqueness of Picard’s theorem when values are truncated multiplicities One of the goals when studying the unigcueness problem is to reduce the number of shared values Accordingly, we pose research problems and improve the results of W Lin, X Lin and A Wu The problem of the algebraic dependence of the meromorphic 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 meromorphic maps, having an inverse image intersecting the 2n + 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 + through the algebraic dependence of 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 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 share a part of the values with its f (z + c) Following that, the dissertation establishes some second main theorems 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 meromorphic mappings from Cm into the projective space Pn(C) intersecting with 2n + hyperplanes in general position 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 meromorphic mapping 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 Scientific and practical significances The dissertation contributes to deepening the results on the uniqueness and finiteness of the meromorphic functions or the meromorphic mappings Besides enriching these problems, the dissertation also gives new results for the algebraic dependence of meromorphic mappings in the projective space with few hyperplanes The dissertation is a useful reference for students, graduate students and postgraduate students in this direction 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 Chapter II The uniqueness for the meromorphic function that has hyper degree less than Chapter III The uniqueness for zero ordcr meromorphic mapping Chapter IV.The algebraic degneracy and finiteness of meromorphic mappings The dissertation is based on three published articles Place of writing the dissertation Hanoi National University of Education Chapter 1: Overview I The uniqueness for the meromorphic function that has hyper order less than 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 Korhonen, there have been many interesting uniqueness theorems similar to Nevanlinna’s point theorem For example, in 2009, J Heittokangas and colleagues considered this problem for a meromorphic function f (z) with a finite order on complex plane sharing 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 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 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 ˆ ) be hyper-order γ(f ) < and c ∈ C \ {0} Let a1, a2, a3, a4 ∈ S(f four distinct periodic functions with period c If δ(a, f ) > for ˆ ) and some a ∈ S(f E(aj , f (z)) ⊆ E(aj , f (z + c)), j = 1, 2, 3, then f (z) = f (z + c) for all z ∈ C In 2018, W Lin, X Lin and A Wu [11] obtained a counterexample 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, " with a positive integer k, even if f (z) and f (z + c) share a3, a4 CM Then, they introduced the following results under a reduced deficiency assump2 tion Θ(0, f ) + Θ(∞, f ) > 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 ) < and c ∈ C \ {0} Let k1, k2 be two positive ˆ ) be four distinct integers, and let a1, a2 ∈ S(f )\{0}, a3, a4 ∈ S(f periodic functions with period c such that f (z) and f (z + c) share a3, a4 CM and E ≤kj (aj , f (z)) ⊆ E ≤kj (aj , f (z + c)), j = 1, 2 If Θ(0, f ) + Θ(∞, f ) > k+1 , where k := min{k1, k2}, then 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 ) = 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, If k ≥ 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 meromorphic function with Θ(∞, f ) = Θ(∞, g) = 1, where f has a nonzero periodic c ∈ C \ {0} with hyper-order γ(f ) < 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, 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 hyperˆ ) be three order γ(f ) < and let c ∈ C \ {0} Let a1, a2, a3 ∈ S(f 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 E ≤k (a3, f (z)) ⊆ E ≤k (a3, f (z + c)) ˆ ) \ {a3, a3(a1+a2)−2a1a2 } then, If Θ(a, f ) > k+1 for some a ∈ S(f 2a3 −(a1 +a2 ) 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 ) = 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)) 11 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 For the purpose of