Algebraic dependences of meromorphic mappings in several complex variables

We give some theorems on algebraic dependence of meromorphic mappings in several complex variables into complex projective spaces.

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Бібліографічні деталі
Дата:	2010
Автори:	Duc Thoan Pham, Viet Duc Pham
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2010
Назва видання:	Український математичний журнал
Теми:	Статті
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Цитувати:	Algebraic dependences of meromorphic mappings in several complex variables / Duc Thoan Pham, Viet Duc Pham // Український математичний журнал. — 2010. — Т. 62, № 7. — С. 923–936. — Бібліогр.: 6 назв. — англ.

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spelling	irk-123456789-1661762020-02-19T01:26:20Z Algebraic dependences of meromorphic mappings in several complex variables Duc Thoan Pham Viet Duc Pham Статті We give some theorems on algebraic dependence of meromorphic mappings in several complex variables into complex projective spaces. Наведено деякі теореми про алгебраїчну залежність мероморфних відображень для багатьох комплексних змінних на комплексні проективні простори. 2010 Article Algebraic dependences of meromorphic mappings in several complex variables / Duc Thoan Pham, Viet Duc Pham // Український математичний журнал. — 2010. — Т. 62, № 7. — С. 923–936. — Бібліогр.: 6 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166176 517.946 en Український математичний журнал Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Статті Статті
spellingShingle	Статті Статті Duc Thoan Pham Viet Duc Pham Algebraic dependences of meromorphic mappings in several complex variables Український математичний журнал
description	We give some theorems on algebraic dependence of meromorphic mappings in several complex variables into complex projective spaces.
format	Article
author	Duc Thoan Pham Viet Duc Pham
author_facet	Duc Thoan Pham Viet Duc Pham
author_sort	Duc Thoan Pham
title	Algebraic dependences of meromorphic mappings in several complex variables
title_short	Algebraic dependences of meromorphic mappings in several complex variables
title_full	Algebraic dependences of meromorphic mappings in several complex variables
title_fullStr	Algebraic dependences of meromorphic mappings in several complex variables
title_full_unstemmed	Algebraic dependences of meromorphic mappings in several complex variables
title_sort	algebraic dependences of meromorphic mappings in several complex variables
publisher	Інститут математики НАН України
publishDate	2010
topic_facet	Статті
url	http://dspace.nbuv.gov.ua/handle/123456789/166176
citation_txt	Algebraic dependences of meromorphic mappings in several complex variables / Duc Thoan Pham, Viet Duc Pham // Український математичний журнал. — 2010. — Т. 62, № 7. — С. 923–936. — Бібліогр.: 6 назв. — англ.
series	Український математичний журнал
work_keys_str_mv	AT ducthoanpham algebraicdependencesofmeromorphicmappingsinseveralcomplexvariables AT vietducpham algebraicdependencesofmeromorphicmappingsinseveralcomplexvariables
first_indexed	2025-07-14T20:54:39Z
last_indexed	2025-07-14T20:54:39Z
