Uniqueness problem of meromorphic mappings with few targets

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LXII, 2008 SECTIO A 123–142

SI DUC QUANG and TRAN VAN TAN

Uniqueness problem of meromorphic mappings with few targets

Abstract. In this paper, using techniques of value distribution theory, we give some uniqueness theorems for meromorphic mappings of C^minto CPⁿ.

1. Introduction. Using the Second Main Theorem of Value Distribution Theory and Borel’s lemma, R. Nevanlinna [11] proved that for two noncon- stant meromorphic functions f and g on the complex plane C, if they have the same inverse images for five distinct values, then f ≡ g, and that g is a special type of linear fractional transformation of f if they have the same inverse images, counted with multiplicities, for four distinct values.

In 1975, H. Fujimoto [5] generalized Nevanlinna’s result to the case of meromorphic mappings of C into CPⁿ. He showed that for two linearly nondegenerate meromorphic mappings f and g of C into CPⁿ, if they have the same inverse images, counted with multiplicities for (3n+2) hyperplanes in CPⁿlocated in general position, then f ≡ g, and there exists a projective linear transformation L of CPⁿ to itself such that g = L · f if they have the same inverse images counted with multiplicities for (3n + 1) hyperplanes in CPⁿ located in general position. Since that time, this problem has been studied intensively for the case of hyperplanes by H. Fujimoto ([7], [8]),

2000 Mathematics Subject Classification. 32H30, 32H04.

Key words and phrases. Meromorphic mappings, value distribution theory, uniqueness problem.

W. Stoll [17], L. Smiley [14], S. Ji [9], M. Ru [13], Z. Ye [20], G. Dethloff–

T. V. Tan ([2], [3], [4]), D. D. Thai–S. D. Quang [15] and others.

Let f be a linearly nondegenerate meromorphic mapping of C^minto CPⁿ. For each hyperplane H, we denote by ν_(f,H) the map of C^m into N₀ whose value ν_(f,H)(a) (a ∈ C^m) is the intersection multiplicity of the image of f and H at f (a).

Take q hyperplanes H₁, . . . , H_q in CPⁿ located in general position with a) dim f⁻¹(H_i) ∩ f⁻¹(H_j)
≤ m − 2 for all 1 ≤ i < j ≤ q.

For each positive integer (or +∞) M , denote by G {H_j}^q_j=1, f, M
the set of all linearly nondegenerate meromorphic mappings g of C^m into CPⁿ such that

b) min{ν_(g,Hj), M } = min{ν_(f,Hj), M }, j ∈ {1, . . . , q} and c) g = f onSq

j=1f⁻¹(H_j).

In 1983, L. Smiley [14] showed that:

Theorem A. If q ≥ 3n + 2 then g1= g2 for any g₁, g2 ∈ G {H_j}^q_j=1, f, 1
.

In 1998, H. Fujimoto [7] obtained the following theorem:

Theorem B. If q ≥ 3n + 1 then G {Hj}^q_j=1, f, 2

contains at most two mappings.

He also gave the open question: Does his result remain valid if the number of hyperplanes is replaced by a smaller one? In 2006, G. Dethloff and T. V.

Tan [4] showed that the above result of Fujimoto remains valid if q ≥ 3n − 1, n ≥ 7. In this paper, by a different approach, we extend Theorem B to the case of

q > max 7(n + 1)

4 ,

√

17n²+ 16n + 3n + 4 4

 . In 1980, W. Stoll [19] obtained the following theorem:

Theorem C. Let f1, . . . , f_k (k ≥ 2) be linearly nondegenerate holomorphic mappings of C into CPⁿ. Let H₁, . . . , H_q (q ≥ (k + 1)n + 2) be hyperplanes in CPⁿ located in general position. Assume that

i) f₁⁻¹(Hj) = · · · = f_k⁻¹(Hj) for all j ∈ {1, . . . , q}, ii) f₁⁻¹(Hi) ∩ f₁⁻¹(Hj) = ∅ for all 1 ≤ i < j ≤ q and iii) f₁∧ · · · ∧ f_k= 0 on Sq

j=1f₁⁻¹(Hj).

Then f₁∧ · · · ∧ f_k≡ 0.

In 2001, M. Ru [13] generalized the above result to the case of moving hyperplanes. In the last part of this paper, we extend Theorem C to the case of moving hypersurfaces.

Acknowledgements. The authors would like to thank Professors D. D.

Thai, G. Dethloff, J. Nugochi for constant help and encouragement.

2. Preliminaries. For z = (z₁, . . . , z_m) ∈ C^m, we set kzk =

 m

X

j=1

|z_j|²

1/2

and define

B(r) = {z ∈ C^m: kzk < r}, S(r) = {z ∈ C^m: kzk = r}, d^c =

√−1

4π (∂ − ∂), V = dd^ckzk²
m−1

, σ = d^clog kzk²∧ dd^clog kzk
m−1

. Let F be a nonzero holomorphic function on C^m. For a set α = (α₁, . . . , αm) of nonnegative integers, we set |α| = α₁+· · ·+αm and D^αF = _∂α1z^D|α|F

1...∂^αmzm. We define the map ν_F : C^m→ N₀ by

ν_F(a) = max{p : D^αF (a) = 0 for all α with |α| < p}.

