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HANOI NATIONAL UNIVERSITY OF EDUCATION—————————– Pham Duc Thoan ON THE RELATION DEFECT AND THE ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS Specialized: Geometry and Topology Code: 62.4

HANOI NATIONAL UNIVERSITY OF EDUCATION

—————————–

Pham Duc Thoan

ON THE RELATION DEFECT AND THE ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS

Specialized: Geometry and Topology

Code: 62.46.10.01

SUMMARY DOCTOR OF PHILOSOPHY IN MATHEMATICS

Hanoi, 01-2011

Thesis was completed at: Hanoi national University of Education

Science instructor: Prof Dr Do Duc Thai

Rewier 1: Prof Dr Nguyen Van Mau, Hanoi University of Vietnam National University

Science-Rewier 2: Prof Dr Le Hung Son, Hanoi University of Technology.Rewier 3: Prof Dr Nguyen Van Khue, Hanoi National University

of Education

Thesis will be approved by School committee at hour date month year

Thesis can be found at: -Viet Nam national library

-Library of Hanoi National University ofEducation

1 Reasons for selecting topics

In the late 20’s last century, Nevanlinna foundated the valuedistribution theory of the meromorphic function of a variable Overthe next decade many mathematicians in the world such as H.Cartan, W Stoll, PA Griffiths, L Carlson, P Vojta, J Noguchi interest in research and develop on Nevanlinna theory for moregeneral object class So far, Nevanlinna theory has become one

of the most important theory of mathematics with many beautifultheorems have been proved The most striking result is that theinequality in terms of defects and unicity theorem By the attractivenature of the geometric theory, we have chosen the theme ”Onthe relation defect and the algebraic dependences ofmeromorphic mappings” Specifically, we focus on research andhas given some results on the defects for meromorphic functions to

P1(C) and meromorphic mappings to Pn(C), and we also study thealgebraic dependences and apply these results to the study unicityproblem with truncated multiplicity for meromorphic mappings ofseveral complex variables

2 The aim and subject of thesis

The main aim of the thesis is to study the meromorphic havingmaximal defect sum and the algebraic dependence of the meromor-phic mappings

Stadying subject is the meromorphic mappings with a maximumwith maximal defect sum and the algebraic dependence of themeromorphic mappings

3 Studying methods used in thesis

Using knowledges about Complex Geomatry and Complex ysis, Nevanlinna theory Simultaneously, we also created new tech-niques to solve the issues raised in the thesis The first is the study

Anal-of the maximal defect sum Anal-of the meromorphic function, we havedevised a ”noise” by ”small” function The second is the study ofunique problems of meromorphic mapping, the authors often proveddirectly and through the second fundamental theorem Here, we ap-proach the problem with the theory of ”algebraic dependence” of themeromorphic mappings of several complex variables that W Stollproposed

4 The results of the thesis

Among the theorems that Nevanlinna proved, the theorem aboutthe relationship of defect to keep a special role Namely, the theorem

is stated the following:

Theorem A If f be a nonconstant meromorphic function on

Toda proved the following theorem:

Theorem B Let f : Cm −→ Pn(C) be a linearly nondegenerate,

and let {Hj}qj=1 be hyperplanes in N -subgeneral position in

Pn(C), where 1 ≤ n < N and 2N − n + 1 < q ≤ +∞ Assume

Then one of the following two statements holds:

(I) There are at least  2N − n + 1

n + 1

+ 1 of the hyperplanes Hj

at which f has deficiency value 1, i.e δ(Hj, f ) = 1,

(II) {Hj}qj=1 has a Borel distribution

Continue the above research , in the first two chapters of the thesis

we study the class meromorphic mappings which has a maximal

de-fect sum Namely, in Chapter 1 we showed the necessary condition

for the class meromorphic function has a maximal defect function,

also indicate that the meromorphic function is very small

Specifi-cally, we have proved the following two theorems

Theorem 1.3.1 Let f : C → P1(C) be a meromorphic function

of finite order For each n ≥ 1, define gn(z) = f (zn), ∀z ∈ C and

hn(z) = fn(z), ∀z ∈ C Then we have necessarily λ := ρf ∈ Z+

and λ equals the lower order of f

1) If there exists n0 ≥ 2 such that P

a∈Cδ(a, gn0) = 2

2) If there exists a sequence {ni}+∞i=1 ⊂ Z+ such that Pa∈Cδ(a, hni) =

2, ∀i ≥ 1

Theorem 1.3.2 Let f : Cm → P1

(C) be a meromorphic tion of finite order satisfying

func-λ := ρf ∈ Z and/ P

a∈C δ(a, f ) = 2

Denote by A the set of all nonconstant meromorphic functions

h : Cm → P1(C) such that Th(r) = o Tf(r)
, TDh(r) = o TDf(r)
.Then, for each h ∈ A, we have

P

a∈Cδ(a, f + h) 6 2 − 2k(λ) < 2,where k(λ) is a positive constant which depends only on λ

Chapter 2 of the thesis has extended the results of N Toda forclass meromorphic mappings of several variables that have maximaldefect sum for moving targets Namely, we have proved the followingtheorem

Theorem 2.3.1 Let f : Cm −→ Pn(C) be a nonconstantmeromorphic mapping, and let {ai}q−1i=0 be ”small” (with respect

to f ) meromorphic mappings of Cm into Pn(C) in N − subgeneralposition such that f is linearly nondegenerate over R({ai}q−1i=0),where 1 ≤ n < N and 2N − n + 1 < q < +∞ Suppose furtherthat f has nonzero deficiency value at ai for each 0 ≤ i ≤ q − 1and Pq−1j=0δ (aj, f ) = 2N − n + 1

Then one of the following two statements holds

(I) There are at least [2N − n + 1

n + 1 ] + 1 of the moving targets aj

at which f has deficiency value 1, i.e δ(aj, f ) = 1 ,

(II) n is odd and the family {aj}q−1j=0 has a Borel distribution

In 1926, Nevanlinna proved that if f and g be two non-constantmeromorphic functions on C such that f−1(ai) = g−1(ai) at 5

distinction point a1, · · · , a5 then f ≡ g In the context of the reviewtheorem 5 point of Nevanlinna for meromorphic function of severalcomplex variables into complex projective space, in 1975 H Fujimotoproved the following important theorem.

Theorem C Let Hi (1 ≤ i ≤ 3N + 2) be 3N + 2 hyperplanes

in general position in PN(C), f and g be two non-constantmeromorphic mappings from Cn to PN(C) such that f (Cn) *

Hi, g(Cn) * Hi Assume that v(f,Hi) = v(g,Hi) with 1 ≤ i ≤ 3N +2.Then, if f or g be linearly nondependence then f ≡ g

In the last decade many works have continued to develop on theresults of H Fujimoto and has formed a research direction in theNevanlinna theory is the study problem unicity (also known as theunicity theorem.) In particular, the unicity theorem has been studiedcontinuously in recent years and has obtained deep results Amongthe approaches to the unicity problem has a method by W Stollproposed, research unicity problem that is through study of the al-gebraic dependence of meromorphic mappings Development of theabove ideas of W Stoll, in 2001 M.Ru showed unicity theorem forholomorphic curves into complex projective space with moving tar-gets Namely, M Ru proved the following:

Theorem D Let f and g be non-constant meromorphic tions if there exists 7 distinction meromorphic functions

func-a1, a2, · · · , a7 such that Taj(r) = o(max{Tf(r), Tg(r)}) (0 ≤ j ≤7) and f (z) = aj(z) ⇔ g(z) = aj(z) then f ≡ g

Continue the above research , in the chapters 3 of the thesis, wehave showed some unicity theorem for meromorphic mappings ofseveral variables to complex projective space through the study of

the algebraic dependence of their mapping The results that weachieved a significant expansion for the theorems of M Ru Namely,

we proved the following theorem:

Theorem 3.2.4 Let f1, · · · , fλ : Cm → Pn(C) be nonconstantmeromorphic mappings Let gi : Cm → Pn(C) (1 ≤ i ≤ q) bemoving targets located in general position such that T (r, gi) =o(max1≤j≤λ T (r, fj)) (1 ≤ i ≤ q) and (fi, gj) 6≡ 0 for 1 ≤

i ≤ λ, 1 ≤ j ≤ q Let κ be a positive integer or κ = +∞and κ = min{κ, n} Assume that the following conditions aresatisfied

i) min{κ, v(f1,gj)} = · · · = min{κ, v(fλ,gj)} for each 1 ≤ j ≤ q,ii) dim{z|(f1, gi)(z) = (f1, gj)(z) = 0} ≤ m − 2 for each

q > n(n + 2)λ − (κ − 1)(λ − 1)

then f1, · · · , fλ are algebraically dependent over C

iii) If fi, 1 ≤ i ≤ λ are linearly nondegenerate over C, gi (1 ≤

i ≤ q) are constant mappings and (q − n − 1)((λ − 1)(κ − 1) +q(λ − l + 1)) ≤ qnλ, then f1, · · · , fλ are algebraically dependent

pos-i) min{κ, v(f1,gj)(z)} = min{κ, v(f2,gj)} for each z ∈ Cm and

5 Structure of the thesis

The layout of the thesis out of the introduction and the conclusion,the thesis consists of three chapters

Chapter I: ”On meromorphic functions with maximaldefect sum”

Chapter II: ”Meromorphic mappings with maximal fect sum for moving targets”

de-Chapter III: ”The algebraic dependence of the phic mappings and applications”

Theorem of classical Nevanlinna deficiency relation was pointedout that if f : Cm −→ P1

(C) be a meromorphic function thenP

As stated in the introduction, the purpose of this chapter is tocontinue to study the problem on the meromorphic function ofthe fixed target Namely, we show the necessary conditions ofthe maximality of defect sum Later, we show that the class ofmeromorphic functions with maximal defect sum is very thin inthe sense that deformations of meromorphic functions with maximaldefect sum by small meromorphic functions are not meromorphicfunctions with maximal defect sum Further more, we can measurethe deviation of defect sum of meromorphic functions before andafter by constant

1.2 Some initial results

The purpose of this section is to prove the Lemmas prepare for two

of the first main result of the thesis

Lemma 1.2.3 Let f, g : Cm → P1(C) be nonconstantmeromorphic functions of finite order Assume that ρf = λ, ρg =

α : f 7→ f1 v βa : f 7→ f + a, ∀a ∈ C

Lemma 1.2.7 Let f, g : Cm → P1

(C) be nonconstant morphic functions of finite order Then, one of the followingtwo assertions holds:

meromorphic function such that δ(∞, f ) = 0 Then

TDg(r) 6 n+1n Tg(r) + O(log rTf(r))

Lemma 1.2.15 Let f : Cm → P1(C) be a nonconstantmeromorphic function of finite order Then, there exists ameromorphic function f1 : Cm → P1

(C) of finite order suchthat

The purpose of this chapter is to prove the following two results:Theorem 1.3.1 Let f : C → P1(C) be a meromorphic function

of finite order For each n ≥ 1, define gn(z) = f (zn), ∀z ∈ C and

hn(z) = fn(z), ∀z ∈ C Then we have necessarily λ := ρf ∈ Z+

and λ equals the lower order of f

1) If there exists n0 ≥ 2 such that P

Then, for each h ∈ A, we have Pa∈Cδ(a, f + h) 6 2 − 2k(λ) < 2,

where k(λ) is a positive constant which depends only on λ

Chapter 2

Meromorphic mapping with maximal with

defect sum for moving targets

This chapter is based on the article [3] The chapter for the study ofthe meromorphic mappings from Cm to Pn(C) with maximal defectsum for moving targets

For about 20 years, the study of Nevanlinna theory for movingtargets were more interested in mathematics This is a natural ex-tension when we replace the hyperplane (or hypersurface) fixed inthe complex projective space with a small coefficient of the function.One of the most important results of this research is the theoremCartan-Nochka for moving targets proved by M Ru and W Stoll.Theorem Let f : Cm −→ Pn

(C) be a nonconstant phic mapping and assume that {ai}q−1i=0 be ”small” meromorphicmappings corresponding to f from Cm to Pn(C) in N-subgeneralposition such that f is linearly nondependence on R({ai}q−1i=0).Then

meromor-Pq−1 j=0δ (aj, f ) ≤ 2N − n + 1

Thus there is a question naturally arises:What we can say about thefunctions f which is the number of defects for maximal? In otherwords, we can extend the results of N Toda for the meromorphicmappings of several variables that have maximal defect sum withmoving targets or not? The main purpose of this chapter is to answerthat question

2.2 The initial results

First we recall two of the Nochka weight Lemma for moving targets.

