Uniqueness problem for meromorphic mappings with truncated multiplicities and few targets

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

GERDDETHLOFF, TRANVANTAN

Uniqueness problem for meromorphic mappings with truncated multiplicities and few targets

Tome XV, n^o2 (2006), p. 217-242.

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Uniqueness problem for meromorphic mappings with truncated multiplicities and few targets

Gerd Dethloff and Tran Van Tan⁽¹⁾

ABSTRACT.— In this paper, using techniques of value distribution theory, we give a uniqueness theorem for meromorphic mappings ofC^mintoC^Pn

withtruncated multiplicities and “few” targets. We also give a theorem of linear degeneration for suchmaps withtruncated multiplicities and moving targets.

R^´ESUM´E.— Dans cet article, on donne un th´eor`eme d’unicit´e pour des applications m´eromorphes deC<sup>m dans</sup> C<sup>Pn</sup> avec multiplicit´es coup´ees et avec«peu de»cibles. On donne aussi un th´eor`eme de d´eg´en´eration lin´eaire pour des telles applications avec multiplicit´es coup´ees et avec des cibles mobiles. Les preuves utilisent des techniques de la distribution des valeurs.

1. Introduction

The uniqueness problem of meromorphic mappings under a condition on the inverse images of divisors was first studied by R. Nevalinna [8]. He showed that for two nonconstant meromorphic functions f and g on the complex plane C, if they have the same inverse images for five distinct values thenf ≡g. In 1975, H. Fujimoto [3] generalized Nevanlinna’s result to the case of meromorphic mappings of C^m into CPⁿ. He showed that for two linearly nondegenerate meromorphic mappingsf andg ofC^m into CPⁿ, if they have the same inverse images counted with multiplicities for (3n+ 2) hyperplanes in general position in CPⁿ, then f ≡g. Since that time, this problem has been studied intensively by H.Fujimoto ([4], [5] ...), L. Smiley [11], S. Ji [6], M. Ru [10], Z. Tu [12] and others.

(∗) Re¸cu le 9 septembre 2004, accept´e le 13 avril 2005

(1) Gerd Dethloff, Tran Van Tan, Universit´e de Bretagne Occidentale, UFR Sciences et Techniques, D´epartement de Math´ematiques, 6, avenue Le Gorgeu, BP 452, 29275 Brest Cedex (France).

E-mail: gerd.dethloff@univ-brest.fr

Letfbe linearly nondegenerate meromorphic mappings ofC^mintoCPⁿ. For each hyperplane H we denote by v_(f,H) the map of C^m into N0 such that v_(f,H)(a) (a∈ C^m ) is the intersection multiplicity of the image off andH atf(a) .

TakeqhyperplanesH1, ..., Hq inCPⁿ in general position and a positive integer l0 .

We consider the family F({H_j}^q_j=1, f, l₀) of all linearly nondegenerate meromorphic mappingsg:C^m−→CPⁿ satisfying the conditions:

(a) min

v_(g,Hj), l0

= min

v_(f,Hj), l0

for allj ∈ {1, ..., q}, (b) dim

f⁻¹(Hi)∩f⁻¹(Hj)

m−2 , for all 1i < j q ,and (c)g=f on

q j=1

f⁻¹(Hj).

In 1983, L.Smiley showed that:

Theorem A. — ([11]) If q 3n+ 2 then g1=g2 for any g1,g2 ∈ F({Hj}^q_j=1, f,1).

For the caseq= 3n+1 in [4],[5],[6] the authors gave the following results:

Theorem B. — ([6])Assume that q= 3n+ 1.Then for three mappings g₁, g₂, g₃ ∈ F({H_j}^q_j=1, f,1), the map g₁×g₂×g₃ : C^m −→ CPⁿ× CPⁿ× CPⁿis algebraically degenerate, namely, {(g₁(z), g₂(z), g₃(z)), z∈C^m}is included in a proper algebraic subset of CPⁿ×CPⁿ×CPⁿ.

Theorem C. — ([4]) Assume that q= 3n+ 1.Then there are at most two distinct mappings in F({Hj}^q_j=1, f,2).

