THAI NGUYEN UNIVERSITY UNIVERSITY OF EDUCATION LE QUANG NINH

THAI NGUYEN UNIVERSITY UNIVERSITY OF EDUCATION LE QUANG NINH ON THE DETERMINATION OF HOLOMORPHIC FUNCTION AND MAPPING THROUGH INVERSE MAPPING OF POINT SETS Specialty: Mathematical Analysis Code: 62.46.01.02 SUMMARY OF DOCTORAL DISSERTATION OF MATHEMATICS THAI NGUYEN - 2017

This disseration is completed at: UNIVERSITY OF EDUCATION - THAI NGUYEN UNIVERSITY Scientific supervisor: 1. Prof. Dr.SC Ha Huy Khoai 2. Dr. Vu Hoai An Reviewer 1:... Reviewer 2:... Reviewer 3:... The disseration will be defended at: UNIVERSITY OF EDUCATION - THAI NGUYEN UNIVERSITY at... time... date...month...year 2017 The disseration can be found at: - National Library of Vietnam; - Learning Resource Center of Thai Nguyen University; - Library of University of Education;

1 Contens Introduction............................................. 2 Chng 1. Determine moromorphic function through inverse mapping of point sets.................................... 5 1.1. Some concepts and supplementary results................... 5 1.2. The equation of Fermat-Waring type for meromorphic functions. 5 1.3. Determine moromorphic function through inverse mapping of point sets.............................................................. 6 Chng 2. Determine holomorphic curves through inverse mapping of point sets............................................ 9 2.1. Some concepts and supplementary results................... 9 2.2. The equation of Fermat-Waring type for holomorphic curves. 9 2.3. Determine holomorphic curves through inverse mapping of point sets............................................................ 12 Chng 3. Determine moromorphic functions and holomorphic curves on a field non-archimedean...................... 14 3.1. Some concepts and supplementary results.................. 14 3.2. The equation of Fermat-Waring type of variables for non-archimedean entire functions................................................. 14 3.3. Determine moromorphic functions and holomorphic curves on a field non-archimedean.......................................... 15

2 Introduction 1. Rationale One of the deepest applications of value distribution theory (complex and p -adic) is the uniquely identifies problem of meromorphic functions (complex and p -adic) through inverse mapping of point sets which is now called Five Value theorem of Nevanlinna (or its similarities for the case p-adic). In 1977, F.Gross presented a novel idea which considered inverse images of a set in C { }. He posed the following two questions: i) Is there exist or not a subset S of C { } such that for arbitrary two non-constant meromorphic functions f and g, with condition E f (S) = E g (S) implies f = g? ii) Is there exist or not subsets S i, i = 1, 2 of C { } such that for arbitrary two non-constant meromorphic functions f and g, with condition E f (S i ) = E g (S i ), i = 1, 2 implies f = g? The function equation P (f) = P (g) (P (f 1,..., f N+1 ) = P (g 1,..., g N+1 ) intimately embedded with the uniqueness determined problems for meromorphic functions (linearly non-degenerate holomorphic curves). Base on this two questions arose: Question 1: problem with non solution, problem with solution, problem with finite solutions, problem with unique solution, describe solutions of functional equation P (f) = P (g) with the inverse image of point sets for meromorphic functions? Question 2: problem with non solution, problem with solution, problem with finite solutions, problem with unique solution, describe solutions of functional equation P (f 1,..., f N+1 ) = Q(g 1,..., g N+1 ) with the inverse image of hypersurfaces for holomorphic curves? In 2007, F.Pakovich [26] presented a vovel idea which considered inverse images of K 1, K 2 C be finite or infinite compact sets. He posed the following question: Under what conditions on the collection f 1, f 2, K 1, K 2 do the preimages f1 1 (K 1) and f2 1 (K 2) coincide, that is

