MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES WITH TRUNCATED MULTIPLICITIES

In 1999, Fujimoto proved that there exists an integer l0 such that, if two meromorphic mappings f and g of Cm into Pn(C) have the same inverse images for (2n + 2) hyperplanes in general position with counting multiplicities to level l0, then the map f × g is algebraically degenerate. The purpose of this paper is to generalize the result of H. Fujimoto to the case where meromorphic mappings have the same inverse images of slowly moving hyperplanes

IMAGES OF MOVING HYPERPLANES WITH TRUNCATED

MULTIPLICITIES

Si Duc Quanga and Le Ngoc Quynhb

a Department of Mathematics, Hanoi National University of Education,

136 Xuan Thuy street, Cau Giay, Hanoi, Vietnam email address: quangsd@hnue.edu.vn

b Faculty of Education, An Giang University, 18 Ung Van Khiem,

Dong Xuyen, Long Xuyen, An Giang, Vietnam email address: nquynh1511@gmail.com

Abstract In 1999, Fujimoto proved that there exists an integer l 0 such that, if two

meromorphic mappings f and g of C m into P n (C) have the same inverse images for

(2n + 2) hyperplanes in general position with counting multiplicities to level l 0 , then the

map f × g is algebraically degenerate The purpose of this paper is to generalize the

result of H Fujimoto to the case where meromorphic mappings have the same inverse

images of slowly moving hyperplanes.

Introduction Let f and g be two meromorphic mappings of Cm into Pn(C) Let H1, , Hq be q hyperplanes of Pn(C) in general position Denote by ν(f,Hi) the pull-back divisor of Hi

by f In 1975, Fujimoto proved the following

Theorem A (Fujimoto [2, Theorem II]) Assume that ν(f,H i ) = ν(g,Hi) (1 ≤ i ≤ q) If

q = 3n + 2 and either f or g is linearly non-degenerate over C, i.e., the image is not included in any hyperplane in Pn(C), then f = g

We note that, in this theorem, the condition ν(f,Hi) = ν(g,Hi) (1 ≤ i ≤ q) means that

f and g have the same inverse images with counting multiplicities for these hyperplanes

In 1999, Fujimoto [3] considered the case where these inverse images are counted with multiplicities truncated by a level l0 He proved the following theorem, in which the number q of hyperplanes is also reduced

2010 Mathematics Subject Classification: Primary 32H04; Secondary 32A22, 32A35.

Key words and phrases: Degenerate meromorphic mapping, truncated multiplicity, hyperplane.

1

2MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES

Theorem B (Fujimoto [3, Theorem 1.5]) Let H1, , H2n+2 be hyperplanes of Pn(C)

in general position Then there exist an integer l0 such that, for any two meromorphic mappings f and g of Cm into Pn(C), if min(ν(f,Hi), l0) = min(ν(g,Hi), l0) (1 ≤ i ≤ 2n + 2) then the mapping f × g into Pn(C) × Pn(C) is algebraically degenerate

Here, f × g is a mapping from Cm into Pn(C) × Pn(C) defined by

(f × g)(z) = (f (z), g(z)) ∈ Pn(C) × Pn(C) for all z outside the union of the indeterminacy loci of f and g We also say that a meromorphic mapping into a projective variety is algebraically degenerate if its image is included in a proper analytic subset of the projective variety, otherwise it is algebraically non-degenerate

Then the following question arises naturally:

Are there any similar results to the above results of Fujimoto in the case where fixed hyperplanes are replaced by moving hyperplanes?

The purpose of the present paper is to give an answer for this questions We shall generalize and improve Theorem B to the case of moving hyperplanes To state our result, we first recall some known results

Let f be a meromorphic mappings of Cm into Pn(C) and let a be a meromorphic mappings of Cm into Pn(C)∗ We say that a is slowly moving hyperplanes or slowly moving target of Pn(C) with respect to f if || T (r, a) = o(T (r, f )) as r → ∞ (see Section

1 for the notations) Similarly, a meromorphic function ϕ on Cm is said to be “small” with respect to f if || T (r, ϕ) = o(T (r, f )) as r → ∞

Let a1, , aq (q ≥ n + 1) be q moving hyperplanes of Pn(C) with reduced represen-tations ai = (ai0 : · · · : ain) (1 ≤ i ≤ q) We say that a1, , aq are located in general position if det(aikl) 6≡ 0 for any 1 ≤ i0 < i1 < · · · < in≤ q We denote by M the field of all meromorphic functions on Cm and R{ai}qi=1the smallest subfield of M which contains