Theorem D. — ([5])Assume that n= 2, q= 7.Then there exist some positive integer l₀ and a proper algebraic set V in the cartesian product of seven copies of the space

CP²_∗

of all hyperplanes in CP² such that, for an arbitrary set (H₁, ..., H₇)∈/ V and two nondegenerate meromorphic mappings f, g of C^m into CP² with min

v_(g,Hj), l₀

= min

v_(f,Hj), l₀ for all j∈ {1, ...,7}, we have f =g.

In [5], H.Fujimoto also gave some open questions:

+) Does Theorem D remain valid under the assumption that theH_j ’s are in general position ?

+) What is a generalization of Theorem D for the casen3 ?

In connection with the above results, it is also an interesting problem to ask whether these results remain valid if the number of hyperplanes is replaced by a smaller one. In this paper, we will try to get some partial answers to this problem. We give a uniqueness theorem for the case q

n+I

2n(n+ 1) +1 and a theorem of the linear degeneration for the case of (2n+ 2) moving targets (where we denoteI(x) := min{k∈N0:k > x} for a positive constantx).

Letf, abe two meromorphic mappings of C^m into CPⁿ with reduced representationsf = (f0:. . .:fn),a= (a0:. . .:an).

Set (f, a) :=a0f0+. . .+anfn. We say thatais “small” with respect tof if T_a(r) =o(T_f(r)) as r→ ∞. Assuming that (f, a)≡0, we denote byv_(f,a) the map of C^minto N0 withv_(f,a)(z) = 0 if (f, a)(z)= 0 andv_(f,a)(z) =k ifzis a zero point of (f, a) with multiplicityk.

Leta1, . . . , a_q (qn+ 1) be meromorphic mappings ofC^m into CPⁿ with reduced representations aj = (aj0 : . . . : ajn), j = 1, . . . , q. We say that

aj

q

j=1 are in general position if for any 1 j0 < . . . < jn q, det(ajki,0k, in)≡0.

For each j ∈ {1, ..., q}, we put aj = (aj0

ajtj

: ... : ajn

ajtj

) and (f,aj) = f₀a_j0

a_jtj +...+f_na_jn

a_jtj wherea_jtj is the first element ofa_j0, ..., a_jn not iden- tically equal to zero. Let M be the field (over C ) of all meromorphic functions on C^m. Denote by R

aj

q

j=1 ⊂ Mthe subfield generated by the set {a_ji

ajtj

,0in,1 j q} overC . This subfield is independant of the reduced representationsaj = (aj0:. . .:ajn), j= 1, . . . , q, and it is of course also independant of our choice of the ajtj, because it contains all quotients of the quotients aji

a_jtj, i= 0, . . . , n.

We say thatf is linearly nondegenerate overR aj

q

j=1 iff0, . . . , fn

are linearly independant overR a_jq

j=1 .

Denote by Ψ the Segre embedding ofCPⁿ×CPⁿ intoCPⁿ²⁺²ⁿwhich is defined by sending the ordered pair ((w0, ..., wn),(v0, ..., vn)) to (..., wivj, ...) in lexicographic order.

Leth:C^m−→ CPⁿ×CPⁿ be a meromorphic mapping. Let (h₀:· · ·: h_n²_+2n) be a representation of Ψ◦h. We say thathis linearly degenerate

(with the algebraic structure inCPⁿ×CPⁿgiven by the Segre embedding) ifh0, ..., h_n²_+2n are linearly dependant overR

aj

q j=1 .

Our main results are stated as follows: Letn, x, y, p be nonnegative in- tegers. Assume that:

2pn, 1y2n, and 0x <min{2n−y+ 1, (p−1)y

n+ 1 +y}.

Letk be an integer or +∞with 2n(n+1+y)(3n+p−x)

(p−1)y−x(n+1+y) k+∞.

Theorem 1.1. —Let f, g be two linearly nondegenerate meromorphic mappings of C^mintoCPⁿ and {H_j}^q_j=1 be q:= 3n+ 1−xhyperplanes in CPⁿ in general position.

Set E_f^j :=

z ∈ C^m : 0 v_(f,Hj)(z) k

,^∗E_f^j :=

z ∈ C^m : 0 <

v(f,Hj)(z)k

,and similarily for E_g^j,^∗E_g^j,j= 1, . . . , q.