3 f1 1 (K 1) = f2 1 (K 2)? In order to answer the questions of Gross, Pakovich, Questions 1, Questions 2 and to enrich the research in theory Nevanlinna, we chose dissertation entitled: On the determination of holomorphic function and mapping through inverse mapping of point sets. The dissertation studies the following problem: Let S i, T i C { }, S i, T i, i = 1,..., k; X i, Y i be hypersurfaces in P N (C), i = 1,..., k. Problem 1: Determine moromorphic function through inverse images of S i, T i. Problem 2: Determine linearly non-degenerate holomorphic curves through inverse images of X i, Y i. Problem 3: Same as Problem 1 and Problem 2 for case p-adic. 2. Purpose of the study 2.1. Find S i, T i, i = 1, dots, k, with condition: There exist nonconstant meromorphic functions f and g satisfying E f (S i ) = E g (T i ) or E f (S i ) = E g (T i ), i = 1,..., k. Then describe f, g and contact the uniqueness determined problems for meromorphic functions. 2.2. Find S i, T i, i = 1,..., k, with condition: There no exist nonconstant meromorphic functions f and g satisfying E f (S i ) = E g (T i ) or E f (S i ) = E g (T i ), i = 1,..., k. 2.3. Find X i, Y i, i = 1,..., k with condition: There exist linearly nondegenerate holomorphic curves f and g satisfying ν X i f = ν Y i g, i = 1,..., k. Then describe f, g and contact the uniqueness determined problems for holomorphic curves. 2.4. Find X i, Y i, i = 1,..., k with condition: There exist linearly nondegenerate holomorphic curves f and g satisfying ν X i f = ν Y i g, i = 1,..., k. Where, νf X be the pull-back of the divisor X by f. The thesis focuses on studying the above aims in the case of i = 1. 2.5. Find the sets S i such that from E f (S i ) = E g (S i ) determine f, g for f, g are p-adic meromorphic functions. 2.6. Find the hypersurface X uniquely determined p-adic non-degenerate holomorphic curves. 3. Object and scope of study Meromorphic functions, holomorphic curves, properties of solutions of some polynomial equations, application of Nevanlinna theory on the value distribution of holomorphic mappings into the problem that uniqueness determined holomorphic mapping through inverse mapping of sets.

4 4. Methodology The tools used to solve the three problems mentioned above are the two Main Theorem of Nevanlinna theory and its similarities, Borel lemma type and the like in the case p-adic. 5. Significance of the dissertation The thesis paticipates in perfect and rich more in the application of Nevanlinna Theory on the value distribution of holomorphic mappings into the problem that uniqueness determined holomorphic mapping through inverse mapping of sets. 6. Structure and findings of the thesis The main content of the thesis consists of three chapters corresponding to three research problems of the thesis: Chapter 1: Studying Problem 1, we obtain result Theorem 1.3.3, Theorem 1.3.4 and Theorem 1.3.7. These results are extension of the Five Value and Four Value theorems of Nevanlinna and in response to questions by F.Gross and Pakovich. The content of Chapter 1 uses some of the results published in [5], [16]. Chapter 2: Studying Problem 2, we obtain result Theorem 2.3.1, Theorem 2.3.2, Theorem 2.3.3, Theorem 2.3.5 and Theorem 2.3.7. Theorem 2.3.1, Theorem 2.3.2, Theorem 2.3.3 and Theorem 2.3.5 contributes to Pkovic s question and F.Gross s question in the case hypersurfaces. The content of Chapter 2 uses some of the results published in [5], [16]. Chapter 3: Studying Problem 3, we obtain result Theorem 3.3.1, Theorem 3.3.4, Theorem 3.3.5 and Theorem 3.3.6. Theorem 3.3.1 and Theorem 3.3.4 are extension of the Four Value and Two Value theorems p-adic. Theorem 3.3.6 contributes to Gross s question for linearly non-degenerate holomorphic curves from K to P N (K). The content of Chapter 3 uses some of the results published in [6], [23].