studying the uniqueness or the Picard theorem of the meromorphic maps from Cm into Pn(C) having order 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 = for all ≤ j ≤ m and let f : Cm → Pn(C) be a meromorphic mapping with zero-order Assume that f is algebraically nondegenerate over the field φ0q Denote by f˜ = (f0 : · · · : 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 p (q − (N − n + 1)(n + 1)) Tf (r) ≤ i=1 − N ˜ (r) di Qi(f ) N −n+1 un+1 (n+1)! + O(un) NCq (f I1 , ,f IM )(r) + o (Tf (r)) on a set of logarithmic density 1, where Ij = (ij0, , ijn), u+d |Ij | = ij0 + · · · + ijn = u and M = u 12 We have the following result similar to the Nochka-Cartan theorem with truncated multiplicities Theorem 3.2.3 Let q = (q1, , qm) ∈ Cm with qj = for all ≤ j ≤ m and let f : Cm → Pn(C) be a meromorphic mapping with zero-order Assume that f is algebraically nondegenerate over the field φ0q Denote by f˜ = (f0 : · · · : 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, for every > 0, we have p (p − (N − n + 1)(n + 1) − ) Tf (r) ≤ j=1 ¯ [M0,q] N (r) + o (Tf (r)) dj Qj (f˜) on a set of logarithmic density 1, where M0 = 4(ed(N − n + 1)(n + 1)2I( −1))n − 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 = 0, for all j ∈ {1, , m} and let f : Cm → Pn(C) be a zeroorder meromorphic mapping Let Q1, , Qp be hypersurfaces in Pn(C), located in N -subgeneral position with common degree d n+d Put M = − Assume that f is forward invariant over n Qj with respect to the rescaling τq (z) = qz If p ≥ M + 2N − n + then the image of f is contained in a linear subspace over the field φ0q of dimension ≤ M − n − + p [ ]+1 In the case of hyperplanes in subgeneral position in Pn(C), we have M = n Moreover, when |qi| = for all i ∈ {1, , m}, then f (z) = f (qz) implies that f must be a constant mapping Immediately, we have the following corollary p−N −1 M −n+1 13 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 = for all j ∈ {1, , m} Assume that τq ((f, Hj )−1) ⊂ (f, Hj )−1 (counting multiplicity) hold for distinct hyperplanes {Hj }pj=1 in N -subgeneral position in Pn(C) If p > 2N then f (qz) = f (z) In particular, if |qi| = for all i ∈ {1, , m}, then f is constant III The finiteness of meromorphic mappings The problem of the algebraic dependence of the complex sereralvariable 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 + meromorphic mappings sharing 2n + hypnplanes with +n Recently, S D Quang has truncated multiplicities to level n(n+1) obtained the algebraic degenerate theorem for three meromorphic maps and used it to give results on the finiteness of meromorphic mapping sharing 2n + 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 Theorem E Let H1, , H2n+2 be hyperplanes in general position in Pn(C) Assume that 2n+2 j=1 n+1 < kj + n(3n + 1) Then for three mappings f 1, f 2, f ∈ F(f, {Hj , kj }2n+2 j=1 , 1), we 14 have f ∧ f ∧ f ≡ on Cm 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 + hyperplanes in general position as follows Theorem F Let f 1, f 2, f ∈ F(f, {Hj }qj=1, n) and let {Hi}qi=1 be a family √ of q hyperplanes of Pn(C) in general position If q > 2n + + 28n2 + 20n + then one of the following assertions holds: (i) There exist q + hyperplanes Hi1 , , Hi [ 3q ]+1 such that (f u, Hi q ) (f , Hi1 ) (f , Hi2 ) [ ]+1 = = · · · = , (f v , Hi1 ) (f v , Hi2 ) (f v , Hi q ) [ ]+1 u u (ii) f ∧ f ∧ f ≡ 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 + 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 : Cm → Pn(C) be 15 meromorphic mappings which belong to F(f, {Hj , kj }2n+1 j=1 , n) If 2n+1 i=1 n−4 < , ki + 2n(2n + 1) then f ∧ f ∧ f ≡ 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 + hyperplanes of Pn(C) in general position and k1, , kn+2 be positive integers or +∞ Assume that 2n+2 i=1 n + 5n − n2 − 1 < , , ki + 3n2 + n 24n + 12 10n2 + 8n Then F(f, {Hi, ki}2n+2 i=1 , 1) ≤ The following question arises naturally at this moment: By using Quang’s techniques in [7] and by Theorem 1.0.11, we obtain a result on finiteness for meromorphic mappings sharing 2n + 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 Theorem 4.3.1 Let f be a linearly nondegenerate meromorphic mappings of Cm into Pn(C) Let H1, , H2n+1 be 2n + hyperplanes of Pn(C) in general