C and all ajk/ajl with ajl 6≡ 0

Let N be a positive integer and let V be a projective subvariety of PN(C) Take a homogeneous coordinates (ω0 : · · · : ωN) of PN(C) Let F be a meromorphic mapping of

Cm into V with a representation F = (F0 : · · · : FN)

Definition C.The meromorphic mapping F is said to be algebraically degenerate over

a subfield R of M if there exists a homogeneous polynomial Q ∈ R[ω0, , ωN] with the form

Q(z)(ω0, , ωN) = X

I∈I d

aI(z)ωI,

where d is an integer, Id = {(i0, , iN) ; 0 ≤ ij ≤ d,PN

j=0ij = d}, aI ∈ R and

ωI = ωi0

0 · · · ωiN

N for I = (i0, , iN), such that (i) Q(z)(F0(z), , FN(z)) ≡ 0 on Cm,

(ii) there exists z0 ∈ Cm with Q(z0)(ω0, , ωN) 6≡ 0 on V

Now let f and g be two meromorphic mappings of Cm into Pn(C) with representations

f = (f0 : · · · fn) and g = (g0 : · · · : gn)

We consider Pn(C) × Pn(C) as a projective subvariety of P(n+1)2−1(C) by Segre embed-ding Then the map f × g into Pn(C) × Pn(C) is algebraically degenerate over a subfield

R of M if there exists a nontrivial polynomial

Q(z)(ω0, , ωn, ω00, , ω0n) = X

I=(i 0 , ,i n )∈Zn+1+

i0+···+in=d

X

J =(j 0 , ,j n )∈Zn+1+

j0+···+jn=d0

aIJ(z)ωIω0J,

where d, d0 are positive integers, aIJ ∈ R, such that

Q(z)(f0(z), , fn(z), g0(z), , gn(z)) ≡ 0

We now generalize Theorem B as follows

Main Theorem Let a1, , a2n+2 be (2n + 2) meromorphic mappings of Cm into

Pn(C)∗ in general position Then there exists a positive integer l0 such that, for any two meromorphic mappings f and g of Cm into Pn(C), if a1, , a2n+2 are slowly with respect to f and min(ν(f,Hi), l0) = min(ν(g,Hi), l0) (1 ≤ i ≤ 2n + 2) then the map f × g into

Pn(C) × Pn(C) is algebraically degenerate over R{ai}2n+2

i=1 Here, for a divisor ν on Cm and a positive integer l0, the function min(ν, l0) is defined

on Cm by

min(ν, l0)(z) = min{ν(z), l0} for all z ∈ Cm Remark: Concerning finiteness or degeneracy problems of meromorphic mappings with fixed or moving hyperplanes, there are many results given by Ru [8], Tu [12], Thai-Quang [10], Dethloff-Tan [1], Giang-Quynh-Quang [4][7] and others However, in all their results, they need an aditional assumption that f and g are agree on the inverse images of all hyperplanes This is a strong condition and it is very hard to examine

Acknowledgments This work was completed during a stay of the first author at Vietnam Institute for Advanced Study in Mathematics He would like to thank the insti-tute for the support The research of the authors was supported in part by a NAFOSTED grant of Vietnam

1 Basic notions and auxiliary results from Nevanlinna theory (a) We set ||z|| = |z1|2+ · · · + |zm|2
1/2

for z = (z1, , zm) ∈ Cm and define B(r) := {z ∈ Cm; ||z|| < r}, S(r) := {z ∈ Cm; ||z|| = r} (0 < r < ∞)

Define

vm−1(z) := ddc||z||2
m−1

and

σm(z) := dclog||z||2∧ ddclog||z||2
m−1on Cm\ {0}

4MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES

Let F be a nonzero meromorphic function on a domain Ω in Cm For a sequence

α = (α1, , αm)

of nonnegative integers, we set |α| = α1+ · · · + αm and

|α|F

∂α 1z1· · · ∂α mzm.

We denote by ν0

F (resp νF∞) the divisor of zeros (resp the divisor of poles) of the function

F

A divisor ν on Cm is given by a formal sum ν = P νµXµ, where {Xµ} is a locally finite family of analytic hypersurfaces in Cm and νµ ∈ Z We may assume that Xµ are irreducible and distinct to each other, and νµ 6= 0 for all µ Then the set Supp(ν) =S

µXµ

is called the support of ν Sometimes, we identify the divisor ν with the function ν(z) from Cm into Z defined as follows: For a point p ∈ Cm, there exist a neighborhood U of

p in Cm with a local coordinate (ω1, , ωm) and two holomorphic functions f and g on