Assume that :

(a) min{v(f,Hj),1}= min{v(g,Hj),1}on E_f^j∩E_g^j for all j∈ {n+ 2 +y, . . . , q},and

min{v(f,Hj), p}= min{v(g,Hj), p}onE_f^j∩E^jgfor allj∈ {1, . . . , n+1+y}, (b) dim_∗

E_fⁱ ∩^∗E_f^j

m−2 , dim_∗

E_gⁱ ∩^∗E_g^j

m−2

for all 1i < j q, (c)f =gon

q j=1

_∗

E_f^j∩^∗E^j_g . Then f =g.

We state some corollaries of Theorem 1.1:

+) Taken2, y = 1, p= 2, x= 0 and kn(n+ 2)(6n+ 4). Then we have:

Corollary 1.2. —Let f, gbe two linearly nondegenerate meromorphic mappings of C^mintoCPⁿ(n2)and {Hj}³ⁿ⁺¹_j=1 be hyperplanes in CPⁿin general position.

Assume that:

(a)min{v(f,Hj),1}= min{v(g,Hj),1} onE_f^j ∩E_g^j for allj ∈ {n+ 3, . . . ,3n+ 1}, and

min{v(f,Hj),2}= min{v(g,Hj),2} onE_f^j∩E_g^j for allj ∈ {1, . . . , n+ 2}, (b) dim_∗

E_fⁱ ∩^∗E_f^j

m−2 , dim_∗

E_gⁱ ∩^∗E_g^j

m−2

for all1i < j3n+ 1, (c)f =g on

3n+1

j=1

_∗

E_f^j∩^∗E_g^j . Thenf =g.

Corollary 1.2 is an improvement of Theorem C. It is also a kind of generalization of Theorem D to the case wheren2 and the hyperplanes are in general position.

+) Taken3, y=n+ 2, p= 3, x= 1 andk= +∞. Then we have:

Corollary 1.3. —Let f, gbe two linearly nondegenerate meromorphic mappings of C^m intoCPⁿ(n3)and {H_j}³ⁿj=1 be hyperplanes in CPⁿ in general position.

Assume that:

(a) min{v(f,Hj),1}= min{v(g,Hj),1} for all j ∈ {2n+ 4, . . . ,3n}, and min{v_(f,Hj),3}= min{v_(g,Hj),3} for all j ∈ {1, . . . ,2n+ 3}, (b) dim

f⁻¹(Hi)∩f⁻¹(Hj)

m−2 for all 1i < j3n, (c)f =gon

3n

j=1

f⁻¹(Hj).

Then f =g.

+) Taken2, y=I

2n(n+ 1) , p=n, x= 2n−I

2n(n+ 1) , k= +∞.Then we have:

Corollary 1.4. — Let f, g be two linearly nondegenerate meromor- phic mappings of C^m into CPⁿ (n 2) and {Hj}^n+I

√2n(n+1)

+1

j=1 be

hyperplanes in CPⁿ in general position.

Assume that:

(a) min{v(f,Hj), n}= min{v(g,Hj), n}

for all j∈ {1, . . . , n+I

2n(n+ 1) + 1}, (b) dim

f⁻¹(Hi)∩f⁻¹(Hj)

m−2

for all 1i < jn+I

2n(n+ 1) + 1,

(c)f =gon

n+I√

2n(n+1) +1 j=1

f⁻¹(H_j).

Thenf =g.

We finally give a result for moving targets:

Theorem 1.5. — Let f, g: C^m −→CPⁿ (n2)be two nonconstant meromorphic mappings with reduced representations f = (f₀:...:f_n)and g= (g₀:...:g_n).

Let {aj}²ⁿ⁺²_j=1 be “small” (with respect to f ) meromorphic mappings of C^m into CPⁿ in general position with reduced representations aj= (aj0:...:ajn), j= 1, ...,2n+ 2. Suppose that (f, aj)≡0,(g, aj)≡0, j= 1, . . . ,2n+ 2. TakeM an integer or +∞with

3n(n+ 1)

2n+ 2 n+ 1

2 2n+ 2

n+ 1

−2

M +∞.

Assume that:

(a) min{v_(f,aj), M}= min{v_(g,aj), M}for all j∈ {1, . . . ,2n+ 2}, (b) dim{z∈C^m: (f, a_i)(z) = (f, a_j)(z) = 0}m−2

for alli=j, i∈ {1, ..., n+ 4}, j∈ {1, ...,2n+ 2}, (c)There exist γ_j∈ R

a_j2n+2

j=1 (j = 1, ...,2n+ 2 )such that γj= ^a_a^{j0f0+...+ajnfn}

j0g0+...+ajngn on _n+4

i=1

{z: (f, ai)(z) = 0}

\ {z: (f, aj)(z) = 0}.