position and let k1, , kn+1 be positive integers or +∞ such that 2n+1 i=1 n−4 < ki + 2n(2n + 1) 16 If n ≥ then F(f, {Hi, ki}2n+1 i=1 , n) ≤ Chapter 2: The uniqueness of the zero rder meromorphic mapping Chapter is written based on the article [1] in the published works related to the thesis 2.1 Some preparation knowledge 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 q (q − 2)T (r, f ) ≤ N r, i=1 + S(r, f ) f − Lemma 2.1.4 Let f be a nonconstant meromorphic function and c ∈ C If f is of finite order, then m r, log r f (z + c) =O T (r, f ) f (z) r 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 > 0, we have T (r, f ) f (z + c) m r, = o 1−γ(f )− f (z) r 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., log+ log+ T (r) γ = lim sup < 1, log r r→∞ 17 Then T (r) , r1−γ− where > and r → ∞ outside a subset of finite logarithmic measure T (r + s) = T (r) + o For each meromorphic function f , we denote fc(z) = f (z + c) and ∆cf := fc − f Lemma 2.1.6 Let c ∈ C and let f be a meromorphic function of hyper-order γ(f ) < such that ∆cf ≡ Let q ≥ and a1(z), , aq (z) be distinct meromorphic periodic small functions of f with period c Then q m(r, f ) + m r, k=1 ≤ 2T (r, f ) − Npair (r, f ) + S1(r, f ), f − ak where Npair (r, f ) = 2N (r, f ) − N (r, ∆cf ) + N r, ∆cf 2.2 The uniqueness for the meromorphic function with hyper order less than 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 In this section, we give some necessary conditions for a meromorphic function less than 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 18 Chapter is written based on the article [2] in the published works related to the thesis 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 ) = Cq f I1 , , f IM ≡ if and only if the functions f0, , fn are alge-braically dependent over the field φ0q Lemma 3.1.2 Let Q1, , Qk+1 be hypersurfaces in Pn(C) of the same degree d such that k+1 Qi = ∅ i=1 Then there exist n hypersurfaces P2, , Pn+1 of the forms k−n+t ctj Qj , ctj ∈ C, t = 2, , n + 1, Pt = j=2 such that n+1 i=1 Pi = ∅ 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 i∈R Qi = ∅ Then there exist positive constants α and β such that α||f˜||d ≤ max |Qi(f˜)| ≤ β||f˜||d i∈R The following lemma is called the q-difference logarithmic derivative 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 = for 19 all j, then f (qz) = o(T (r, f (z))) f (z) on a set of logarithmic density m r, 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 Corollary 1.0.10 To prove Theorem 1.0.9, we need the following main lemma Lemma 3.3.2 Let Q1, , Qp be hypersurfaces of the common degree d in Pn(C), located in N -subgeneral position Put M = n+d p−N −1 − and let p˜ = M −n+1 + N + If p ≥ M + 2N − n + 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 have span{Q∗j }j∈R ∩ span{Q∗i }i∈R∗ = {0}, where R∗ = U \ R and Q∗j is a homogeneous polynomial defining Qj 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 = for all j Assume that σ(f ) = and all zeros of f0, , fnare forward invariant with respect to the n+d rescaling τq (z) = qz Let d ∈ N∗ and put M = − If n f Ii for each d, I ∈ φ0q for all i, j ∈ {0, , M } such that Ii = Ij , f j then f0, , fn are algebraically nondegenerate over the field φ0q Lemma 3.3.4 Let f = (f0 : · · · : fn) be a meromorphic mapping of Cm to Pn(C) such that σ(f ) = and all q = (q1, , qm) ∈ Cm 20 with qj = 0, for all j Assume that all zeros of f0, , fn are forward invariant with respect to the rescaling τq (z) = qz Let n+d d ∈ N∗, put M = − Let S1 ∪ · · · ∪ Sl be a portion of n {0, , M } formed in such a way that i and j are in the same f Ii class Sk if and only if I ∈ φ0q If f I0 + · · · + f IM = then f j f Ij = for all k ∈ {1, , l} j∈Sk Chapter 4: The algebraic degeneracy and finiteness of meromorphic mappings Chapter 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 + 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 > Theorem 4.1.2 [The second main theorem] Let f : Cm → Pn(C) be a linearly nondegenerate meromorphic mapping and 21 H1, , Hq be hyperplanes in general position in Pn(C) Then q [n] ||(q − n − 1)Tf (r) ≤ N(f,Hi)(r) + o(Tf (r)) i=1 Lemma 4.1.3 If Φα (F, G, H) = and Φα F1 , G1 , H1 = for all α with |α| ≤ 1, then one of the following assertions