Let ϕ be a nonzero meromorphic function on C^m. For each a ∈ C^m, we choose nonzero holomorphic functions F and G on a neighborhood U of a such that ϕ = ^F_G on U and dim F⁻¹(0) ∩ G⁻¹(0)
≤ m − 2 and we define the map ν_ϕ: C^m−→ N₀ by ν_ϕ(a) = νF(a). Set

|ν_ϕ| = {z : ν_ϕ(z) 6= 0}.

Let k be positive integer or +∞. Set ν_ϕ^(k)(z) = min{ν_ϕ(z), k}, and

N_ϕ^(k)(r) :=

r

Z

1

n^(k)(t)

t^2m−1dt (1 < r < +∞) where

n^(k)(t) = Z

|νϕ|∩B(t)

ν_ϕ^(k)· V for m ≥ 2

and

n^(k)(t) = X

|z|≤t

ν_ϕ^(k)(z) for m = 1.

We simply write N_ϕ(r) for N_ϕ^(+∞)(r). We have the following Jensen’s for- mula:

Nϕ(r) − N¹

ϕ(r) = Z

S(r)

log |ϕ|σ − Z

S(1)

log |ϕ|σ.

Let f be a meromorphic mapping of C^m into CPⁿ. For arbitrary fixed homogeneous coordinates (w₀ : · · · : wn) of CPⁿ, we take a reduced rep- resentation f = (f₀ : · · · : fn) which means that each fi is holomor- phic function on C^m and f (z) = (f₀(z) : · · · : f_n(z)) outside the ana- lytic I(f ) := {z : f₀(z) = · · · = fn(z) = 0} of codimension ≥ 2. Set kf k = max{|f₀|, . . . , |f_n|}.

The characteristic function of f is defined by T_f(r) :=

Z

S(r)

log kf kσ − Z

S(1)

log kf kσ, 1 < r < +∞.

For a meromorphic function ϕ on C^m, the characteristic function T_ϕ(r) of ϕ is defined as ϕ is a meromorphic map of C^m into CP¹. The proximity function m(r, ϕ) is defined by

m(r, ϕ) = Z

S(r)

log⁺|ϕ|σ,

where log⁺x = max{log x, 0} for x ≥ 0.

Then

Tϕ(r) = N¹

ϕ(r) + m(r, ϕ) + O(1).

We state the First and the Second Main Theorems of Value Distribution Theory:

Let f be a nonconstant meromorphic mapping of C^m into CPⁿ. We say that a meromorphic function ϕ on C^m is “small” with respect to f if T_ϕ(r) = o(T_f(r)) as r → ∞ (outside a set of finite Lebesgues measure).

Denote by R_f the field of all “small” (with respect to f ) meromorphic functions on C^m.

Theorem D (First Main Theorem). Let f be a nonconstant meromorphic mapping of C^m into CPⁿ and Q be a homogeneous polynomial of degree d in R_f[x0, . . . , xn] such that Q(f ) 6≡ 0 then

N_{Q(f )}(r) ≤ d · T_f(r) + o(T_f(r)) for all r > 1.

For a hyperplane H : a₀w0+ · · · + anwn= 0 in CPⁿ with im f 6⊆ H, we denote (f, H) := a₀f₀+ · · · + a_nf_n, where (f₀ : · · · : f_n) again is a reduced representation of f .

As usual, by the notation “|| P” we mean the assertion P holds for all r ∈ (1, +∞) excluding a subset E of (1, +∞) of finite Lebesgue measure.

Theorem E (Second Main Theorem). Let f be a linearly nondegenerate meromorphic mapping of C^m into CPⁿ and H₁, . . . , H_q (q ≥ n + 1) hyper- planes in CPⁿ located in general position, then

|| (q − n − 1)T_f(r) ≤

q

X

j=1

N_(f,H⁽ⁿ⁾

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

3. Uniqueness problem for hyperplanes. First of all, we give the fol- lowing lemma, which is an extension of uniqueness theorem to the case of few hyperplanes.

Lemma 1. Let f, g : C^m → CPⁿ be two linearly nondegenerate mero- morphic mappings with reduced representations f = (f₀ : · · · : f_n), g = (g0 : · · · : gn). Let {Hi}^q_i=1 be q hyperplanes located in general position with dim f⁻¹(Hi) ∩ f⁻¹(Hj)
≤ m − 2 for all 1 ≤ i < j ≤ q. Assume that

q >

√

17n²+ 16n + 3n + 4 4

and

(i) min{ν_(f,Hi)(z), n} = min{ν_(g,Hi)(z), n}, for all i ∈ {1, . . . , q}, (ii) Zero (f_j) ∩ f⁻¹(H_i) = Zero (g_j) ∩ f⁻¹(H_i), for all 1 ≤ i ≤ q, 0 ≤ j ≤

n, (iii) D^{α f}_f^k

s
= D^{α g}_g^k

s
on S^q_i=1f⁻¹(Hi)
\ Zero (f_s)
, for all |α| ≤ 1, 0 ≤ k 6= s ≤ n.