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fulltext	UDC 517.946 Duc Thoan Pham (Hanoi Nat. Univ. Education, Vietnam), Viet Duc Pham (Thai Nguyen Univ. Education, Vietnam) ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES* АЛГЕБРАЇЧНА ЗАЛЕЖНIСТЬ МЕРОМОРФНИХ ВIДОБРАЖЕНЬ ДЛЯ БАГАТЬОХ КОМПЛЕКСНИХ ЗМIННИХ In this article, some algebraic dependence theorems of meromorphic mappings in several complex variables into the complex projective spaces are given. Наведено деякi теореми про алгебраїчну залежнiсть мероморфних вiдображень для багатьох комплекс- них змiнних на комплекснi проективнi простори. 1. Introduction. The theory on algebraic dependences of meromorphic mappings in several complex variables into the complex projective spaces for fixed targets is studied by Wilhelm Stoll [1]. Later, Min Ru [2] generalized Stoll’s result to holomorphic curves into the complex projective spaces for moving targets and show some unicity theorems of holomorphic curves into the complex projective spaces for moving targets. As far as we know, they are the first results on the unicity problem for moving targets. We now state his remarkable results. Let g0, . . . , gq−1, q ≥ N, be q meromorphic mappings of Cn into PN (C) with reduced representations gj = (gj0 : . . . : gjN , 0 6 j 6 q − 1. We say that g0, . . . , gq−1 are located in general position if det(gjkl) 6≡ 0 for any 0 6 j0 < j1 < . . . < jN 6 q−1. LetMn be the field of all meromorphic functions on Cn. Denote byR ({ gj }q−1 j=0 ) ⊂ ⊂Mn the smallest subfield which contains C and all gjk gjl with gjl 6≡ 0. Let f be a meromorphic mapping of Cn into PN (C) with reduced representation f = (f0 : . . . : fN ). We say that f is linearly nondegenerate over R ({ gj }q−1 j=0 ) if f0, . . . , fN are linearly independent over R ({ gj }q−1 j=0 ) . Let ft : Cn → PN (C), 1 6 t 6 λ, be meromorphic mappings with reduced repre- sentations ft := (ft0 : . . . : ftN ). Let gj : Cn → PN (C), 0 6 j 6 q − 1, be moving targets located in general position with reduced representations gj := (gj0 : . . . : gjN ). Assume that (ft, gj) := ∑N i=0 ftigji 6= 0 for each 1 6 t 6 λ, 0 6 j 6 q − 1 and (f1, gj) −1{0} = . . . = (fλ, gj) −1{0}. Put Aj = (f1, gj) −1{0} for each 0 6 j 6 q − 1. Assume that every analytic set Aj has the irriducible decomposition as follows Aj = = ∪tji=1Aji, 1 6 tj 6 ∞. Set A = ∪Aji 6≡Akl{Aji ∩ Akl} with 1 6 i 6 tj , 1 6 l 6 tk, 0 6 j, k 6 q − 1. Denote by T [N+1, q] the set of all injective maps from {1, . . . , N+1} to {0, . . . , q− − 1}. For each z ∈ Cn \ {∪β∈T [N+1,q]{z\|gβ(1)(z) ∧ . . . ∧ gβ(N+1)(z) = 0} ∪ A ∪ ∪ ∪λi=1I(fi)}, we define ρ(z) = ]{j\|z ∈ Aj}. Then ρ(z) 6 N. Indeed, suppose that z ∈ Aj for each 0 6 j 6 N. Then ∑N i=0 f1i(z) · gji(z) = 0 for each 0 6 j 6 N. Since *The research of the authors is supported in part by an NAFOSTED grant of Vietnam. c© DUC THOAN PHAM, VIET DUC PHAM, 2010 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 923 924 DUC THOAN PHAM, VIET DUC PHAM gβ(1)(z) ∧ . . . ∧ gβ(N+1)(z) 6= 0, it implies that f1i(z) = 0 for each 0 6 i 6 N. This means that z ∈ I(f1). This is impossible. For any positive number r > 0, define ρ(r) = sup{ρ(z) \| \|z\| 6 r}, where the supremum is taken over all z ∈ Cn \ { ∪β∈T [N+1,q] {z\|gβ(1)(z) ∧ . . . ∧ gβ(N+1)(z) = = 0} ∪A ∪ ∪λi=1I(fi) } . Then ρ(r) is a decreasing function. Let d := lim r→+∞ ρ(r). Then d 6 N. If for each i 6= j, dim{Ai ∩Aj} 6 n− 2, then d = 1. Theorem A (see [2], Theorem 1). Let f1, . . . , fλ : C → PN (C) be nonconstant holomorphic curves. Let gi : C → PN (C), 0 6 i 6 q − 1, be moving targets located in general position and T (r, gi) = o(max16j6λ T (r, fj)), 0 6 i 6 q − 1. Assume that (fi, gj) 6≡ 0 for 1 6 i 6 λ, 0 6 j 6 q − 1, and Aj := (f1, gj) −1{0} = . . . . . . = (fλ, gj) −1{0} for each 0 6 j 6 q − 1. Denote A = ∪q−1j=0Aj . Let l, 2 6 l 6 λ, be an integer such that for any increasing sequence 1 6 j1 < . . . < jl 6 λ, fj1(z)∧ . . . . . . ∧ fjl(z) = 0 for every point z ∈ A. If q > dN2(2N + 1)λ λ− l + 1 , then f1, . . . , fλ are algebraically dependent over C, i.e., f1 ∧ . . . ∧ fλ ≡ 0 on C. Theorem B (see [2], Theorem 2). In addition to the assumption in Theorem A we assume further that fi, 1 6 i 6 λ, are linearly nondegenerated. Then f1, . . . , fλ are algebraically dependent over C, i.e., f1 ∧ . . . ∧ fλ ≡ 0 on C, if q > dN(N + 2)λ λ− l + 1 . With the same assumption on the nondegeneracy of small moving targets, it is our main purpose of the present paper to show some algebraic dependence theorems of meromorphic mappings from Cn into PN (C) for moving targets in more general situ- ations. Namely, we are going to prove the following. Theorem 1. Let f1, . . . , fλ : Cn → PN (C) be nonconstant meromorphic map- pings. Let gi : Cn → PN (C), 0 6 i 6 q − 1, be moving targets located in general position and T (r, gi) = o(max16j6λ T (r, fj)), 0 6 i 6 q− 1. Assume that (fi, gj) 6≡ 0 for 1 6 i 6 λ, 0 6 j 6 q−1, and Aj := (f1, gj) −1{0} = . . . = (fλ, gj) −1{0} for each 0 6 j 6 q − 1. Denote A = ∪q−1j=0Aj . Let l, 2 6 l 6 λ, be an integer such that for any increasing sequence 1 6 j1 < . . . < jl 6 λ, fj1(z) ∧ . . . ∧ fjl(z) = 0 for every point z ∈ A. Then f1, . . . , fλ are algebraically dependent over C, i.e., f1 ∧ . . . ∧ fλ ≡ 0 on C, if q > dN(2N + 1)λ λ− l + 1 . Theorem 2. In addition to the assumption in Theorem 1 we assume further that fi, 1 6 i 6 λ, are linearly nondegenerate over R ( {gj}q−1j=0 ) . Then f1, . . . , fλ are algebraically dependent over C, i.e., f1 ∧ . . . ∧ fλ ≡ 0 on C, if q > dN(N + 2)λ λ− l + 1 . Theorem 3. Let f1, . . . , fλ : Cn → PN (C) be nonconstant meromorphic map- pings. Let gi : Cn → PN (C), 0 6 i 6 q − 1, be moving targets located in general position such that T (r, gi) = o(max16j6λ T (r, fj)), 0 6 i 6 q−1, and (fi, gj) 6≡ 0 for 1 6 i 6 λ, 0 6 j 6 q − 1. Let κ be a positive integer or κ =∞ and κ = min{κ, N}. Assume that the following conditions are satisfied: (i) min{κ, ν(f1,gj)} = . . . = min{κ, ν(fλ,gj)} for each 0 6 j 6 q − 1, (ii) dim{z\|(f1, gi)(z) = (f1, gj)(z) = 0} 6 n− 2 for each 0 6 i < j 6 q − 1, ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS . . . 925 (iii) there exists an integer number l, 2 6 l 6 λ, such that for any increasing sequence 1 6 j1 < . . . < jl 6 λ, fj1(z) ∧ . . . ∧ fjl(z) = 0 for every point z ∈ ∈ ∪q−1i=0 (f1, gi) −1{0}. Then (i) If q > N(2N + 1)λ− (κ − 1)(λ− 1) λ− l + 1 , then f1, . . . , fλ are algebraically depen- dent over C, i.e., f1 ∧ . . . ∧ fλ ≡ 0 on C; (ii) if fi, 1 6 i 6 λ, are linearly nondegenerate over R{gj}q−1j=0 and q > N(N + 2)λ− (κ − 1)(λ− 1) λ− l + 1 , then f1, . . . , fλ are algebraically dependent over C; (iii) if fi, 1 6 i 6 λ, are linearly nondegenerate over C, gi, 0 6 i 6 q − 1, are constant mappings and (q − N − 1)((λ − 1)(κ − 1) + q(λ − l + 1)) 6 qNλ, then f1, . . . , fλ are algebraically dependent over C. 2. Basic notions and auxiliary results from Nevanlinna theory. 2.1. We set ‖z‖ = ( \|z1\|2 + . . .+ \|zn\|2 )1/2 for z = (z1, . . . , zn) ∈ Cn and define B(r) := {z ∈ Cn : ‖z‖ < r}, S(r) := {z ∈ Cn : ‖z‖ = r}, 0 < r <∞. Define vn−1(z) := ( ddc‖z‖2 )n−1 and σn(z) := dclog‖z‖2 ∧ ( ddclog‖z‖2 )n−1 on Cn \ {0}. 2.2. Let F be a nonzero holomorphic function on a domain Ω in Cn. For a set α = (α1, . . . , αn) of nonnegative integers, we set \|α\| = α1 + . . . + αn and DαF = = ∂\|α\|F ∂α1z1 . . . ∂αnzn . We define the map νF : Ω→ Z by νF (z) := max { m : DαF (z) = 0 for all α with \|α\| < m}, z ∈ Ω. We mean by a divisor on a domain Ω in Cn a map ν : Ω → Z such that, for each a ∈ Ω, there are nonzero holomorphic functions F and G on a connected neighbourhood U ⊂ Ω of a such that ν(z) = νF (z)− νG(z) for each z ∈ U outside an analytic set of dimension 6 n− 2. Two divisors are regarded as the same if