How to prove they are repeated all the claims corresponding to the

fixed hyperplane

Lemma 2.2.1 Let {ai}i∈Q be q moving targets in Pn(C) in N

-subgeneral position, and assume that q > 2N − n + 1 Then there

are positive rational constants ωj, j ∈ Q satisfying the following:

The above ωj are called N ochka weights, and ˜ω the N ochka constant

We will denote θ = ˜ω−1 for later convenience

Lemma 2.2.2 Let q > 2N − n + 1, and let {ai}i∈Q be q

moving targets in PnC in N -subgeneral position Let {ωj}j∈Q

be its Nochka weights Let Ej ≥ 1, j ∈ Q be arbitrarily given

numbers Then for every subset R ⊂ Q with 0 < |R| ≤ N + 1,

there is a subset Ro ⊂ R such that |Ro| = rank{ai}i∈R and

Q

i∈REωi

i ≤ Q

i∈Ro Ei.The purpose of this section is to prove lemma plays an important

role in proving the theorem on the deficiency of meromorphic

map-pings with maximal deficiency sum for moving targets

Lemma 2.2.7 Let f be a meromorphic mapping of Cm into

Pn(C) with a reduced representation f = (f0 : · · · : fn) Consider

N > n and q be any integer satisfying 2N −n+1 < q < +∞ Put

Q = {0, 1 , q − 1} Let X = {aj : j ∈ Q} be the set of ”small”(with respect to f ) meromorphic mappings from Cm into Pn(C)

in N -subgeneral position Assume that f is nondegenerate overR({ai}q−1i=0) and any function ω : Q → (0, 1] satisfy the condition(iv) in Lemma 2.2.1, we have

Pq−1 j=0ω(j) · δ (aj, f ) ≤ n + 1

2.3 The meromorphic mapping with maximal ciency sum

defi-In this section, we use the above lemmas to prove the theorems

on the deficiency of meromorphic mappings with maximal deficiencysum for moving targets

Theorem 2.3.1 Let f : Cm −→ Pn

(C) be a nonconstantmeromorphic mapping, and let {ai}q−1i=0 be ”small” (with respect

to f ) meromorphic mappings of Cm into Pn(C) in N − subgeneralposition such that f is linearly nondegenerate over R({ai}q−1i=0),where 1 ≤ n < N and 2N − n + 1 < q < +∞ Suppose furtherthat f has nonzero deficiency value at ai for each 0 ≤ i ≤ q − 1and

Pq−1 j=0δ (aj, f ) = 2N − n + 1

Then one of the following two statements holds

(I) There are at least [2N − n + 1

n + 1 ] + 1 of the moving targets aj atwhich f has deficiency value 1, i.e δ(aj, f ) = 1 ,

(II) n is odd and the family {aj}q−1j=0 has a Borel distribution

Chapter 3

The algebraic dependences of meromorphic

mappings and applications

This chapter for the study of algebraic dependence of the morphic mapping from Cm to Pn(C) with moving targets in generalposition and application to the study of unicity problem The chapter

mero-is based on the article [1] end [2]

The theory on algebraic dependences of meromorphic mappings

in several complex variables into the complex projective spacesfor fixed targets is studied by W Stoll in 1989 Later, Min Rugeneralized Stoll’s result to holomorphic curves into the complexprojective spaces for moving targets and show some unicity theorems

of holomorphic curves into the complex projective spaces for movingtargets As far as we know, they are the first results on the unicityproblem for moving targets We now state his remarkable results.Let g1, , gq (q ≥ n) be q meromorphic mappings of Cm into

Pn(C) with reduced representations gj = (gj0 : · · · : gjn) (1 ≤

j ≤ q) We say that g1, , gq are located in general position ifdet(gjkl) 6≡ 0 for any 1 ≤ j0 < j1 < < jN ≤ q

Let Mn be the field of all meromorphic functions on Cm Denote