Then f ≡ g.

Proof. Assume that f 6≡ g. We write H_i:Pn

j=0a_ijω_j = 0.

For any fixed index i, (1 ≤ i ≤ q), it is easy to see that there exists j ∈ {1, . . . , q}\{i} (depending on i) such that

P_ij := (f, H_i)

(f, Hj) − (g, H_i) (g, Hj) 6≡ 0.

Set

I := I(f ) ∪ I(g) ∪[

1≤k<s≤q

{z ∈ C^m: ν_(f,Hk)(z) > 0 and ν_(f,Hs)(z) > 0}.

Then I is an analytic subset of codimension ≥ 2.

Case 1. n ≥ 2.

Let t be an arbitrary index in {1, . . . , q}\{i, j}. For any fixed point z₀ 6∈ I satisfying ν_(f,Ht)(z0) > 0, there exists l ∈ {0, . . . , n} such that f_l(z0)g_l(z0) 6=

0. It follows that

D^αP_ij(z₀) = D^α (f, H_i)

(f, Hj) − (g, H_i) (g, Hj)

 (z₀)

= D^α Pn

v=0 fv

fla_iv Pn

v=0 fv

f_lajv

− Pn

v=0 gv

gla_iv P_n

v=0 gv

gla_jv

!

(z0) = 0, for all α with |α| < 2. So

ν_Pij(z₀) ≥ 2.

(3.1)

For any fixed point z₁ 6∈ I satisfying ν_(f,Hi)(z1) > 0, we have ν_Pij(z₁) ≥ min{ν_(f,Hi)(z₁), ν_(g,Hi)(z₁)} ≥ min{ν_(f,Hi)(z₁), n}.

(3.2)

From (3.1) and (3.2), we have

νPij ≥ min{n, ν_(f,Hi)} + X

t∈{1,...,q}\{i,j}

2 min{1, ν_(f,Ht)}, (outside an analytic subset of codimension two).

It yields that

N_Pij(r) ≥ N_(f,H⁽ⁿ⁾

i)(r) + X

t∈{1,...,q}\{i,j}

2N_(f,H⁽¹⁾

t)(r) (3.3)

It is clear that

N 1

Pij(r) ≤ N (r, ν_j), (3.4)

where ν_j(z) := max{ν_(f,Hj)(z), ν_(g,Hj)(z)}.

We have m



r,(f, Hi) (f, H_j)



= T_(f,Hi)

(f,Hj )

(r) − N_(f,Hj)(r) + O(1)

≤ T_f(r) − N_(f,Hj)(r) + O(1), and

m



r, (g, Hi) (g, H_j)



≤ T_g(r) − N_(g,Hj)(r) + O(1), This implies that

m(r, Pij) ≤ m



r,(f, Hi) (f, Hj)

 + m



r, (g, Hi) (g, Hj)



+ O(1)

= T_f(r) + T_g(r) − N_(f,Hj)(r) − N_(g,Hj)(r) + O(1).

Combining with (3.3) and (3.4) we get N_(f,H⁽ⁿ⁾

i)(r) + X

t∈{1,...,q}\{i,j}

2N_(f,H⁽¹⁾

t)(r) ≤ NPij(r) ≤ TPij(r) + O(1)

= N ¹

Pij(r) + m(r, Pij) + O(1)

≤ T_f(r) + Tg(r) + N (r, νj) − N_(f,Hj)(r)

− N_(g,Hj)(r) + o(T_f(r) + T_g(r)).

This gives

N_(f,Hj)(r) + N_(g,Hj)(r) − N (r, νj) + N_(f,H⁽ⁿ⁾

i)(r) + X

t∈{1,...,q}\{i,j}

2N_(f,H⁽¹⁾

t)(r)

≤ T_f(r) + Tg(r) + o(Tf(r) + Tg(r)).

On the other hand, since

ν_j(z) − ν_(f,Hj)− ν_(g,Hj)+ min{n, ν_(f,Hj)} ≤ 0

(outside an analytic subset of codimension two), we have N (r, νj) − N_(f,Hj)(r) − N_(g,Hj)(r) + N_(f,H⁽ⁿ⁾

j)(r) ≤ 0.

Hence

N_(f,H⁽ⁿ⁾

i)(r) + N_(f,H⁽ⁿ⁾

j)(r) + X

t∈{1,...,q}\{i,j}

2N_(f,H⁽¹⁾

t)(r)

≤ T_f(r) + Tg(r) + o(T_f(r) + Tg(r)).

It implies that

(3.5)

N_(f,H⁽ⁿ⁾

i)(r) + 2 n

X

t∈{1,...,q}\{i}

N_(f,H⁽ⁿ⁾

t)(r)

≤ T_f(r) + T_g(r) + o(T_f(r) + T_g(r)), (note that n ≥ 2).