they are identical outside an analytic set of dimension 6 n−2. For a divisor ν on Ω we set \|ν\| := {z : ν(z) 6= 0}, which is a purely (n− 1)-dimensional analytic subset of Ω or empty. Take a nonzero meromorphic function ϕ on a domain Ω in Cn. For each a ∈ Ω, we choose nonzero holomorphic functions F and G on a neighbourhood U ⊂ Ω such that ϕ = F G on U and dim(F−1(0) ∩G−1(0)) 6 n− 2, and we define the divisors νϕ, ν∞ϕ by νϕ := νF , ν ∞ ϕ := νG, which are independent of choices of F and G and so globally well-defined on Ω. 2.3. For a divisor ν on Cn and for a positive integer M or M =∞, we define the counting function of ν by ν(M)(z) = min {M,ν(z)}, ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 926 DUC THOAN PHAM, VIET DUC PHAM n(t) =  ∫ \|ν\| ∩B(t) ν(z)vn−1 if n ≥ 2,∑ \|z\|6t ν(z) if n = 1. Similarly, we define n(M)(t). Define N(r, ν) = r∫ 1 n(t) t2n−1 dt, 1 < r <∞. Similarly, we define N(r, ν(M)) and denote them by N (M)(r, ν) respectively. Let ϕ : Cn → C be a meromorphic function. Define Nϕ(r) = N(r, νϕ), N (M) ϕ (r) = N (M)(r, νϕ). For brevity we will omit the character (M) if M =∞. 2.4. Let f : Cn → PN (C) be a meromorphic mapping. For arbitrarily fixed ho- mogeneous coordinates (w0 : . . . : wN ) on PN (C), we take a reduced representation f = (f0 : . . . : fN ), which means that each fi is a holomorphic function on Cn and f(z) = ( f0(z) : . . . : fN (z) ) outside the analytic set {f0 = . . . = fN = 0} of codimension ≥ 2. Set ‖f‖ = ( \|f0\|2 + . . .+ \|fN \|2 )1/2 . The characteristic function of f is defined by T (r, f) = ∫ S(r) log‖f‖σn − ∫ S(1) log‖f‖σn. Let a be a meromorphic mapping of Cn into PN (C) with reduced representation a = (a0 : . . . : aN ). We define mf,a(r) = ∫ S(r) log ‖f‖‖a‖ \|(f, a)\| σn − ∫ S(1) log ‖f‖‖a‖ \|(f, a)\| σn, where ‖a‖ = ( \|a0\|2 + . . .+ \|aN \|2 )1/2 . If f, a : Cn → PN (C) are meromorphic mappings such that (f, a) 6≡ 0, then the first main theorem for moving targets in value distribution theory (see [3]) states T (r, f) + T (r, a) = mf,a(r) +N(f,a)(r). 2.5. Let ϕ be a nonzero meromorphic function on Cn, which are occationally regarded as a meromorphic map into P1(C). The proximity function of ϕ is defined by m(r, ϕ) := ∫ S(r) log max (\|ϕ\|, 1)σn. 2.6. As usual, by the notation ′′‖P ′′ we mean the assertion P holds for all r ∈ ∈ [0,∞) excluding a Borel subset E of the interval [0,∞) with ∫ E dr <∞. ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS . . . 927 2.7. The First Main Theorem for general position [1, p. 326]. Let fi : Cn → → PN (C), 1 6 i 6 λ, be meromorphic mappings located in general position. Assume that 1 6 λ 6 N + 1. Then N(r, µf1∧...∧fλ) +m(r, f1 ∧ . . . ∧ fλ) 6 ∑ 16i6λ T (r, fi) +O(1). Let V be a complex vector space of dimension N ≥ 1. The vectors {v1, . . . , vk} are said to be in general position if for each selection of integers 1 6 i1 < . . . < ip 6 k with p 6 N, then vi1 ∧ . . . ∧ vip 6= 0. The vectors {v1, . . . , vk} are said to be in special position if they are not in general position. Take 1 6 p 6 k. Then {v1, . . . , vk} are said to be in p-special position if for each selection of integers 1 6 i1 < . . . < ip 6 k, the vectors vi1 , . . . , vip are in special position. 2.8. The Second Main Theorem for general position ([1, p. 320], Theorem 2.1). 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 6 p 6 k 6 n+ 1. Let fj : M → P (V ), 1 6 j 6 k, be meromorphic mappings. Assume that f1, . . . , fk are in general position. Also assume that f1, . . . , fk are in p-special position on A. Then we have µf1∧...∧fk ≥ (k − p+ 1)νA. 2.9. The Second Main Theorem for moving target. 