5 Chapter 1 Determine moromorphic function through inverse mapping of point sets 1.1. Some concepts and supplementary results In this section, we review some concepts and results of value distribution theory: counting functions, The Second Main Theorem, some The Second Main Theorem type,... 1.2. The equation of Fermat-Waring type for meromorphic functions Theorem 1.2.1. Let n > 2m + 3, a 1, b 1, c, a 2, b 2 are nonzero on C. Suppose that (n, m) = 1, m 2, or m 4. Then the following equation f n + a 1 f n m + b 1 = c(g n + a 2 g n m + b 2 ) (1.1) has non-constant moromorphic solutions (f, g), iff c = b 1 b 2, there exist h C such that h n = c, h m = a 1 a 2 and f = hg. Let P (z) = (z a 1 )... (z a q ) and Q(z) = (z b 1 )... (z b q ) be two polynomials of degree q, where a i a j, b i b j. Let the derivative of P (z) and Q(z) be given by P (z) = q(z d 1 ) m 1... (z d k ) m k, Q (z) = q(z e 1 ) n 1... (z e k ) n k, (1.2)

6 where d 1,..., d k are mutually distinct, e 1,..., e k, too, and d 1 + +d k = e 1 + + e k = k. Let P (z) and Q(z) be two polynomials satisfying the following conditions: P (d i ) P (d j ) with every i j, i, j {1,..., k}, Set Q(e i ) Q(e j ) with every i j, i, j {1,..., k}. (H) A = {i, j : i {1,..., k}, j {1,..., k}, P (d i ) = cq(e j ), c 0}, m = #A. In the case A =, set m = 0. Theorem 1.2.8. Let P and Q be two polynomials satisfying condition (H), k 4 or k > m. With h is nonzero constant, suppose that the functional equation P (f) = hq(g) has non-constant moromorphic solutions (f, g). Then f = ag + b, ad bc 0. cg + d Lemma 1.2.9. Let n, n 1, n 2,..., n q N, a 1, a 2,..., a q be distinct points of C, c C, c 0 and q > 2 + q n i. Then the functional equations n i=1 (f a 1 ) n 1 (f a 2 ) n 2... (f a q ) n q = cg n, (1.3) (f a 1 ) n 1 (f a 2 ) n 2... (f a q ) n q g n = c (1.4) has no non-constant moromorphic solutions (f, g). 1.3. Determine moromorphic function through inverse mapping of point sets Theorem 1.3.3. Let a 1, b 1, a 2, b 2 0, P (z) = z n + a 1 z n m + b 1 and Q(z) = z n + a 2 z n m + b 2 have no multiple roots, (m, n) = 1, m 2, or m 4, let S, T be them zero sets, respectively. Then i) Let n 2m+9, the following statement hold: There exist non-constant meromorphic functions f and g satisfying E f (S) = E g (T ), iff f = hg, where h m = a 1 a 2, h n = b 1 b 2.

7 ii) Let n 5m + 15, the following statement hold: There exist nonconstant meromorphic functions f and g satisfying E f (S) = E g (T ), iff f = hg, where h m = a 1 a 2, h n = b 1 b 2. Theorem 1.3.4. Let a 1, a 2 0, a 1, a 2 C, let S 1, T 1 are sets of all n th roots of a 1, a 2, respectively. Then 1. Let n > 8, the following statement hold: There exist non-constant meromorphic functions f and g satisfying E f (S 1 ) = E g (T 1 ), iff f = l g, or f = hg, where l n = a 1 a 2, h n = a 1 a 2. 2. Vi n > 14, the following statement hold: There exist non-constant meromorphic functions f and g satisfying E f (S 1 ) = E g (T 1 ), iff f = l g, or f = hg, where l n = a 1 a 2, h n = a 1 a 2. Let P (z) and Q(z) be two polynomials of form (1.2) satisfying the following conditions: P (d 1 ) + P (d 2 ) + + P (d k ) 0, Q(e 1 ) + Q(e 2 ) + + Q(e k ) 0. (H 1 ) P (d 1 ) + + P (d k ) = Q(e 1 ),..., Q(e k ). (H 2 ) Theorem 1.3.5. Let f and g be two meromorphic functions, P (z) and Q(z) be two polynomials satisfying the following condition (H), (H 1 ) and (H 2 ), S and T be them zero sets, respectively. Suppose that k 4, q > 2k + 6 and E f (S) = E g (T ). Then 1. If S = T then f = g; 2. If S T then f = ag + b, ad bc 0. cg + d Theorem 1.3.6. Let f and g be two non-constant meromorphic functions, P (z) and Q(z) be two polynomials satisfying the following condition (H), (H 1 ) and (H 2 ), S and T be them zero sets, respectively. Suppose that k 4 q > 2k + 10, and E f (S) = E g (T ). Then 1. If S = T then f = g. 2. If S T then f = ag + b, ad bc 0. cg + d