Taking summing-up of both sides of (3.5) over all i ∈ {1, . . . , q}, we obtain

(3.6)



1 +2(q − 1) n

 ^q X

i=1

N_(f,H⁽ⁿ⁾

i)(r)

≤ q(T_f(r) + T_g(r)) + o(T_ff (r) + T_g(r)).

On the other hand, by Theorem E we have

|| (q − n − 1)(T_f(r) + T_g(r)) ≤ 2

q

X

i=1

N_(f,H⁽ⁿ⁾

i)(r) + o(T_f(r) + T_g(r)).

(3.7)

From (3.6) and (3.7), letting r −→ ∞ we have 1 +2(q − 1)

n ≤ 2q

q − n − 1. This contradicts to

q >

√17n²+ 16n + 3n + 4

4 .

Thus f ≡ g.

Case 2. n = 1. We have q ≥ 4. If ^(f,H_(f,H¹⁾

4) ≡ ^(g,H_(g,H¹⁾

4), then f ≡ g.

We now assume that

P14:= (f, H1)

(f, H₄) −(g, H1) (g, H₄) 6≡ 0.

Let t be an arbitrary index in {1, 2, 3}. For any fixed point z₀ 6∈ I satisfying ν_(f,Ht)(z0) > 0, there exists l ∈ {0, 1} such that fl(z0)gl(z0) 6= 0. It follows

that

D^αP14(z0) = D^α (f, H₁)

(f, H4) −(g, H₁) (g, H4)

 (z0)

= D^α

a₁₀^f_f⁰

l + a₁₁^f_f¹

l

a₄₀^f_f⁰

l + a₄₁^f_f¹

l

−a₁₀^g_g⁰

l + a₁₁^g_g¹

l

a40g0

g_l + a41g1

g_l

!

(z0) = 0,

for all α with |α| < 2. It implies that ν_P14(z₀) ≥ 2. Hence, we have ν_P14 ≥ 2 min{1, ν_(f,H1)} + min{1, ν_(f,H2)} + min{1, ν_(f,H3)}
, (outside an analytic subset of codimension two). It implies that

N_P14(r) ≥ 2 N_(f,H⁽¹⁾

1)(r) + N_(f,H⁽¹⁾

2)(r) + N_(f,H⁽¹⁾

3)(r) . (3.8)

Let z₁ be an arbitrary pole of P₁₄ such that z₁6∈ I. Then z₁ is a zero of (f, H4) and there exists l ∈ {0, 1} such that fl(z1)gl(z1) 6= 0. Then

D^α

 a10

f0

f_l + a11

f1

f_l



a40

g0

g_l + a41

g1

g_l



−

 a₄₀f₀

fl

+ a₄₁f₁ fl



a₁₀g₀ gl

+ a₁₁g₁ gl

!

(z₁) = 0,

for all α with |α| < 2. This implies that

ν_{((f,H1)(g,H4)−(f,H4)(g,H1))}(z1) ≥ 2.

Then, we have ν ¹

P14(z1) ≤ ν_(f,H4)(z1) + ν_(g,H4)(z1) − 2.

Hence we see ν ¹

P14

≤ ν_(f,H4)+ ν_(g,H4)− 2 min{1, ν_(f,H4)},

(outside an analytic subset of codimension two). This implies that N ¹

P14(r) ≤ N_(f,H4)(r) + N_(g,H4)(r) − 2N_(f,H⁽¹⁾

4)(r).

Combining with (3.8) we have 2

 N_(f,H⁽¹⁾

1)(r) + N_(f,H⁽¹⁾

2)(r) + N_(f,H⁽¹⁾

3)(r)



≤ N_P14(r) ≤ T_P14(r) + O(1)

= m(r, P₁₄) + N ¹

P14(r) + O(1)

≤ m



r,(f, H1) (f, H4)

 + m



r,(g, H1) (g, H4)



+ N_(f,H4)(r) + N_(g,H4)(r) − 2N_(f,H⁽¹⁾

4)(r) + O(1)

= T_(f,H1)

(f,H4)

(r) + T_(g,H1)

(g,H4)

(r) − 2N_(f,H⁽¹⁾

4)(r) + O(1)

≤ T_f(r) + Tg(r) − 2N_(f,H⁽¹⁾

4)(r) + o(T_f(r) + Tg(r)).

It implies that 2

N_(f,H⁽¹⁾

1)(r) + N_(f,H⁽¹⁾

2)(r) + N_(f,H⁽¹⁾

3)(r) + N_(f,H⁽¹⁾

4)(r)

≤ T_f(r) + T_g(r) + o(T_f(r) + T_g(r)).

On the other hand, by Theorem E, we also have

|| 2T_f(r) ≤ N_(f,H⁽¹⁾

1)(r) + N_(f,H⁽¹⁾

2)(r) + N_(f,H⁽¹⁾

3)(r) + N_(f,H⁽¹⁾

4)(r) + o(Tf(r)) and

|| 2T_g(r) ≤ N_(g,H⁽¹⁾

1)(r) + N_(g,H⁽¹⁾

2)(r) + N_(g,H⁽¹⁾

3)(r) + N_(g,H⁽¹⁾

4)(r) + o(Tg(r))

= N_(f,H⁽¹⁾

1)(r) + N_(f,H⁽¹⁾

2)(r) + N_(f,H⁽¹⁾

3)(r) + N_(f,H⁽¹⁾

4)(r) + o(Tg(r)) Hence, we have

|| 2(T_f(r) + Tg(r)) ≤ Tf(r) + Tg(r) + o(Tf(r) + Tg(r)).