2.9.1 ([4], Theorem 3.1). Let f : Cn → PN (C) be a meromorphic mapping. Let {a1, . . . , aq}, q 6= 2, be a set of q meromorphic mappings of Cn into PN (C) in general position such that f is linearly nondegenerate over R ({ aj }q j=1 ) . Then q N + 2 T (r, f) 6 q∑ i=1 N (N) (f,ai) (r) +O ( max 06i6q−1 T (r, ai) ) +o ( T (r, f) ) . 2.9.2 ([5], Corollary 1) . Let f : Cn → PN (C) be a meromorphic mapping. Let A = {a1, . . . , aq}, q ≥ 2N+1, be a set of q meromorphic mappings of Cn into PN (C) located in general position such that (f, ai) 6≡ 0 for each 1 6 i 6 q. Then q 2N + 1 T (r, f) 6 q∑ i=1 N (N) (f,ai) (r) +O ( max 16i6q T (r, ai) ) +O ( log+ T (r, f) ) . 3. Proofs of main theorems. 3.1. Proof of Theorem 1. It suffices to prove Theo- rem 1 in the case of λ 6 N + 1. Assume that f1 ∧ . . . ∧ fλ 6≡ 0. We denote by µf1∧...∧fλ the divisor associated with f1 ∧ . . . ∧ fλ. Denote by N(r, µf1∧...∧fλ) the counting function associated with the divisor µf1∧...∧fλ . We now prove the following. Claim 3.1.1. For every 1 6 t 6 λ, we have q∑ j=1 min{N, ν(ft,gj)(z)} 6 dN λ− l + 1 µf1∧...∧fλ(z) + qN ∑ β µgβ(1)∧...∧gβ(N+1) (z) for each z 6∈ A∪∪λi=1I(fi), where the sum is over all injective maps β : {1, 2, . . . , N + + 1} → {1, 2, . . . , q}. ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 928 DUC THOAN PHAM, VIET DUC PHAM We now prove Claim 3.1.1. For each regular point z0 ∈ A \ (A ∪ ∪λi=1I(fi) ∪ ∪β∈T [N+1,q]{z\|gβ(1)(z) ∧ . . . . . .∧gβ(N+1)(z) = 0}), let S be an irriducible analytic subset of A containing z0. Since z0 6∈ A and A = ∪Aij 6≡Akl{Aji ∩ Akl}, where Aji are the irriducible components of Aj = (f1, gj) −1{0}, it implies that S is a pure (n− 1)-dimentional analytic subset and hence, S is only contained in at most d sets of Aj . Thus ν(ft,gj)(z0) 6= 0 at most d indices. We have q∑ j=1 min{N, ν(ft,gj)(z0)} 6 dN. For each increasing sequence 1 6 j1 < . . . < jl 6 λ, we have fj1(z) ∧ . . . ∧ fjl(z) = 0 ∀z ∈ S. This implies that the fimily {f1, . . . , fλ} is in l-special position on S. By the Second Main Theorem for general position [1, p. 320] (Theorem 2.1), we have µf1∧...∧fλ(z) ≥ (λ− (l − 1))νS . By the properties of divisor, we have µf1∧...∧fλ(z0) ≥ λ− l + 1. Hence q∑ j=1 min{N, ν(ft,gj)(z0)} 6 dN 6 dN λ− l + 1 µf1∧...∧fλ(z0). If z0 ∈ ∪β∈T [N+1,q]{z\|gβ(1)(z) ∧ . . . ∧ gβ(N+1)(z) = 0}, then we have q∑ j=1 min{N, ν(ft,gj)(z0)} 6 qN 6 qN ∑ β∈T [N+1,q] µgβ(1)∧...∧gβ(N+1) (z0). From the above cases and by the properties of divisor, for each z 6∈ A ∪ ∪λi=1I(fi), we have q∑ j=1 min{N, ν(ft,gj)(z)} 6 6 dN λ− l + 1 µf1∧...∧fλ(z) + qN ∑ β∈T [N+1,q] µgβ(1)∧...∧gβ(N+1) (z). Claim 3.1.1 is proved. The above assertions and The First Main Theorem for general position [1, p. 326], yield that q∑ j=1 N (N) (ft,gj) (r) 6 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS . . . 929 6 dN λ− l + 1 N(r, µf1∧...∧fλ) + qN ∑ β∈T [N+1,q] N(r, µgβ(1)∧...∧gβ(N+1) ) 6 6 dN λ− l + 1 λ∑ i=1 T (r, fi) + qN ∑ β∈T [N+1,q] N+1∑ i=1 T (r, gβ(i)) = = dN λ− l + 1 λ∑ i=1 T (r, fi) + o ( max 16i6λ {T (r, fi)} ) . Thus, by summing them up, we have λ∑ t=1 q∑ j=1 N (N) (ft,gj) (r) 6 dNλ λ− l + 1 λ∑ i=1 T (r, fi) + o ( max 16i6λ {T (r, fi)} ) . (1) By using the Second Main Theorem for moving targets [5] (Corollary 1), it implies that λ∑ t=1 q 2N + 1 T (r, ft) 6 dNλ λ− l + 1 λ∑ i=1 T (r, fi) + o ( max 16i6λ {T (r, fi)} ) . Letting r → +∞, we get q 6 dN(2N + 1)λ λ− l + 1 . This is a contradiction. Thus, the family {f1, . . . , fλ} is algebraically dependent over Cn, i.e., f1 ∧ . . . ∧ fλ = 0. Theorem 1 is proved. 