Then the mapping f ×g : C^m −→ CPⁿ×CPⁿ is linearly degenerate (with the algebraic structure in CPⁿ×CPⁿ given by the Segre embedding) over R

a_j2n+2 j=1 .

Remark. — The condiction (c) is weaker than the following easier one:

(c’) f =gon

n+4

i=1

{z: (f, a_i)(z) = 0}.

We finally remark that we also obtained uniqueness theorems with mov- ing targets (in [1]), and with fixed targets, but not taking into account, at all, truncations from some fixed order on (in [2]). But in both cases the number of targets has to be bigger than in our results above.

Acknowledgements. — The second author would like to thank Professor Do Duc Thai for valuable discussions, the Universit´e de Bretagne Occi- dentale (U.B.O.) for its hospitality and for support, and the PICS-CNRS ForMathVietnam for support.

2. Preliminaries We set z =

|z₁|²+. . .+|z_m|²1/2

for z = (z₁, . . . , z_m) ∈ C^m and define

B(r) :={z∈C^m:|z|< r}, S(r) :={z∈C^m:|z|=r} for all 0< r∞.

Define d^c:=

√−1

4π (∂−∂), υ:= (dd^cz²)^m−1 and σ:=d^clogz²∧(dd^clogz²)^m−1.

Let F be a nonzero holomorphic function on C^m. For every a ∈ C^m, ex- pandingF asF =

P_i(z−a) with homogeneous polynomialsP_i of degree iarounda, we define

vF(a) := min{i:Pi≡0}.

Let ϕ be a nonzero meromorphic function on C^m. We define the map v_ϕ as follows: for each z ∈ C^m, we choose nonzero holomorphic func- tions F and G on a neighborhood U of z such that ϕ = F

G on U and dim

F⁻¹(0)∩G⁻¹(0)

m−2, and then we put vϕ(z) :=vF(z).

Set |vϕ|:=

z∈C^m:vϕ(z)= 0 . Letk,M be positive integers or +∞.

Set

Mv_ϕ^[k](z) = 0 if vϕ(z)> M and^Mv_ϕ^[k](z) = min{vϕ(z), k} if vϕ(z)M

>Mv^[k]_ϕ (z) = 0 if v_ϕ(z)M and^>Mv^[k]_ϕ (z) = min{v_ϕ(z), k} ifv_ϕ(z)> M.

We define

MN_ϕ^[k](r) :=

r 1

Mn(t) t^2m−1 dt and

>MN_ϕ^[k](r) :=

r 1

>Mn(t)

t^2m−1 dt (1r <+∞) where,

Mn(t) :=

|vϕ|∩B(r)

Mv^[k]_ϕ .υ form2,^Mn(t) :=

|z|t

Mv^[k]_ϕ (z) form= 1

>Mn(t) :=

|vϕ|∩B(r)

>Mv_ϕ^[k].υform2,^>Mn(t) :=

|z|t

>Mv^[k]_ϕ (z) form= 1.

Set N_ϕ(r) :=^∞Nϕ^[∞](r), Nϕ^[k](r) :=^∞Nϕ^[k](r).

We have the following Jensen’s formula (see [5], p.177, observe that his definition of Nϕ(r) is a different one than ours):

Nϕ(r)−N¹

ϕ(r) =

S(r)

log|ϕ|σ−

S(1)

log|ϕ|σ, 1r∞.

Letf : C^m −→ CPⁿ be a meromorphic mapping. For arbitrary fixed homogeneous coordinates (w0 : . . . : wn) of CPⁿ, we take a reduced rep- resentation f = (f0 :. . . : fn) which means that each fi is a holomorphic function on C^m and f(z) = (f0(z) : . . . : fn(z)) outside the analytic set {f0=. . .=fn= 0}of codimension2. Setf=

|f0|²+. . .+|fn|²1/2

. The characteristic function off is defined by

Tf(r) =

S(r)

logfσ−

S(1)

logfσ, 1r <+∞.

For a meromorphic functionϕonC^m, the proximity function is defined by m(r, ϕ) :=

S(r)

log⁺|ϕ|σ

and we have, by the classical First Main Theorem that (see [4], p.135) m(r, ϕ)Tϕ(r) +O(1).