8 Theorem 1.3.7. Let n, m be positive intergers, P (z) = z n m (a m z m + + a 1 z + a 0 ) + a be a polynomials of degree n with only simple zeros, R(z) = cz n + b, a m, a m 1,..., a 1, a 0, c, b C, cba m a 0 0, S and T be them zero sets, respectively. Write a m z m + + a 1 z + a 0 = a m (z e 1 ) n 1... (z e q) n q. Suppose that q > 2 + q n i i=1 n + n m m. Then 1. If n > m + 8 then there no exist non-constant meromorphic functions f and g satisfying E f (S) = E g (T ). 2. If n > 3m + 14 then there no exist non-constant meromorphic functions f and g satisfying E f (S) = E g (T ).

9 Chapter 2 Determine holomorphic curves through inverse mapping of point sets 2.1. Some concepts and supplementary results In this section, we review some concepts and results: characteristic functions of holomorphic curves, hypersurfaces in general position, linearly non-degenerate holomorphic curves, a uniqueness polynomial and a strong uniqueness polynomial for linearly non-degenerate holomorphic curves, some Borel lemma type,... 2.2. The equation of Fermat-Waring type for holomorphic curves (B 1 ): Let u i, v j, (i = 1,..., q, j = 1,..., N + 1) be two row vectors in general position on C N+1, q N + 1. Let α 1,..., α N+1 are distinct integers, define the set Q = {α = (α 1,..., α N+1 ) : 1 α 1,..., α N+1 q}. For each element α = (α 1,..., α N+1 ) Q, we set α = {α 1,..., α N+1 } and associate the matrix u α1 u α2 A α =,. u αn+1

B α = 10 v α1 v α2. v αn+1. Denote σ be a bijective from {1, 2,..., q} to {1, 2,..., q} ; σ(α) = (σ(α 1 ),..., σ(α N+1 )), σ(α) = {σ(α 1 ),..., σ(α N+1 )}. For two vectors w = (w 1,..., w N+1 ), x = (x 1,..., x N+1 ), we define w x = w 1 x 1 + + w N+1 x N+1. Let ω 1,..., ω q C such that ωi d = 1, d N, Ω α = ω 1 0... 0 0 ω 2... 0........ 0... 0 ω αn+1. For N + 1 entire functions f 1,..., f N+1, put f = (f 1,..., f N+1 ), we denote f 1 f t f 2 =.. f N+1 (B 2 ): Suppose u j (j = 1,..., q) be a row vectors in general position on C N+1. Let d be a positive integer and α, α, β, β Q. (B 3 ): If α α or β β and if α β or α β then ( ) d ( ) d detaα detaβ. deta α deta β (B 4 ) : If ᾱ ᾱ, β β and if ᾱ β or ᾱ β then ( ) nd ( ) nd detaα detaβ. deta α deta β Theorem 2.2.5. Let q, d, N N, d (2q 1) 2, q N + 1, the functional equation q q (u j f) d = (v j g) d, (2.1) j=1 where f = (f 1,..., f N+1 ), g = (g 1,..., g N+1 ). Then the equation (2.1) has solutions (f 1,..., f N+1, g 1,..., g N+1 ), with {f 1,..., f N+1 }, {g 1,..., j=1