Letting r −→ ∞, we have 2 ≤ 1. This is a contradiction, hence f ≡ g. We

have completed the proof of Lemma 1. 

Let f be a linearly nondegenerate meromorphic mapping of C^minto CPⁿ with reduced representation f = (f₀ : · · · : f_n). Let d be a positive integer and let H₁, . . . , Hqbe q hyperplanes in CPⁿlocated in general position with

dimz ∈ C^m : ν_(f,Hi)(z) > 0 and ν_(f,Hj)(z) > 0 ≤ m − 2 (1 ≤ i < j ≤ q).

Consider the set F (f, {H_j}^q_j=1, d) of all linearly nondegenerate meromor- phic mappings g : C^m → CPⁿ with reduced representation g = (g₀ : · · · : g_n) satisfying the conditions:

(a) min(ν_(f,Hi), d) = min(ν_(g,Hi), d) (1 ≤ i ≤ q),

(b) Zero (f_j) ∩ f⁻¹(Hi) = Zero (gj) ∩ f⁻¹(Hi), for all 1 ≤ i ≤ q, 0 ≤ j ≤ n,

(c) D^{α f}_f^k

s
= D^{α g}_g^k

s
on S^q_i=1f⁻¹(Hi)
\ Zero (f_s)
, for all |α| < d, 0 ≤ k 6= s ≤ n.

Take M + 1 maps f⁰, . . . , f^M ∈ F (f, {H_j}^q_j=1, d) with reduced representa- tions

f^k:= f₀^k : · · · : f_n^k
and set T (r) :=PM

k=0T_fk(r). For each c = (c₀, . . . , c_n) ∈ Cⁿ⁺¹\ {0} we put (f^k, c) :=

n

X

i=0

cif_i^k (0 ≤ k ≤ M ).

Denote by C the set of all c ∈ Cⁿ⁺¹\ {0} such that

dim{z ∈ C^m : (f^k, Hj)(z) = (f^k, c)(z) = 0} ≤ m − 2 (1 ≤ j ≤ q, 0 ≤ k ≤ M ).

Lemma A ([9], Lemma 5.1). C is dense in Cⁿ⁺¹.

Lemma B ([7]). For each c ∈ C, we put Fc^jk = ^(f_(f^kk^,H,c)^j). Then T_Fjk c (r) ≤ T_fk(r) + o(T (r)).

Definition 1. Let F₀, . . . , F_M be meromorphic functions on C^m, where M ≥ 1. Take a set α := (α⁰, . . . , α^{M −1}) whose components α^kare composed of n nonnegative integers, and set |α| = |α⁰| + · · · + |α^{M −1}|. We define Cartan’s auxiliary function by

Φ^α(F₀, . . . , F_M) := F₀· F₁· · · F_M

×         

1 1 · · · 1

D^α0(_F¹

0) D^α0(_F¹

1) · · · D^α0(_F¹

M)

... ... ... ...

D^{αM −1}(_F¹

0) D^{αM −1}(_F¹

1) · · · D^{αM −1}(_F¹

M)          .

Lemma C ([7], Proposition 3.4). If Φ^α(F, G, H) = 0 and Φ^α(_F¹,_G¹,_H¹) = 0 for all α with |α| ≤ 1, then one of the following conditions holds:

i) F = G or G = H or H = F . ii) ^F_G,_H^G and ^H_F are all constant.

Lemma 2. Assume that there exists Φ^α := Φ^α Fc^j00, . . . , Fc^j0M

6≡ 0 for some c ∈ C, |α| ≤ ^{M (M −1)}₂ , d ≥ |α|. Then, for each 0 ≤ i ≤ M, the following holds:

|| N_(f^(d−|α|)_i,H

j0)(r) + M dX

j6=j0

N_(f⁽¹⁾i,Hj)(r) ≤ N_Φ^α(r) ≤ T (r) + o(T (r)).

Proof. Denote by P the set of all β with |β| ≤ ^{M (M −1)}₂ , d ≥ |β| such that Φ^β = Φ^β F_c^j00, . . . , F_c^j0M
6≡ 0 for some c ∈ C. Let α be the minimal multi-index in P (in the lexicographic order). Set

I :=

M

[

t=0

I(f^t) ∪ [

1≤t<j≤q

(f, H_t)⁻¹{0} ∩ (f, H_j)⁻¹{0}

∪

q

[

t=1

(f, Ht)⁻¹{0} ∩ (f, c)⁻¹{0}
.

Then I is an analytic subset of codimension ≥ 2.

Assume that a is a zero of some (fⁱ, Hj), j 6= j0 such that a 6∈ I. Let Γ be an irreducible component of the zero-divisor of the function (fⁱ, Hj) which contains a. We take a holomorphic function h on C^m satisfying: ν_h|

Γ = 1 and ν_h|

(Cn\Γ) = 0.