Here, the characteristic functionT_ϕ(r) ofϕis defined asϕcan be considered as a meroromorphic mapping ofC^minto CP¹.

We state the First and Second Main Theorem of Value Distribution Theory. Let a be a meromorphic mapping of C^m into CPⁿ such that (f, a) ≡ 0, then for reduced representations f = (f0 : . . . : fn) and a= (a0:. . .:an), we have:

First Main Theorem. — (Moving target version, see [12], p.569) N(f,a)(r)Tf(r) +Ta(r) +O(1) for r1.

For a hyperplane H : a₀w₀+. . .+a_nw_n = 0 in CPⁿ with imf H, we denote (f, H) =a0f0+. . .+anfn, where (f0 :. . .:fn) again is a reduced representation off.

Second Main Theorem. — (Classical version) Let f be a linearly nondegenerate meromorphic mapping of C^m into CPⁿ and H₁, . . . , H_q (qn+ 1) hyperplanes ofCPⁿ in general position, then

(q−n−1)T_f(r) q j=1

N_(f,H^[n]

j)(r) +o(T_f(r)) for allrexcept for a set of finite Lebesgue measure.

3. Proof of Theorem 1.1 First of all, we need the following:

Lemma 3.1. —Let f, gbe two linearly nondegenerate meromorphic map- pings of C^m intoCPⁿ and {Hj}^q_j=1 be hyperplanes in CPⁿ in general po- sition. Then there exists a dense subset C ⊂ Cⁿ⁺¹{0} such that for any

c = (c₀, ..., c_n) ∈ C, the hyperplane H_c defined by c₀ω₀+...+c₀ω_n = 0 satifies:

dim

f⁻¹(Hj)∩f⁻¹(Hc)

m−2

and dim

g⁻¹(Hj)∩g⁻¹(Hc)

m−2

for all j ∈ {1, ..., q}. Proof. — We refer to [6], Lemma 5.1.

We now begin to prove Theorem 1.1.

Assume thatf ≡g .

Letj0 be an arbitrarily fixed index, j0 ∈ {1, ..., n+ 1 +y}.Then there exists a hyperplaneH in CPⁿ such that:

dim

f⁻¹(H_j)∩f⁻¹(H)

m−2 , dim

g⁻¹(H_j)∩g⁻¹(H)

m−2

for all j∈ {1, ..., q}and (f, H_j0)

(f, H) ≡(g, H_j0)

(g, H) : (3.1)

Indeed, suppose that this assertion does not hold. Then by Lemma 3.1 we have ^(f,H_(f,H)^j0) ≡ ^(g,H_(g,H)^j0) for all hyperplanes H in CPⁿ. In particular,

(f,Hj0)

(f,H_ji) ≡ ^(g,H_(g,H^j0)

ji), i ∈ {1, ..., n} where {j1, ..., jn} is an arbitrary subset of {1, ..., q} \ {j₀}.After changing the homogeneous coordinates (w₀:...:w_n) onCPⁿ we may assume thatH_ji :w_i= 0,(i= 0, ..., n). Then ^f_f⁰

i = ^g_g⁰

i for alli ∈ {1, ..., n}.This means that f ≡g.This is a contradiction. Thus we get (3.1).

Since min{v(f,Hj0), p} = min{v(g,Hj0), p} on E^j_f⁰ ∩E^j_g⁰, f = g on q

j=1

_∗

E_f^j∩^∗E_g^j

and by (3.1) we have that a zero pointz0of (f, Hj0) with multiplicity k is either a zero point of ^(f,H_(f,H)^j0) −^(g,H_(g,H)^j0) with multiplic- ity min{v(f,Hj0)(z0), p}or a zero point of (g, Hj0) with multiplicity> k (outside an analytic set of codimension2). (3.2) For any j ∈ {1, ..., q}{j₀}, by the asumptions (a),(c) and by (3.1), we have that a zero point of (f, H_j) with multiplicitykis either a zero point of ^(f,H_(f,H)^j0)− ^(g,H_(g,H)^j0) or zero point of (g, H_j) with multiplicity > k (outside

an analytic set of codimension2). (3.3)

By (3.2) and (3.3), the assumption (b) and by the First Main Theorem we have

kN_(f,H^[p]

j0)+ q j=1,j=j0

kN_(f,H^[1]

j)(r) N₍

f,Hj0) (f,H) −^(g,Hj(g,H)⁰⁾

(r) +^>kN_(g,H^[p]

j0)