11 g N+1 } be two families of linearly independent entire functions, iff there exists bijective σ from {1,..., q} to {1,..., q} such that A 1 α Ω α B σ(α) = A 1 β Ω βb σ(β) (α, β Q.) More over, if the necessary condition, where α = (1,..., N + 1). f = A 1 α Ω α B σ(α) g t, (2.2) (B 5 ): Let d, m, n, N be positive integers, m < n, consider homogeneous polynomials: A i = z n i+1 a i z n m i+1 zm 1 + b i z n 1, B i = z n i+1 c i z n m i+1 zm 1 + d i z n 1, i = 1,..., N + 1, A(z 1,..., z N+1 ) = A d 1 + + A d N, B(z 1,..., z N+1 ) = B d 1 + + B d N. Theorem 2.2.8. Let the assumptions as in (B 5 ) and n 2m + 9, (n, m) = 1, d (2N 1) 2. Then the functional equation A(f 1,..., f N+1 ) = B(g 1,..., g N+1 ) (2.3) has solutions (f 1,..., f N+1, g 1,..., g N+1 ), where {f 1,..., f N+1 }, {g 1,..., g N+1 } be two families of linearly independent entire functions on C, iff there exists bijective σ from {1, 2,..., N} to {1, 2,..., N} and c iσ(i), l 1, l σ(i) +1 C, satisfying b i c iσ(i) d σ(i) = l n 1, l n σ(i)+1 = 1 c iσ(i), l n m σ(i)+1 lm 1 = g i = l i f i, i = 1,..., N + 1, c d iσ(i) = 1. a i c iσ(i) a σ(i), i = 1,..., N, (B 6 ): Let a 0 0, P (z) = z n + a n 1 z n 1 + + a 0 has no multiple roots, let S be its zero sets. Set G(z 1, z 2 ) = z n 1 + a n 1 z n 1 z 2 + + a 0 z n 2, H(z 1,..., z N+1 ) = G d (L 2, L 1 ) + + G d (L q, L 1 ), where L 1,..., L q are linear forms on C N+1 (q N + 2), which satisfy conditions (B 2 ) and (B 4 ).

12 Theorem 2.2.14. Let signs, assumptions as in (B 6 ) and S be a unique range set for meromorphic functions, d (2q 3) 2, L i (f 1,..., f N+1 ), L 1 (f 1,..., f N+1 ) (L i (g 1,..., g N+1 ), L 1 (g 1,..., g N+1 )), (i = 2,..., q) have no common zeros, the functional equation H(f 1,..., f N+1 ) = H(g 1,..., g N+1 ), (2.4) with {f 1,..., f N+1 }, {g 1,..., g N+1 } are families of linearly independent entire functions. Then the functional equation (2.4) has solution (cg 1,..., cg N+1, g 1,..., g N+1 ) with c nd = 1. 2.3. Determine holomorphic curves through inverse mapping of point sets Let X, Y be two hypersurfaces of P N (C), f and g be two nondegenerate holomorphic curves from C to P N (C) with reduced representation f = (f 1,..., f N+1 ), g = (g 1,..., g N+1 ), respectively. Let X 1, Y 1 be two hypersurfaces of P n (C), which is defined by the q q equations X 1 : (u j x) d = 0 and Y 1 : (ν j x) d = 0, respectively. j=1 j=1 Theorem 2.3.1. Let q, d, N N, d (2q + 1) 2, q N + 1. Then the following statements are equivalent: i) There exist non-degenerate holomorphic curves f and g satisfying ν X 1 f = ν Y 1 f. ii) There exist a bijective σ from {1,..., q} to {1,..., q} such that B 1 σ(α) Ω 1 α A α = B 1 σ(β) Ω 1 β A β, (α, β Q). More over, if the necessary condition, f t = A 1 α Ω σ(α) B σ(α) (hg) t, where h are entire function which has no zeros and α = (1,..., N + 1). Theorem 2.3.2. Let q, d, N N, d (2q + 1) 2, q N + 2, if α β and α α hoc β β then A 1 α Ω 1 α A α A 1 β Ω 1 β A β, where α, β, α, β Q. Suppose that f, g be two linearly non-degenerate holomorphic curves satisfying ν X 1 f = ν X 1 g. Khi f = g. Theorem 2.3.3. Let n 2m + 9, (n, m) = 1, d (2N 1) 2. Then, the following statements are equivalent: i) There exist linearly non-degenerate holomorphic curves f and g satisfying ν X 2 f = ν Y 2 g.