By the condition (c), we have that ϕ_i:= _hdF¹_j0i −_hdF¹_j0M
is a holomor- phic function on a neighborhood U of a for all i ∈ {0, . . . , M − 1}. Since α := min P, we have

Φ^α := h^{M d}F^j00· · · F^j0M ×       

D^α0ϕ0 · · · D^α0ϕM −1

... ... ... D^{αM −1}ϕ₀ · · · D^{αM −1}ϕ_{M −1}

       .

It implies that

νΦ^α(a) ≥ M d.

(3.9)

Assume that b is a zero of (fⁱ, H_j0) such that b 6∈ I. If ν_(fi,H_j0)(b) ≥ d, we write

Φ^α = X

σ∈SM +1

sign(σ)F^j00· · · F^j0M

× D^α0 1 F^{j0(σ(2)−1)}



· · · D^{αM −1} 1 F^j0(σ(M +1)−1)

 . Then

ν_Φ^α(b) ≥ d − |α|.

(3.10)

If ν_(fi,H_j0)(b) < d, then ν_(f0,H_j0)(b) = · · · = ν_(fM,H_j0)(b) < d. There exists a holomorphic function h on an open neighborhood U of b such that ν_h = ν_(fi,H_j0)|U

.

We write

Φ^α= h^−MF_c^j00· · · F_c^j0M

×        

D^{α0 h}

Fc^j00

− D^{α0 h}

Fc^j0M

· · · D^{α0 h}

Fc^j0(M−1)

− D^{α0 h}

Fc^j0M

... ... ...

D^{αM −1 h}

Fc^j00

− D^{αM −1 h}

Fc^j0M

· · · D^{αM −1 h}

Fc^j0(M−1)

− D^{αM −1 h}

Fc^j0M

        .

Then

νΦ^α(b) ≥ ν_(fⁱ_,Hj0)(b).

(3.11)

From (3.9), (3.10) and (3.11), we have mind − |α|, ν_(fi,H_j0) + M d X

j∈{1,...,q}\{j0}

min1, ν_(fi,Hj) ≤ ν_Φ^α,

(outside an analytic subset of codimension two). It immediately follows the first inequality in the lemma.

It is easy to see that a pole of Φ^α is a zero or a pole of some F_c^j0k. By (3.9), (3.10) and (3.11) we have that Φ^αis holomorphic at all zeros of F_c^j0i, (0 ≤ i ≤ M ). Then

N ¹

Φα(r) ≤

M

X

i=0

N ¹

Fj0i c

(r).

On the other hand, it is easy to see that m(r, Φ^α) ≤

M

X

i=0

m(r, F_c^j0i) + O



Xm



r,D^αi(ϕ^jc^0k) ϕ^jc^0k



+ O(1)

≤

M

X

i=0

m(r, F_c^j0i) + o(T (r)), where ϕ^j_c^0k = 1/Fc^j0k. Hence, we have

N_Φ^α(r) ≤ T_Φ^α(r) + O(1) ≤ m(r, Φ^α) + N ¹

Φα(r) + O(1)

≤

M

X

i=0

N 1 Fj0i

c

(r) + m(r, F_c^j0i)
+ o(T (r))

=

M

X

i=0

T_Fj0i

c (r) + o(T (r)) ≤ T (r) + o(T (r)).  Theorem 1. If

q > max 7(n + 1)

4 ,

√17n²+ 16n + 3n + 4 4



then F (f, {H_i}^q_i=1, 2) contains at most two mappings.

Proof. If n = 1, by Lemma 1 we have ]F (f, {H_i}^q_i=1, 1) = 1.

We prove the theorem for the case of n ≥ 2. Assume that there exist three distinct mappings f⁰, f¹, f²∈ F (f, {H_i}^q_i=1, 2).

Denote by Q the set of all indices j ∈ {1, 2, . . . , q} satisfying the following:

There exist c ∈ C and α ∈ Zⁿ₊with |α| ≤ 1 such that Φ^α Fc^j0, Fc^j1, Fc^j2
6≡ 0.

Set T (r) = T_f0(r) + T_f1(r) + T_f2(r).

We now prove that Q = ∅. Suppose that there exists j₀ ∈ Q. By Lem- ma 2, we have

(3.12)

|| N_(f⁽¹⁾_i,H

j0)(r) + 4 X

j∈{1,...,q}\{j0}

N_(f⁽¹⁾i,Hj)(r)

≤ N (r, ν_Φ^α) ≤ T (r) + o(T (r)).

(0 ≤ i ≤ 2).

By Theorem E, we have

|| X

j6=j0

N_(f⁽¹⁾i,Hj)(r) ≥ q − n − 2

3n T (r) + o(T (r)) and

q

X

j=0

N_(f⁽¹⁾i,Hj)(r) ≥ q − n − 1

3n T (r) + o(T (r)).