+ q j=1,j=j0

>kN_(g,H^[1]

j)(r)

T₍

f,Hj0) (f,H) −(g,Hj0)

(g,H)

(r)+ p

k+ 1N_(g,Hj

0)(r)+ 1 k+ 1

q j=1,j=j0

N_(g,Hj)(r)+O(1) T_(f,Hj

0) (f,H)

(r) +T(g,Hj0) (g,H)

(r) +p+q−1

k+ 1 Tg(r) +O(1) (3.4) Since dim(f⁻¹(Hj0)∩f⁻¹(H))m−2 we have:

T(f,Hj0) (f,H)

(r) =

S(r)

log (|(f, Hj0)|²+|(f, H)|²)¹²σ +O(1)

S(r)

log fσ+O(1) =T_f(r) +O(1). Similarly,

T(g,Hj0) (g,H)

(r) Tg(r) +O(1). So by (3.4) we have

kN_(f,H^[p]

j0)(r) + q j=1,j=j0

kN_(f,H^[1]

j)(r) Tf(r) +Tg(r) +p+q−1

k+ 1 T_g(r) +O(1).

Similarly,

kN_(g,H^[p]

j0)(r) + q j=1,j=j0

kN_(g,H^[1]

j)(r) Tf(r) +Tg(r) +p+q−1

k+ 1 T_f(r) +O(1).

Thus,

kN_(f,H^[p]

j0)(r) +^kN_(g,H^[p]

j0)(r) + q j=1,j=j0

kN_(f,H^[1]

j)(r) +^kN_(g,H^[1]

j)(r)

2 +p+q−1 k+ 1

(Tf(r) +Tg(r)) +O(1).

⇒ p n

kN_(f,H^[n]

j0)(r) +^kN_(g,H^[n]

j0)(r) +1

n q j=1,j=j0

kN_(f,H^[n]

j)(r) +^kN_(g,H^[n]

j)(r) 2(k+ 1) + (p+q−1)

k+ 1 (Tf(r) +Tg(r)) +O(1), (note thatpn).

⇒ p−1 n

kN_(f,H^[n]

j0)(r) +^kN_(g,H^[n]

j0)(r) 2(k+ 1) + (p+q−1)

k (Tf(r) +Tg(r))

−1 n

q j=1

kN_(f,H^[n]

j)(r) +^kN_(g,H^[n]

j)(r) +O(1) (3.5) By the First and the Second Main Theorem, we have:

(q−n−1)T_f(r) q j=1

N_(f,H^[n]

j)(r) +o(T_f(r))

= k

k+ 1 q j=1

kN_(f,H^[n]

j)(r) + q j=1

1 k+ 1

kN_(f,H^[n]

j)(r) +^>kN_(f,H^[n]

j)(r)

+o(T_f(r))

k

k+ 1 q j=1

kN_(f,H^[n]

j)(r) + n k+ 1

q j=1

N_(f,Hj)(r) +o(T_f(r))

k

k+ 1 q j=1

kN_(f,H^[n] _j)(r) + nq

k+ 1Tf(r) +o(Tf(r))

⇒ q j=1

kN_(f,H^[n]

j)(r) (q−n−1)(k+ 1)−qn

k T_f(r) +o(T_f(r)) Similarly,

q j=1

kN_(g,H^[n]

j)(r)(q−n−1)(k+ 1)−qn

k Tg(r) +o(Tg(r))

So, q j=1

kN_(f,H^[n]

j)(r) +^kN_(g,H^[n]

j)(r) (q−n−1)(k+ 1)−qn

k (Tf(r) +Tg(r))

+o(T_f(r) +T_g(r)) (3.6)

By (3.5) and (3.6) we have p−1

n

kN_(f,H^[n]

j0)(r) +^kN_(g,H^[n]

j0)(r) +o(T_f(r) +T_g(r))

2(k+ 1) + (p+q−1)

k −(q−n−1)(k+ 1)−qn nk

(T_f(r) +T_g(r))