13 ii) There exist a bijective σ d from {1,..., N} to {1,..., N}, such that with the numbers c iσ(i), l 1, l σ(i)+1 satisfying c d iσ(i) = 1, b i ln 1 =, c iσ(i) d σ(i) lσ(i)+1 n = 1, l n m a i σ(i)+1 c lm 1 =, i = 1,..., N; g i = l i f i, i = iσ(i) c iσ(i) a σ(i) 1,..., N + 1. Let q, d, m, n be positive integers, m < n and q linear forms: L i (z 1,..., z N+1 ), i = 1,..., q, a i, b i 0, i = 1,..., q 1. Consider the following homogeneous polynomials: T i (z 1,..., z N+1 ) = L n i+1(z 1,..., z N+1 ) a i L n m (z 1,..., z N+1 )L m 1 (z 1,..., z N+1 ) + b i L n 1(z 1,..., z N+1 ), T (z 1,..., z N+1 ) = T d 1 (z 1,..., z N+1 ) + + T d q 1(z 1,..., z N+1 ). Denote by X 3 the hypersurface in P N (C), which is defined by the equation T (z 1,..., z N+1 ) = 0. Theorem 2.3.4. The polynomial T (z 1,..., z N+1 ) is an uniqueness polynomial for holomorphic curves. Theorem 2.3.5. Let f and g be two linearly non-degenerate holomorphic mappings from C to P N (C). Let X 3 be the Fermat-Waring hypersurface defined as above. Assume that: 1. ν X 3 f =ν X 3 g, 2. n 2m + 9, m 2, (m, n) = 1, q > n; d (2q 3) 2, 3. b 2d i b d j bd l vi i j, i l. Then f = g. Theorem 2.3.6. Let d (2q 1) 2, n 2m + 9, (n, m) = 1, m 2. Then there no exists be two linearly non-degenerate holomorphic curves f, g satisfying ν X 4 f = ν Y 4 g Theorem 2.3.7. Let S be a uniqueness range set for meromorphic functions, d (2q 3) 2, f and g be two linearly non-degenerate holomorphic curves satisfying f 1 (H i ) f 1 (H 1 ) =, g 1 (H i ) g 1 (H 1 ) =, i = 2,..., q, and ν X 4 f = ν X 4 g. Then f = g.

14 Chapter 3 Determine moromorphic functions and holomorphic curves on a field non-archimedean 3.1. Some concepts and supplementary results In this section, we review some concepts and results same as Section 1.1 and 2.1 for case p-adic. 3.2. The equation of Fermat-Waring type of variables for non-archimedean entire functions Lemma 3.2.1. [29] Let d, N N, q i N and z d q i i D i (z 1, z 2,..., z N+1 ) be a family in general position of homogeneous polynomials with coefficients in K of degree d such that f d q i i D i (f 1,..., f N+1 ) 0, 1 i N + 1. Suppose N+1 i=1 N+1 f d q i i D i (f 1,..., f N+1 ) = 0, d N 2 1 + i=1 q i, N > 1. Then f d q 1 1 D 1 (f 1,..., f N+1 ),..., f d q N N D N (f 1,..., f N+1 ) are linearly dependent on K. Lemma 3.2.2. Let n, n 1, n 2,..., n q, q N, a 1,..., a q, c K, c 0, and q 2 + q n i i=1. Then the functional equation n (f a 1 ) n 1 (f a 2 ) n 2... (f a q ) n q = cg n has no non-constant meromorphic solutions (f, g).