U such that ν0

f − ν0

g = ν|U We define ν(p) : = max{d; Dαf (p) = 0 for all α with |α| < d}

− max{d; Dαg(p) = 0 for all α with |α| < d}, where α = (α1, , αm) ∈ Zm+, |α| =Pm

i=1αi and

|α|ϕ

∂α 1ω1· · · ∂α mωm

for a holomorphic function ϕ We note that the above definition of ν(z) is independent from the choices of neighborhood U , local coordinate on U and holomorphic functions f and g

For a divisor ν on Cm and positive integers k, M or M = ∞, we define the counting function of ν by

ν[M ](z) = min {M, ν(z)},

ν>k[M ](z) =







ν[M ](z) if ν(z) > k,

0 if ν(z) ≤ k,

n(t) =











R

|ν| ∩B(t)

ν(z)vm−1 if m ≥ 2, P

|z|≤t

Similarly, we define n[M ](t) and n[M ]>k(t)

Define

N (r, ν) =

Z r 1

n(t)

t2m−1dt (1 < r < ∞)

Similarly, we define N (r, ν[M ]), N (r, ν>k[M ]) and denote them by N[M ](r, ν), N>k[M ](r, ν), respectively

Let ϕ : Cm → C be a meromorphic function Define

Nϕ(r) = N (r, νϕ0), Nϕ[M ](r) = N[M ](r, νϕ0), Nϕ,>k[M ] (r) = N>k[M ](r, νϕ0)

For brevity, we will omit the character [M ] if M = ∞

(b) Let f : Cm → Pn(C) be a meromorphic mapping For arbitrarily fixed homogeneous coordinates (ω0 : · · · : ωn) on Pn(C), we take a reduced representation f = (f0 : · · · : fn), which means that each fi is a holomorphic function on Cm and f (z) = f0(z) : · · · :

fn(z)
outside the analytic set {f0 = · · · = fn = 0} of codimension at least 2 Set

kf k = |f0|2+ · · · + |fn|2
1/2

The characteristic function of f is defined by

T (r, f ) =

Z

S(r)

logkf kσn−

Z

S(1)

logkf kσn

By Jensen’s fomula, we see that the above definition of the characteristic function of f does not depend on the choice of its reduced representation

Let a be a meromorphic mapping of Cm into Pn(C)∗ with reduced representation

a = (a0 : · · · : an) Setting (f, a) := a0f0+ · · · + anfn If (f, a) 6≡ 0, then we define

mf,a(r) =

Z

S(r)

log||f || · ||a||

|(f, a)| σm−

Z

S(1)

log||f || · ||a||

|(f, a)| σm,

where kak = |a0|2+ · · · + |an|2
1/2

The first main theorem for moving hyperplanes in value distribution theory (see [5]) states

T (r, f ) + T (r, a) = mf,a(r) + N(f,a)(r)

Let ϕ be a nonzero meromorphic function on Cm, which are occasionally regarded as

a meromorphic map into P1(C) The proximity function of ϕ is defined by

m(r, ϕ) :=

Z

S(r)

log max (|ϕ|, 1)σm

(c) 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 REdr < ∞

The following play essential roles in Nevanlinna theory

Theorem 1.1 ([9, Theorem 2.1] and [11, Theorem 2]) Let f = (f0 : · · · : fn) be a reduced representation of a meromorphic mapping f of Cm into Pn(C) Assume that

fn+1 is a holomorphic function with f0 + · · · + fn + fn+1 = 0 If P

i∈Ifi 6= 0 for all

6MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES

I ( {0, , n + 1}, then

|| T (r, f ) ≤

n+1

X

i=0

Nf[n]

i (r) + o(T (r, f ))

Theorem 1.2 ([5] and [3, Theorem 5.5]) Let f be a nonzero meromorphic function on

Cm Then

m



r,Dα(f ) f



= O(log+T (r, f )) (α ∈ Zm+)

Theorem 1.3 ([5, Theorem 5.2.29]) Let f be a nonzero meromorphic function on Cm with a reduced representation f = (f0 : · · · : fn) Suppose that fk 6≡ 0 Then

T (r,fj

fk) ≤ T (r, f ) ≤

n

X

j=0

T (r,fj

fk) + O(1).