3.2. Proof of Theorem 2. From (1), we have λ∑ t=1 q∑ j=1 N (N) (ft,gj) (r) 6 dNλ λ− l + 1 λ∑ i=1 T (r, fi) + o ( max 16i6λ {T (r, fi)} ) . By using the Second Main Theorem for moving targets [4] (Theorem 3.1), it implies that λ∑ t=1 q N + 2 T (r, ft) 6 dNλ λ− l + 1 λ∑ i=1 T (r, fi) + o ( max 16i6λ {T (r, fi)} ) . Letting r → +∞, we get q 6 dN(N + 2)λ λ− l + 1 . This is a contradiction. Thus, the family {f1, . . . , fλ} is algebraically dependent over Cn, i.e. f1 ∧ . . . ∧ fλ = 0. Theorem 2 is proved. 3.3. Proof of Theorem 3. It suffices to prove Theorem 3 in the case of λ 6 N+1. Assume that f1 ∧ . . . ∧ fλ 6≡ 0. We now prove the following. Claim 3.3.1. For any λ− 1 moving targets gj1 , . . . , gjλ−1 ∈ {gj}qj=1, there exists gj0 6∈ {gj1 , . . . , gjλ−1 } such that det  (f1, gj1) . . . (fλ, gj1) ... ... ... (f1, gjλ−1 ) . . . (fλ, gjλ−1 ) (f1, gj0) . . . (fλ, gj0)  6≡ 0. ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 930 DUC THOAN PHAM, VIET DUC PHAM We now prove Claim 3.3.1. Suppose on contrary. Without loss of generality, we assume gj1 = g1, . . . , gjλ−1=gλ−1. Then rank  (f1, g1)(z) . . . (fλ, g1)(z) ... ... ... (f1, gN+1)(z) . . . (fλ, gN+1)(z) 6 λ− 1 for each z ∈ Cn. By f1 ∧ . . . ∧ fλ 6≡ 0, there exists z0 ∈ Cn such that f1(z0) ∧ . . . . . . ∧ fλ(z0) 6= 0 and z0 6∈ {g1 ∧ . . . ∧ gN+1}−1(0). On the other hand, we have (f1, g1)(z0) . . . (fλ, g1)(z0) ... ... ... (f1, gN+1)(z0) . . . (fλ, gN+1)(z0)  = =  g10(z0) . . . g1N (z0) ... ... ... gN+10(z0) . . . gN+1N (z0)   f10(z0) . . . fλ0(z0) ... ... ... f1N (z0) . . . fλN (z0) . Since the family {gj}qj=1 is located in general position, is implies that the matrix f10(z0) . . . fλ0(z0) ... ... ... f1N (z0) . . . fλN (z0)  is of rank 6 λ− 1. This is a contradiction. The Claim 3.3.1 is proved. We now consider λ − 1 moving targets g1, . . . , gλ−1. Then, by Claim 3.3.1, there exists gj0 with j0 > λ− 1 such that det  (f1, g1) . . . (fλ, g1) ... ... ... (f1, gλ−1) . . . (fλ, gλ−1) (f1, gj0) . . . (fλ, gj0)  6≡ 0. Without loss of generality, we may assume that j0 = λ. Now we putA := ∪λj=1(f1, gj) −1{0}, A = ∪16i<j6λ ( (f1, gi) −1{0}∩(f1, gj) −1{0} ) . We now show the following. Claim 3.3.2. For each 1 6 t 6 λ, we have λ∑ i=1 ( min { κ, ν(ft,gi)(z) } + (λ− l) min { 1, ν(ft,gi)(z) }) + + q∑ i=λ+1 (λ− l + 1) min{1, ν(ft,gi)(z)} 6 6 µf̃1∧...∧f̃λ(z) + (λ(κ + λ− l) + (q − λ)(λ− l + 1))µg1∧...∧gλ(z) (2) ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS . . . 931 for every z ∈ Cn \ (A ∪ ∪λi=1I(fi)), where f̃i := ((fi, g1) : . . . : (fi, gλ)) for each 1 6 i 6 λ. Furthermore we have λ∑ i=1 ( N (κ) (ft,gi) (r) + (λ− l)N (1) (ft,gi) (r) ) + + q∑ i=λ+1 (λ− l + 1)N (1) (ft,gi) (r) 6 6 λ∑ i=1 T (r, fi) + o ( max 16i6λ {T (r, fi)} ) . (3) We now prove Claim 3.3.2. By the properties of divisor, we only consider three cases for regular points. Case 1. Let z0 ∈ A\ (A∪∪λi=1I(fi)∪{z\|g1∧ . . .∧gλ(z) = 0}) be a regular point of A. Then z0 is only a zero of one of the meromorphic functions {(ft, gj)}λj=1. Without loss of generality, we may assume that z0 is a zero of (ft, g1). Let S be an irriducible analytic subset of A, containing z0. Then the pure dimension of S is n − 1. Suppose that U is an open neighbourhood of z0 in Cn such that U ∩ {A \ S} = ∅. Choose a holomorphic function h on Cn such that νh = min{κ, ν(ft,g1)} if z ∈ S and νh = 0 if z 6∈ S. Then (fi, g1) = aih, 1 6 i 6 λ, where ai are holomorphic functions. Since the matrix (f1, g2)(z) . . . (fλ, g2)(z) ... ... ... (f1, gλ)(z) . . . (fλ, gλ)(z)  is of rank 6 λ − 1 for each z ∈ Cn, it implies that there exist holomorphic functions b1, . . . , bλ such that there is at least bi 6≡ 0 and λ∑ i=1 bi(fi, gj) = 0, 2 6 j 6 λ. Without loss of generality, we may assume that the set of common zeros of {bi}λi=1 is an analytic subset