This implies that

(3.13)

|| N⁽¹⁾

(fⁱ,H_j0)(r) + 4 X

j∈{1,...,q}\{j0}

N_(f⁽¹⁾i,Hj)(r)

≥ 4(q − n − 2) + 1

3n T (r) + o(T (r)).

From (3.12) and (3.13), letting r → ∞ we get

4(q − n − 2) + 1 ≤ 3n ⇔ q ≤ 7(n + 1)

4 .

This is a contradiction. Hence Q = ∅. Then for each 1 ≤ j ≤ q, c ∈ C, α ∈ Zⁿ₊, |α| < 2 we have Φ^α Fc^j0, Fc^j1, Fc^j2
≡ 0. Since C is dense in Cⁿ⁺¹, we have that

Φ^α F_i^j0, F_i^j1, F_i^j2
≡ 0 (1 ≤ i, j ≤ q), for all |α| < 2,

where F_i^jt := ^(f_(f^tt^,H,H^ji⁾), 0 ≤ t ≤ 2. By Lemma C, for each 1 ≤ i, j ≤ q, there exists a nonzero constant χ_ij such that F_i^j0= χ_ijF_i^j1, F_i^j1= χ_ijF_i^j2or F_i^j2 =

χ_ijF_i^j0. We now show that χ_ij = 1. Indeed, if χ_ij 6= 1, without loss of gener- ality we may assume that F_i^j0= χ_ijF_i^j1. ThenS

t∈{1,...,q}\{i,j}f⁻¹(H_t) = ∅.

Thus, by Theorem E, we have

|| (q − n − 3)T_f(r) ≤ X

t∈{1,...,q}\{i,j}

N_(f,H⁽ⁿ⁾

t)(r) + o(Tf(r)) = o(Tf(r)).

Letting r −→ +∞, we obtain q − n − 3 ≤ 0. This contradicts to n ≥ 2.

Thus,

χij = 1 (1 ≤ i, j ≤ q).

We take an arbitrary element k ∈ {0, 1, 2} and an index i ∈ {1, . . . , q}.

We will show that ν_(fk,Hi) = ν_(fl,Hi) or ν_(fk,Hi) = ν_(f^t_,Hi), where {l, t} :=

{0, 1, 2} \ {k}. In fact, if there is no index j 6= i such that F_i^jk = F_i^jl or F_i^jk = F_i^jt, then since χ_ij = 1 we have F_i^jl= F_i^jt for all j 6= i. This implies that f^k ≡ f^l. This is a contradiction. Hence there exists j 6= i such that F_i^jk = F_i^jl or F_i^jk = F_i^jt. This yields that

ν_(fk,Hi)= ν_(fl,Hi) or ν_(fk,Hi)= ν_(f^t_,Hi) (3.14)

for all k ∈ {0, 1, 2}, i ∈ {1, . . . , q}. For any fixed index i ∈ {1, . . . , q}, by (3.14) (with k = 0) we may assume that ν_(f0,Hi) = ν_(f¹_,Hi). By (3.14) (with k = 2) we obtain ν_(f2,Hi)= ν_(f0,Hi) or ν_(f2,Hi)= ν_(f1,Hi). This implies that ν_(f⁰_,Hi) = ν_(f¹_,Hi) = ν_(f²_,Hi) for all i ∈ {1, . . . , q}. By Lemma 1, we have f⁰≡ f¹ ≡ f² . This is a contradiction.

Thus, ]F (f, {H_i}^q_i=1, 2) ≤ 2 if q > max 7(N + 1)

4 ,

√

17N²+ 16N + 3N + 4 4



. 

4. Uniqueness problem for hypersurfaces. Let f be a nonconstant meromorphic mapping of C^m into CPⁿ. We say that a meromorphic func- tion ϕ on C^m is “small” with respect to f if T_ϕ(r) = o(Tf(r)) as r → ∞ (outside a set of finite Lebesgues measure). Denote by R_f the field of all

“small” (with respect to f ) meromorphic functions on C^m.

Take a reduced representation (f₀ : · · · : fn) of f . We say that f is algebraically nondegenerate over R_f if there is no nonzero homogeneous polynomial Q ∈ R_f[x0, . . . , xn] such that Q(f ) := Q(f0, . . . , fn) ≡ 0.

For a homogeneous polynomial Q ∈ R_f[x₀, . . . , x_n], denote by Q(z) the homogeneous polynomial over C obtained by substituting a specific point z ∈ C^m into the coefficients of Q.

We say that a set {Q_j}ⁿ_j=0of homogeneous polynomials of the same degree in R_f[x₀, . . . , x_n] is admissible if there exists z ∈ C^m such that the system

of equations

Q_j(z)(w₀, . . . , w_n) = 0 0 ≤ j ≤ n

has only the trivial solution w = (0, . . . , 0) in Cⁿ⁺¹. First of all, we give the following lemma:

Lemma 3. Let f be a nonconstant meromorphic mapping of C^m into CPⁿ and {Q_j}ⁿ_j=0 be an admissible set of homogeneous polynomials of degree d in R_f[x0, . . . , xn]. Let γ0, . . . , γn be (n + 1) nonzero meromorphic functions in R_f.