15 Lemma 3.2.3. Let n, m N, n 2m + 8, a 1, b 1, a 2, b 2, c K, a 1 0, b 1 0, a 2 0, b 2 0, c 0, and f 1, f 2, g 1, g 2 be non-zero entire functions. 1. Suppose that f 1 f 2 is a non-constant moromorphic function, and f n 1 + a 1 f n m 1 f m 2 + b 1 f n 2 = b 2 g n 2. (3.1) Then there exists c 1 0 such that c 1 b 2 g n 2 = b 1 f n 2, g 2 = hf 2 vi b 1 = c 1 b 2 h n, h K. 2. Suppose that f 1 and g 1 are non-constant meromorphic functions, f 2 g 2 and f n 1 + a 1 f n m 1 f m 2 + b 1 f n 2 = c(g n 1 + a 2 g n m 1 g m 2 + b 2 g n 2 ). (3.2) i. If m 2 then ii. If m 3 then cb 2 g n 2 = b 1 f n 2, g 2 = hf 2 vi b 1 = cb 2 h n, h K. g 1 = lf 1, g 2 = hf 2 vi 1 = cl n, a 1 = ca 2 l n m h m, b 1 = cb 2 h n, l, h K. 3.3. Determine moromorphic functions and holomorphic curves on a field non-archimedean Let S i = {c i, d i }, where c i, d i be roots of polynomial P i = z 2 +a i z+b i = 0, a i a j, b i b j, b i a2 i 4, q 5 and a i a k b i b k a j a k b j b 0, (C 1 ) k with i, j, k are distinct in {1, 2,..., q}. Theorem 3.3.1. With the condition (C 1 ) and let f, g be two meromorphic functions on K satisfying E f (S i ) = E g (S i ), i = 1,..., q. Then f = g. Lemma 3.3.2. Let distinct values a, b, 0, in K { } and T (z) = z ac z + ac, (b = ac2 ). Then T ( ) + T (0) = 0 = T (a) + T (b).

16 ) ) 1 1 With a K { }, denote N N (r,, N N (r, are the f a f a kj without multiplicity counting functions of f such that f(z) = a = g(z), f(z) = a g(z). Lemma 3.3.3. Let {a i }, i = 1,..., p; S k = {a k1, a k2 }, k = 1,..., q; l cc tp phn bit gm cc phn t phn bit; f and g be two non-constant meromorphic functions on K satisfying i = 1,..., p, k = 1,..., q. Then E f (a i ) = E g (a i ), E f (S k ) = E g (S k ), q 2 ) 1 (q + 2p 4)(T (r, f) + T (r, g)) [N N (r, f a k=1 j=1 kj ) 1 + N N (r, ] log r + O(1). g a kj Theorem 3.3.4. Let distinct element a i, (i = 1,..., p); a k1, a k2, (k = 1,..., q) in K. Set S k = {a k1, a k2 }, k = 1,..., q. Suppose f and g be two meromorphic functions over K satisfying E f (a i ) = E g (a i ), E f (S k ) = E g (S k ), i = 1,..., p, k = 1,..., q. Then p + q 3, p, q 1 implies f l mt bin i phn tuyn tnh ca g. Let given q linear forms of N + 1 variables (q > N + 1) in general position in K N+1 : L i = L i (z 1,..., z N+1 ) = α i,1 z 1 +α i,2 z 2 + +α i,n+1 z N+1, i = 1, 2,..., q. Let n, m, be positive integers, m < n, a, b K, a, b 0. The following polynomial is called a Y i -polynomial: Y (m,n) (z 1, z 2 ) = z n 1 az n m 1 z m 2 + bz n 2. Now consider q homogeneous polynomials: P 1 = P 1 (z 1,..., z N+1 ) = Y (m,n) (L 1, L 2 ) = L n 1 al n m 1 L m 2 + bl n 2, and for q i 2, set: P i = P i (z 1,..., z N+1 ) = Y (m,n) (P i 1, L ni 1 i+1 ).