of codimension ≥ 2. Then there exist an index i1, 1 6 i1 6 λ such that S 6⊂ b−1i1 {0}. We can assume that i1 = λ. Then for each z ∈ (U ∩ S) \ b−1λ {0}, we have f̃1(z) ∧ . . . ∧ f̃λ(z) = f̃1(z) ∧ . . . ∧ f̃λ−1(z) ∧ ( f̃λ(z) + λ−1∑ i=1 bi bλ f̃i(z) ) = = f̃1(z) ∧ . . . ∧ f̃λ−1(z) ∧ (V (z)h(z)) = = h(z) ( f̃1(z) ∧ . . . ∧ f̃λ−1(z) ∧ V (z) ) , where V (z) := ( aλ + ∑λ−1 i=1 bi bλ ai, 0, . . . , 0 ) . By assumption, for any increasing sequence 1 6 j1 < . . . < jl 6 λ − 1, we have fj1 ∧ . . . ∧ fjl ≡ 0 on S. Then ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 932 DUC THOAN PHAM, VIET DUC PHAM rank(fj1(z), . . . , fjl(z)) = rank  fj10(z) . . . fjl0(z) ... ... ... fj1N (z) . . . fjlN (z) 6 l − 1 ∀z ∈ S. On the other hand (fj1 , g1)(z) . . . (fjl , g1)(z) ... ... ... (fj1 , gλ)(z) . . . (fjl , gλ)(z)  = = g10(z) . . . g1N (z) ... ... ... gλ0(z) . . . gλN (z)   fj10(z) . . . fjl0(z) ... ... ... fj1N (z) . . . fjlN (z) . Hence rank ( f̃j1(z), . . . , f̃jl(z) ) = = rank (fj1 , g1)(z) . . . (fjl , g1)(z) ... ... ... (fj1 , gλ)(z) . . . (fjl , gλ)(z) 6 l − 1 ∀z ∈ S. Therefore, f̃j1 ∧ . . . ∧ f̃jl ≡ 0 on S. This implies that the family {f̃1, . . . f̃λ−1} is in l-special position on S, and {f̃1, . . . f̃λ−1, V } is in (l + 1)-special position on S. By using The Second Main Theorem for general position [1, p. 320] (Theorem 2.1), we have µf̃1∧...∧f̃λ−1∧V (z) ≥ (λ− l)νS ∀z ∈ S. Hence µf̃1∧...∧f̃λ(z) ≥ νh(z) + (λ− l)νS = = min{κ, ν(ft,g1)(z)}+ (λ− l)νS ,∀z ∈ (U ∪ S) \ b−1i1 {0}. By the properties of divisors, we have µf̃1∧...∧f̃λ(z0) ≥ min{κ, ν(ft,g1)(z0)}+ λ− l. This implies that λ∑ i=1 ( min{κ, ν(ft,gi)(z0)}+ (λ− l) min{1, ν(ft,gi)(z0)} ) + + q∑ i=λ+1 (λ− l + 1) min{1, ν(ft,gi)(z0)} = = min{κ, ν(ft,g1)(z0)}+ λ− l 6 6 µf̃1∧...∧f̃λ(z0) + (λ(κ + λ− l) + (q − λ)(λ− l + 1))µg1∧...∧gλ(z0). ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS . . . 933 Case 2. Let z0 ∈ A\ (A∪∪λi=1I(fi)∪{z\|g1∧ . . .∧gλ(z) = 0}) be a regular point of A. Then z0 is only a zero of (ft, gi), i > λ. By the assumption, we have the family {f̃1, . . . , f̃λ} is in l-special position on an irriducible analytic subset of codimension 1 of A which containing z0. By using The Second Main Theorem for general position [1, p. 320] (Theorem 2.1), we have µf̃1∧...∧f̃λ(z0) ≥ λ− l + 1. Hence λ∑ i=1 ( min{κ, ν(ft,gi)(z0)}+ (λ− l) min{1, ν(ft,gi)(z0)} ) + + q∑ i=λ+1 (λ− l + 1) min{1, ν(ft,gi)(z0)} = = (λ− l + 1) min{1, ν(ft,gi)(z0)} = = λ− l + 1 6 µf̃1∧...∧f̃λ(z0) + (λ(κ + λ− l)+ +(q − λ)(λ− l + 1))µg1∧...∧gλ(z0)). Case 3. Assume that z0 ∈ (g1 ∧ . . . ∧ gλ)−1{0}. Then λ∑ i=1 ( min{κ, ν(ft,gi)(z0)}+ (λ− l) min{1, ν(ft,gi)(z0)} ) + + q∑ i=λ+1 (λ− l + 1) min{1, ν(ft,gi)(z0)} 6 6 λ(κ + (λ− l)) + (q − λ)(λ− l + 1) 6 6 µf̃1∧...∧f̃λ(z0) + (λ(κ + λ− l) + (q − λ)(λ− l + 1))µg1∧...∧gλ(z0)). From the above cases and by the properties of divisors, for each z 6∈ A ∪λi=1 I(fi), we have λ∑ i=1 ( min{κ, ν(ft,gi)(z)}+ (λ− l) min{1, ν(ft,gi)(z)} ) + + q∑ i=λ+1 (λ− l + 1) min{1, ν(ft,gi)(z)} 6 6 µf̃1∧...∧f̃λ(z) + (λ(κ + λ− l) + (q − λ)(λ− l + 1))µg1∧...∧gλ(z). The first assertion of Claim 3.3.2 is proved. ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 934 DUC THOAN PHAM, VIET DUC PHAM By the assumption and definiton of the characteristic function, for each 1 6 j 6 λ, we have T (r, f̃j) 6 T (r, fj) + o ( max 16j6λ {T (r, fi)} ) . By The First Main Theorem for general position [1, p. 326], it implies that λ∑ i=1 ( N (κ) (ft,gi) (r) + (λ− l)N (1) (ft,gi) (r) ) + q∑ i=λ+1 (λ− l + 1)N (1) (ft,gi) (r) 6 6 N(r, µf̃1∧...∧f̃λ)(r) + ( λ(κ + λ− l) + (q − λ)(λ− l + 1) ) Nµg1∧...