Put P = γ₀Q^p₀+ · · · + γnQ^pn, where p is a positive integer, p > n(n + 1).

Assume that f is algebraically nondegenerate over R_f. Then

|| d(p − n(n + 1))T_f(r) ≤ N_{P (f )}⁽ⁿ⁾ (r) + o(T_f(r)).

Proof. Set Td:=I := (i₀, . . . , in) ∈ Nⁿ⁺¹₀ : i0+ · · · + in= d . Assume that

Q_j = X

I∈Td

a_jIx^I (j = 0, . . . , n).

where a_jI ∈ R_f, x^I = xⁱ₀⁰· · · xⁱ_nⁿ. Set

F = γ0Q^p₀(f ) : · · · : γnQ^p_n(f )
: C^m −→ CPⁿ.

Since f is algebraically nondegenerate over R_f we have that F is linearly nondegenerate (over C).

Assume that _γ

0Q^p₀(f )

h : · · · : ^γnQ_h^{pn(f )}

is a reduced representation of F, where h is a meromorphic function on C^m. Put F_i = ^γiQ

p i(f )

h , i ∈ {0, . . . , n}.

We have

0≤j≤nmax |Q^p_j(f )| ≤ |h| ·

 n

X

i=0

1 γ_i   



· max

1≤i≤n+1|F_i|.

(4.1)

Let t = (. . . , t_kI, . . . ) be a family of variables, (k ∈ {0, . . . , n}, I ∈ T_d).

Set

Qe_j = X

I∈T_d

t_jIx^I ∈ Z[t, x], j = 0, . . . , n.

Let eR ∈ Z[t] be the resultant of eQ₀, . . . , eQ_n. Since Q_j n

j=0 is an admissible set, R := eR(. . . , akI, . . . ) 6≡ 0. It is clear that R ∈ R_f since a_kI ∈ R_f.

By Theorems 3.4 and 3.5 in [10], there exists a positive integer s > d and polynomials

Re_ij

0≤i,j≤n in Z[t, x] which are zero or homogeneous in x of

degree s − d such that x^s_i · eR =

n

X

j=0

Re_ij· eQ_j for all i ∈ {0, . . . , n}.

Set

Rij = eRij (. . . , akI, . . . ), (f0, . . . , fn)
, 0 ≤ i, j ≤ n.

Then,

f_i^s· R =

n

X

j=0

Rij· Q_j(f0, . . . , fn) for all i ∈ {0, . . . , n}.

(4.2) So,

(4.3)

|f_i^s· R| =    

n

X

j=0

R_ij · Q_j(f₀, . . . , f_n)    

≤

n

X

j=0

|R_ij| · max

k∈{0,...,n}|Q_k(f0, . . . , fn)|

for all i ∈ {0, . . . , n}.

We write,

Rij = X

I∈Ts−d

β^ij_If^I, β_I^ij ∈ R_f.

By (4.3), we have

|f_i^s· R| ≤

 X

0≤j≤n I∈Ts−d

|β_I^ij| · kf k^s−d



· max

k∈{0,...,n}

|Q_k(f₀, . . . , f_n)|,

i ∈ {0, . . . , n}. So,

|f_i|^s kf k^s−d ≤

 X

0≤j≤n I∈T_s−d

β^ij_I R   



· max

k∈{0,...,n}|Q_k(f₀, . . . , f_n)|

for all i ∈ {0, . . . , n}.

Thus

kf k^d≤

 X

0≤i,j≤n I∈Ts−d

β_I^ij R

 max

k∈{0,...,n}

|Q_k(f0, . . . , fn)|.

(4.4)

By (4.1) and (4.4) we have

kf k^dp ≤

 X

0≤i,j≤n I∈Ts−d

β_I^ij R

p

· |h| ·

 n

X

i=0

1 γi



· kF k.

(4.5)

By (4.2) and since_γ

0Q^p₀(f )

h : · · · : ^γnQ

p n(f ) h



is a reduced representation of F, we have

N_h(r) ≤ pNR(r) +

n

X

i=0

Nγi(r) = o(T_f(r))

and

N¹

h(r) ≤ X

0≤j≤n I∈Td

N ¹

ajI(r) +

n

X

i=0

N¹

γi = o(T_f(r)).

By (4.5), we have

(4.6)

dp · T_f(r) = pd Z

S(r)

log kf kσ + O(1)

≤ Z

S(r)

log

 X

0≤i,j≤n I∈T_s−d

β_I^ij R

p

|h|

 n

X

i=0

1 γi



σ + TF(r) + O(1)

≤ p Z

S(r)

log⁺

 X

0≤i,j≤n I∈Ts−d

β^ij_I R   

 σ +

Z

S(r)

log⁺

 n

X

i=0

1 γ_i   

 σ

+ Z

S(r)

log |h|σ + TF(r) + O(1)

≤ p X

0≤i,j≤n I∈Ts−d

m

 r,β_I^ij

R

 +

n

X

i=0

m

 r, 1

γi



+ N_h(r) − N¹

h(r) + T_F(r) + O(1)

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