17 Then we consider the following polynomial of Fermat-Waring type of degree n q : P (z 1, z 2,..., z N+1 ) = P q (z 1,..., z N+1 ). The polynomial P (z 1, z 2,..., z N+1 ) is called a q-iteration of Y i -polynomials. For entire functions f 1,..., f N+1 and g 1,..., g N+1 over K, we consider the following equation P (f 1,..., f N+1 ) = P (g 1,..., g N+1 ). Denote by X the hypersurface of Fermat-Waring type in P N (K), which is defined by the equation P (z 1,..., z N+1 ) = 0. We shall prove the following theorems Theorem 3.3.5. Let P (z 1, z 2,..., z N+1 ) be a q-iteration of Y i -polynomials, n 2m + 8, m 3, and f 1,..., f N+1 ; g 1,..., g N+1 be two families of linearly independent entire functions over K, satisfying the equation P (f 1,..., f N+1 ) = P (g 1,..., g N+1 ). Then g i = cf i, c nq = 1, i = 1,..., N+ 1. Theorem 3.3.6. Let f and g be two linearly non-degenerate holomorphic mappings from K to P N (K). Let X be the Fermat-Waring hypersurface defined by the equation P (z 1,..., z N+1 ) = 0, where P (z 1, z 2,..., z N+1 ) be a q-iteration of Y i -polynomials, and n 2m + 8, m 3. Then µ X f = µx g implies f = g.

18 Final conclusion and further recommendations The dissertation investigates the problem of the determination of holomorphic function and mapping through inverse mapping of point sets (complex and p -adic). The objective of the thesis is to establish a class of unique range set and uniqueness polynomial in the above cases. The mail results of the thesis. 1. The dissertation has attempted to give some conditions for some functional equations that have solutions and describe solutions of some functional equations. Moreover, the thesis has established uniqueness theorems for non-constant meromorphic functions and linearly non-degenerate holomorphic curves; Two pairs of hypersurfaces were determined linearly non-degenerate holomorphic curves; Three hypersurfaces were uniquely determined linearly non-degenerate holomorphic curves. These results are extension of the Five Value and Four Value theorems of Nevanlinna and in response to questions by F.Gross and Pakovich. 2. The dissertation has established uniqueness theorems for non Archimedean meromorphic functions under the inverse images of the four sets of two points, ignoring multiplicities and two sets of two points counting multiplicities. We establish a class of unique polynomials and hypersurfaces of Fermat- Waring type were uniquely determined non Archimedean holomorphic mappings.

19 The list of published works related to thesis 1. Vu Hoai An and Le Quang Ninh (2012), Uniqueness polynomials for linearly non-degenerate holomorphic curves, Proc.20th Intern. Conf. Finite or Infinite Dimensional Complex Analysis and Applications, Science and Technics publishing house. 2. Vu Hoai An and Le Quang Ninh (2016), On functional equations of the Fermat-Waring type for non-archimedean vectorial entire functions, Bull. Korean Math, 53(4), pp.1185-1196. 3. Ha Huy Khoai, Vu Hoai An and Le Quang Ninh (2014), Uniqueness Theorems for Holomorphic Curves with Hypersurfaces of Fermat- Waring Type, Complex Analysis and Operator Theory, 8, pp 1747-1759. 4. Le Quang Ninh (2015), Uniqueness polynomials for linearly nondegenerate p-adic holomorphic curves, Journal of science and technology Thai Nguyen university, 144 (14).