⇒

kN_(f,H^[n] _j

0)(r) +^kN_(g,H^[n] _j

0)(r) +o(Tf(r) +Tg(r)) (3n+ 1−q)(k+ 1) + (2q+p−1)n

k(p−1) (Tf(r) +Tg(r)) for allj0∈ {1, ..., n+ 1 +y}

So,

n+1+y

j=1

kN_(f,H^[n]

j)(r) +^kN_(g,H^[n]

j)(r) +o(Tf(r) +Tg(r)) (n+ 1 +y) [(3n+ 1−q)(k+ 1) + (2q+p−1)n]

k(p−1) (Tf(r) +Tg(r)) (3.7) By the First and the Second Main Theorem, we have:

yTf(r)

n+1+y

j=1

N_(f,H^[n]

j)(r) +o(Tf(r))

= k

k+ 1

n+1+y

j=1

kN_(f,H^[n]

j)(r) +

n+1+y

j=1

1 k+ 1

kN_(f,H^[n]

j)(r) +^>kN_(f,H^[n]

j)(r)

+o(Tf(r))

k k+ 1

n+1+y

j=1

kN_(f,H^[n]

j)(r) + n k+ 1

n+1+y

j=1

N_(f,Hj)(r) +o(Tf(r))

k

k+ 1

n+1+y

j=1

kN_(f,H^[n]

j)(r) +n(n+ 1 +y)

k+ 1 Tf(r) +o(Tf(r))

⇒ y(k+ 1)−n(n+ 1 +y)

k Tf(r)

n+1+y

j=1

kN_(f,H^[n]

j)(r) +o(Tf(r)).

Similarly,

y(k+ 1)−n(n+ 1 +y)

k Tg(r)

n+1+y

j=1

kN_(g,H^[n] _j)(r) +o(Tg(r)).

So,

y(k+ 1)−n(n+ 1 +y)

k (Tf(r) +Tg(r))

n+1+y

j=1

kN_(f,H^[n] _j)(r) +^kN_(g,H^[n] _j)(r) +o(Tf(r) +Tg(r)) (3.8)

By (3.7) and (3.8) we have y(k+ 1)−n(n+ 1 +y)

k (Tf(r) +Tg(r)) +o(Tf(r) +Tg(r)) (n+ 1 +y) [(3n+ 1−q)(k+ 1) + (2q+p−1)n]

k(p−1) (T_f(r) +T_g(r)) So,

(p−1) [y(k+ 1)−n(n+ 1 +y)](n+1+y) [x(k+ 1) + (6n+p+ 1−2x)n]

⇒ k+ 1 2n(n+ 1 +y)(3n+p−x) (p−1)y−x(n+ 1 +y)

(note that (p−1)y−x(n+ 1 +y)>0).This is a contradiction. Thus, we

have f ≡g.

4. Proof of Theorem 1.5

LetG be a torsion free abelian group and A= (x₁, ..., x_q) be aq−tuple of elements xi in G. Let 1 < s < r q. We say that A has the property Pr,s if any r elements xp1, ..., xpr in A satisfy the condition that for any subset I⊂ {p1, ..., pr} with #I =s,there exists a subsetJ ⊂ {p1, ..., pr}, J =I,#J =ssuch that

i∈I

xi=

j∈J

xj.

Lemma 4.1. —If A has the property P_r,s, then there exists a subset {i₁, ..., i_q−r+2} ⊂ {1, ..., q}such that x_i1 =...=x_iq−r+2.

Proof. — We refer to [3], Lemma 2.6.

Lemma 4.2. — Let f : C^m −→ CPⁿ be a nonconstant meromorphic mapping and

a_in

i=0be “small” (with respect tof) meromorphic mappings of C^m intoCPⁿ in general position.

Denote the meromorphic mapping, F =

c0·(f,˜a0) :· · ·:cn·(f,a˜n)

:C^m−→CPⁿ where

c_in

i=0 are “small” (with respect to f) nonzero meromorphic func- tions on C^m.

Then,

T_F(r) =T_f(r) +o(T_f(r)).

Moreover, if

f = (f₀:· · ·:f_n), ai = (ai0:· · ·:ain),

F =

c0·(f,˜a0)

h :· · ·: cn·(f,a˜n) h

are reduced representations, wherehis a meromorphic function onC^m, then Nh(r)o(Tf(r))

and

N¹

h(r)o(Tf(r)).

Proof. — Set

F_i= ci·(f,˜ai)

h , (i= 0, . . . , n).