holds: (i) F = G, G = H or H = F (ii) F G G, H and H F are all constants Theorem 4.1.4 Let f 1, f 2, f be three mappings in F(f, {Hi, ki}qi=1, 1) Suppose that there exist s, t, l ∈ {1, , q} such that (f 1, Hs) (f 1, Ht) (f 1, Hl ) P := (f 2, Hs) (f 2, Ht) (f 2, Hl ) ≡ (f 3, Hs) (f 3, Ht) (f 3, Hl ) Then we have [1] T (r) ≥ NP (r) ≥ (N (r, {ν(f u,Hi),≤ki }) − N(f 1,H ),≤k (r)) i 1≤u≤3 i=s,t,l q i [1] +2 N(f 1,H ),≤k (r), i i i=1 u=1 Tf u (r) 4.1.5 If 2n+1 i=1 ki +1 where T (r) = Lemma < is linearly nondegenerate and n−1 n , then every g ∈ F(f, {Hi, ki}qi=1, n) ||Tg (r) = O(Tf (r)) ||Tf (r) = O(Tg (r)) 4.2 The algerbraic degeneracy of thrre meromorphic mappings sharing 2n + hypreplanes In this section, we prove Theorem 1.0.11 However, we need the following Lemma 22 Lemma 4.2.2 Let q, N be two integers satisfying q ≥ 2N + 2, N ≥ 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 = for any subset R ⊂ Q = {1, , q} with cardinality R = N + Then q/2 there exists a partition j=1 Ij of {1, , q} satisfying Ij = and rank{ai}i∈Ij = for all j = 1, , q/2 4.3 The finitenees of meromorphic mappings sharing 2n + 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 (g, Hi) (h, Hi) =λ (h, Hj ) (g, Hj ) then λ = 2n+1 Lemma 4.3.3 Let f 1, f 2, f be three elements of F(f, {Hi, ki}i=1 , n), where ki (1 ≤ i ≤ 2n + 1) are positive integers or +∞ Suppose that f ∧ f ∧ f ≡ and Vi ∼ Vj for some distinct indices i and j Then f 1, f 2, f are not distinct Lemma 4.3.4 With the assumption of Theorem 1.0.12, let f 1, f 2, f be three maps in F(f, {Hi, ki}2n+1 i=1 , n) Suppose that f 1, f 2, f are distinct and there are two indices i, j ∈ {1, 2, , 2n+ 1} (i = j) such that Vi ∼ = Vj and Φαij := Φα (F1ij , F2ij , F3ij ) ≡ for every α = (α1, , αm) ∈ Z+ m with |α| = Then for every t ∈ {1, , 2n + 1} \ {i}, the following assertions hold: (i) Φαit ≡ for all |α ≤ 1|, 23 (ii) if Vi ∼ = Vt, then F1ti, F2ti, F3ti are distinct and [1] [1] N(f,Hi),≤ki (r) ≥ [1] N(f,Hs),≤ks (r) − N(f,Ht),≤kt (r) s=i,t [1] −2 N(f u,Hs),>ks (r) + o(T (r)) u=1 s=i,t Lemma 4.3.5 With the assumption of Theorem 1.0.12, let f 1, f 2, f be three maps in F(f, {Hi, ki}2n+1 i=1 , n) Suppose that f 1, f 2, f are distinct and there are two indices i, j ∈ {1, 2, , 2n+ α 1} (i = j) and α ∈ Z+ m with |α| = such that Φij ≡ Then we have 3 [n] N(f u,Hi),≤ki (r) T (r) ≥ u=1 [n] + [1] N(f u,Hj ),≤kj (r) + u=1 N(f,Ht),≤kt (r) t=1,t=i,j 1 − (2n + 1)N(f,H (r) − (n + 1)N(f,H (r) + N (r, νj ) i ),≤ki j ),≤kj − 1+ u=1 2n − [1] n − [1] N(f u,Hj ),>kj + + N(f u,Hi),>ki 3 + o(T (r)), where νj := {z : ki ≥ ν(f u,Hi)(z) = ν(f t,Hi)(z)} for each permutation (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 mappings The dissertation has achieved some of the following results: • Prove some uniqueness theorem for the meromorphic function with hyper order less than in the complex plane • Prove the second main theorem for the meromorphic mapping from 24 Cn into the project space Pn(C) of zero order intersecting the hyperaces in subgeneral position Applies to wide mining of Picard’s unique theorem for meromorphic mappings intersecting hypersurfaces • Prove the theorem of the algebraic dependence of three meromorphic maps from Cn into Pn(C) intersecting the 2n + 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 functions with hyper oeder less than one in the complex plane • We research to try to generalize the theorem of algebraic dependence 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 under more general terms of multiplicities and meromorphic 25 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 theorems for meromorphic maps of several complex variables intersecting hypersurfaces, Asian-European J Math., Vol 14, No (2021) 2150040 (21 pages) [3] P D Thoan and N Vangty, Algebraic dependences and finitenness of meromorphic mappings sharing 2n + hyperplanes with truncated multiplicities, Kodai Math J., 43 (2020), 504-523 ... 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... p-differences for the meromorphic mapping from Cm into the the piojective Pn(C) intersecting the superspattice at the hyperplanes in subgeral position In the one-dimensional case, since R Halburd and R... 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 distinct