∧gλ (r) 6 6 λ∑ i=1 T (r, f̃i) + ( λ(κ + λ− l) + (q − λ)(λ− l + 1) ) λ∑ i=1 T (r, gi) +O(1) 6 6 λ∑ i=1 T (r, fi) + o ( max 16i6λ {T (r, fi)} ) . The second assertion of Claim 3.3.2 is proved. Thus, for any increasing sequence 1 6 i1 < . . . < iλ−1 6 q, we have λ−1∑ j=1 ( N (κ) (ft,gij ) (r) + (λ− l)N (1) (ft,gij ) (r) ) + ∑ i∈I (λ− l + 1)N (1) (ft,gi) (r) 6 6 λ∑ i=1 T (r, fi) + o ( max 16i6λ {T (r, fi)} ) , where I = {1, 2, . . . , q} \ {i1, . . . , iλ−1}. Thus, by summing-up them over all sequences 1 6 i1 < . . . < iλ−1 6 q, we have q∑ i=1 ((λ− 1)N (κ) (ft,gi) (r) + ((λ− 1)(λ− l)+ +(q − λ+ 1)(λ− l + 1))N (1) (ft,gi) (r)) 6 6 q λ∑ i=1 T (r, fi) + o ( max 16i6λ {T (r, fi)} ) . Since κN (N) f (r) 6 NN (κ) f (r) ∀κ, we have q∑ i=1 ( (λ− 1)κN (N) (ft,gi) (r) + ((λ− 1)(λ− l)+ +(q − λ+ 1)(λ− l + 1))N (N) (ft,gi) (r) ) 6 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS . . . 935 6 qN λ∑ i=1 T (r, fi) + o ( max 16i6λ {T (r, fi)} ) . This implies that q∑ i=1 ( (λ− 1)κ + (λ− 1)(λ− l) + (q − λ+ 1)(λ− l + 1) ) N (N) (ft,gi) (r) 6 6 qN λ∑ i=1 T (r, fi) + o ( max 16i6λ {T (r, fi)} ) . Thus, by summing them up over all t (1 6 t 6 λ), we have q∑ i=1 λ∑ t=1 ( (λ− 1)κ + (λ− 1)(λ− l) + (q − λ+ 1)(λ− l + 1) ) N (N) (ft,gi) (r) 6 6 qNλ λ∑ i=1 T (r, fi) + o ( max 16i6λ T (r, fi) ) . (4) We now prove the assertions of Theorem 3. i) By applying the Second Main Theorem for moving targets [5] (Corollary 1) to the left-hand side of (4), it implies that q 2N + 1 λ∑ i=1 ( (λ− 1)κ + (λ− 1)(λ− l) + (q − λ+ 1)(λ− l + 1) ) T (r, fi) 6 6 qNλ λ∑ i=1 T (r, fi) + o ( max 16i6λ {T (r, fi)} ) . Letting r → +∞, we have q 6 λ− 1 + (2N + 1)Nλ− (λ− 1)κ − (λ− 1)(λ− l) λ− l + 1 = = (2N + 1)Nλ− (λ− 1)(κ − 1) λ− l + 1 . This is a contradiction. Thus, we have f1 ∧ . . . ∧ fλ ≡ 0. ii) By applying the Second Main Theorem for moving targets [4] (Theorem 3.1) to the left-hand side of (4), it implies that q N + 2 λ∑ i=1 ((λ− 1)κ + (λ− 1)(λ− l) + (q − λ+ 1)(λ− l + 1))T (r, fi) 6 6 qNλ λ∑ i=1 T (r, fi) + o ( max 16j6λ {T (r, fi)} ) . ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 936 DUC THOAN PHAM, VIET DUC PHAM Letting r → +∞, we have q 6 λ− 1 + (N + 2)Nλ− (λ− 1)κ − (λ− 1)(λ− l) λ− l + 1 = = (N + 2)Nλ− (λ− 1)(κ − 1) λ− l + 1 . This is a contradiction. Thus, we have f1 ∧ . . . ∧ fλ ≡ 0. iii) By applying the Second Main Theorem for hyperplanes in general position [6, p. 304] to the left-hand side of (4), it implies that (q −N − 1) λ∑ i=1 ((λ− 1)κ + (λ− 1)(λ− l) + (q − λ+ 1)(λ− l + 1))T (r, fi) 6 6 qNλ λ∑ i=1 T (r, fi) + o ( max 16j6λ {T (r, fi)} ) . Letting r → +∞, we have (q −N − 1)((λ− 1)(κ − 1) + q(λ− l + 1)) 6 qNλ. This is a contradiction. Thus, we have f1 ∧ . . . ∧ fλ ≡ 0. Theorem 3 is proved. 1. Stoll W. On the propagation of dependences // Pacif. J. Math. – 1989. – 139. – P. 311 – 337. 2. Ru M. A uniqueness theorem with moving targets without counting multiplicity // Proc. Amer. Math. Soc. – 2001. – 129. – P. 2701 – 2707. 3. Ru M., Stoll W. The second main theorem for moving targets // J. Geom. Anal. – 1991. – 1. – P. 99 – 138. 4. Do Duc Thai, Si Duc Quang. Uniqueness problem with truncated multiplicities of meromorphic mappings in several complex variables for moving targets // Int. J. Math. – 2005. – 16. – P. 903 – 942. 5. Do Duc Thai, Si Duc Quang. Second main theorem with truncated counting function in several complex variables for moving targets // Forum Math. – 2008. – 20. – P. 145 – 179. 6. Stoll W. Value distribution theory for meromorphic maps // Aspects Math. E. – 1985. – 7. Received 03.01.10 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7

Algebraic dependences of meromorphic mappings in several complex variables

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