We have 





a00f0+. . .+a0nfn= _c^h

0F0a0t0

. . . . . . . . . . . . . . . an0f0+. . .+annfn= _c^h

nFnantn

(4.1) Since (F₀:· · ·:F_n) is a reduced representation ofF, we have

N¹

h(r) n i=0

Na_iti(r) + n i=0

N¹

ci(r) =o(Tf(r)).

Set

P =





a₀₀ . . . a_0n ... . .. ... an0 . . . ann





and matrices P_i (i ∈ {0, . . . , n}) which are defined from P after changing

the (i+ 1)^th column by





 F0

a0t0

c0

... F_na_ntn

c_n





.

Putui= det(Pi),u= det(P), thenuis a nonzero holomorphic function onCⁿ and

N_u(r) = o(T_f(r)), N¹

ui(r)

n j=0

Ncj(r) =o(Tf(r)), i= 1, . . . , n.

By (4.1) we have, 













f0= h·u₀ ..u . f_n =h·u_n

u

(4.2)

On the other hand (f₀:· · ·:f_n) is a reduced representation off. Hence,

Nh(r)Nu(r) + n i=0

N¹

ui(r) =o(Tf(r)).

We have T_F(r) =

S(r)

log ⁿ

i=0

|F_i|^{2 1/2}σ+ 0(1)

=

S(r)

log ⁿ

i=0

c_i(f,˜a_i) h

^{2 1/2}σ+ 0(1)

=

S(r)

log ⁿ

i=0

|c_i(f,˜a_i)|^{2 1/2}σ−

S(r)

log|h|σ+ 0(1)

S(r)

logfσ+

S(r)

log ⁿ

i=0

|c_i|²· ˜a_i^{2 1/2}σ

−Nh(r) +N¹

h(r) + 0(1) Tf(r) +1

2

S(r)

log^{+ n}

i=0

ci

ai0

aiti

²+· · ·+ci

ain

aiti

² σ

+o(T_f(r)) T_f(r) +

n i,j=0

m

r, c_iaij

a_iti +o(T_f(r))

= T_f(r) +o(T_f(r)). (4.3)

(4.2) can be written as













f₀=h·ⁿ

i=0

b_i0F_i . . . . . . . . . fn=h·ⁿ

i=0

binFi

where bij

n

i,j=0 are “small” (with respect tof) meromorphic functions on C^m.

So, Tf(r) =

S(r)

logfσ+ 0(1)

=

S(r)

log ⁿ

j=0

n i=0

bijFi^{2 1/2}σ+

S(r)

log|h|σ+ 0(1)

S(r)

logFσ+

S(r)

log

i,j

|bij|^{2 1/2}σ+Nh(r)−N¹

h(r) + 0(1) T_F(r) +

S(r)

log⁺

i,j

|b_ij|^{2 1/2}σ+o(T_f(r)) T_F(r) +

i,j

m(r, b_ij) +o(T_f(r))

= TF(r) +o(Tf(r)) (4.4)

By (4.3) and (4.4), we get Lemma 4.2.

We now begin to prove Theorem 1.5.

The assertion of Theorem 1.5 is trivial if f or g is linearly degener- ate overR

a_j2n+2

j=1 . So from now we assume thatf andg are linearly nondegenerate overR

aj

2n+2 j=1 . Define functions

hj := (a_j0f₀+...+a_jnf_n)

(aj0g0+...+ajngn), j∈ {1, . . . ,2n+ 2}.

For each subsetI⊂ {1, . . . ,2n+ 2}, #I=n+ 1, sethI =

i∈I

hi, γI =

i∈I

γi.

LetM^∗be the abelian multiplication group of all nonzero meromorphic functions on C^m. Denote by H ⊂ M^∗ the set of all h ∈ M^∗ with h^k ∈ R

a_j2n+2

j=1 for some positive integer k. It is easy to see that H is a subgroup ofM^∗ and the multiplication groupG:=M^∗/His a torsion free abelian group. We denote by [h] the class inG containingh∈ M^∗.

We now prove that:

A:= ([h₁], ...,[h_2n+2]) has the property P_2n+2,n+1 . (4.5) We have

aj0f0+. . .+ajnfn=hj(aj0g0+. . .+ajngn) j ∈ {1, ...,2n+ 2}

⇒ aj0f0+. . .+ajnfn−hjaj0g0−. . .−hjajngn= 0 1j2n+ 2