(d) We recall the following notion of the rational function of weight ≤ d in logarithmic derivatives of some function due to Fujimoto [3] as follows A polynimial Q( , Xα

j, ) in variables , Xα

j, , where j = 1, 2, and α = (α1, , αn) with nonnegative integers

αl, is said to be of weight ≤ d if

˜ Q(t1, t2, ) := Q( , t|α|j , )

is of degeree ≤ d as a polynomial in t1, t2,

Let h1, h2, , hpbe finitely many nonzero meromorphic functions on Cm By a rational function of weight ≤ d in logarithmic derivatives of hj’s, we mean a nonzero meromorphic function ϕ on Cm which is represented as

ϕ = P (· · · , D

αhj/hj, · · · ) Q(· · · , Dαhj/hj, · · · ) with polynomials P (· · · , Xα, · · · ) and Q(· · · , Xα, · · · ) in variables , Xjα, of weight

≤ d

Proposition 1.4 ([3, Proposition 3.4]) Let h1, h2, , hp (p ≥ 2) be nonzero mero-morphic functions on Cm with h1+ h2+ · · · + hp = 0 Then, the set {1, , p} of indices has a partition

{1, , p} = J1∪ J2∪ · · · ∪ Jk, ]Jα ≥ 2 for all α, Jα∩ Jβ = ∅ for α 6= β

such that, for each α,

(i) P

i∈J αhi = 0,

(ii) hi0/hi (i, i0 ∈ Jα) are rational functions in logarithmic derivatives of hj’s with weight

≤ D(p), where D(p) is a constant depending only on p

(e) Let I = {1, , q} For 1 ≤ s ≤ q, we set

Iq,s:= {(i1, , is); 1 ≤ i1 < i2 < · · · < is ≤ q}

Definition 1.5 A relation∼ on IR q,sis said to be a pre-quivalence relation if it satisfies; (i) I ∼ I for all I ∈ IR q,s,

(ii) I ∼ J, then JR ∼ I.R

We now consider a pre-quivalence relation ∼ on IR q,s For I = (i1, , is) and J = (j1, , js) in Iq,s, we set

RI,J = δi1 + · · · + δis − δj1 − · · · − δjs ∈ Zq, where δi := (0, , 0,i−th1 , 0 , 0) ∈ Zq (1 ≤ i ≤ q) We denote by R the submodule of

Zq generated by all elements RI,J with I ∼ J.R

Definition 1.6 (see [3, Definition 2.2]) For two elements I and J in Iq,s, by the notation I ∼ J we mean that there exists a positive integer m such that mRI,J ∈ R Proposition 1.7 (see [3, Proposition 2.4]) There are q real numbers p1, p2, , pq

satisfying the following conditions;

(i) for i = (i1, , is), J = (j1, , js) ∈ Iq,s, pi1 + · · · + pis = pj1 + · · · + pjs if and only if I ∼ J ,

(ii) for 1 ≤ i < j ≤ q, pi = pj if and only if there is a nonzero integer m0 such that

(0, , 0,i−thm0, 0, , 0,−mj−th0, 0, , 0) ∈ R

Proposition 1.8 (see [3, Proposition 2.7]) Take real numbers p1, p2, , pq satisfying the conditions of Proposition 1.7 and q elements g1, , gq in a torsion free abelian group

G If pi = pj for some i, j with 1 ≤ i < j ≤ q, then there are some positive integer m0 and I1, J1, , Ik0, Jk0 ∈ Iq,s with Il ∼ JR l (1 ≤ l ≤ k0) such that

(gi/gj)m0 =

k 0 Y

l=1

GIl/GJl,

where GI := gi 1· · · gi s for I = (i1, , is) ∈ Iq,s, and the number k0 is taken so as to be bounded by a constant k(q) which depends only on q

2 Proof of Main Theorem

In order to prove the main theorem, we need the following algebraic propositions Let H1, , H2n+1 be (2n + 1) hyperplanes of Pn(C) in general position given by

Hi : xi0ω0+ xi1ω1+ · · · + xinωn= 0 (1 ≤ i ≤ 2n + 1)

We consider the rational map Φ : Pn(C) × Pn(C) → P2n(C) as follows:

For v = (v0 : v1· · · : vn), w = (w0 : w1 : · · · : wn) ∈ Pn(C), we define the value Φ(v, w) = (u1 : · · · : u2n+1) ∈ P2n(C) by

ui = xi0v0+ xi1v1+ · · · + xinvn

xi0w0+ xi1w1+ · · · + xinwn.

8MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES

Proposition 2.1 (see [3, Proposition 5.9]) The map Φ is a birational map of Pn(C) ×

Pn(C) to P2n(C)

Now let b1, , b2n+1 be (2n + 1) moving hyperplanes of Pn(C) in general position with reduced representations

bi = (bi0: bi1 : · · · : bin) (1 ≤ i ≤ 2n + 1)

Let f and g be two meromorphic mappings of Cminto Pn(C) with reduced representations

f = (f0 : · · · : fn) and g = (g0 : · · · : gn)

Define hi = (f, bi)/(g, bi) (1 ≤ i ≤ 2n + 1) and hI = Q

i∈Ihi for each subset I of {1, , 2n + 1} Set I = {I = (i1, , in) ; 1 ≤ i1 < · · · < in ≤ 2n + 1} Let R{bi}2n+1

i=1

be the smallest subfield of M which contains C and {bil/bis; bis 6≡ 0, 1 ≤ i ≤ 2n + 1, 0 ≤

l, s ≤ n} We have the following proposition

Proposition 2.2 If there exist functions AI ∈ R{bi}2n+1i=1 (I ∈ I), not all zero, such that

X

I∈I

AIhI ≡ 0, then the map f × g into Pn(C) × Pn(C) is algebraically degenerate over R{bi}2n+1

i=1 Proof By changing the homogeneous coordinates of Pn(C), we may assume that bi0 6≡

0 (1 ≤ i ≤ 2n + 1) Since b1, , b2n+1 are in general position, we have det(bijk)0≤j,k≤n 6≡ 0 for 1 ≤ i0 < · · · in ≤ 2n + 1 Therefore, the set

I∈I

{z ∈ Cm ; AI(z) = 0}
∪ [

1≤i 0 <···i n ≤2n+1

{z ∈ Cm ; det(bijk(z))0≤j,k≤n = 0}

is a proper analytic subset of Cm Take z0 6∈ S and set xij = bij(z0)

For v = (v0 : v1· · · : vn), w = (w0 : w1 : · · · : wn) ∈ Pn(C), we define the map Φ(v, w) = (u1 : · · · : u2n+1) ∈ P2n(C) as above By Proposition 2.1, Φ is a birational function This implies that the function

X

I∈I

AI(z0)Y

i∈I

Pn j=0bij(z0)vj

Pn j=0bij(z0)wj

is a nonzero rational function It follows that

Q(z)(v0, , vn, w0, , wn) = 1

Q2n+1 i=1 bi0

X

I∈I

AI(z) Y

i∈I

n

X

j=0

bij(z)vj

!

i∈I c

n

X

j=0

bij(z)wj

!

is a nonzero polynomial with coefficients in R{bi}2n+1

i=1 , where Ic = {1, , 2n + 1} \ I

By the assumption of the proposition, it is clear that

Q(z)(f0(z), , fn(z), g0(z), , gn(z)) ≡ 0

Hence f × g is algebraically degenerate over R{bi}2n+1

Proposition 2.3 Let f be a meromorphic mapping of Cminto Pn(C) and let b1, , bn+1

be moving hyperplanes of Pn(C) in general position with reduced representations

f = (f0 : · · · : fn), bi = (bi0 : · · · : bin) (1 ≤ i ≤ n + 1)

Then for each regular point z0 of the analytic subset Sn+1

i=1{z ; (f, bi)(z) = 0} with z0 6∈ I(f ), we have

min

1≤i≤n+1ν(f,b0 i)(z0) ≤ νdet Φ0 (z0), where I(f ) denotes the indeterminacy set of f and Φ is the matrix (bij; 1 ≤ i ≤ n + 1, 0 ≤

j ≤ n)

Proof Since z0 6∈ I(f ), we may assume that f0(z0) 6= 0 We consider the system of equations

bi0f0+ · · · + binfn = (f, bi) (1 ≤ i ≤ n + 1)

Solving these equations, we obtain

f0 = det Φ

0

det Φ, where Φ0 is the matrix obtained from Φ by replacing the first column of Φ by ((f, bi))1≤i≤2n+1 Therefore, we have

νdet Φ0 (z0) = νdet Φ0 0(z0) ≥ min

1≤i≤n+1ν(f,b0

i )(z0)

Proof of Main Theorem We choose l0 := 2n3{(n+1)q2(q −1)(q −2)+d(n)(2n+2)} with q = 2n+2n+1
, d(n) = k(2n + 2) · D(q), where D(q) and k(2n + 2) are constants given

in Propositions 1.4 and 1.8 respectively, which depend only on n

Assume that f, g, ai have reduced representations

f = (f0 : · · · : fn), g = (g0 : · · · : gn), ai = (ai0 : · · · : ain)

Without loss of generality, we may assume that f and g are linearly non-degenerate over R{ai}2n+2i=1 , otherwise the map f ×g will be algebraically degenerate over R{ai}2n+2i=1 Then

by Theorem 1.1, we have

2n + 2

n + 2 T (r, f ) ≤

2n+2

X

i=1

N(f,a[n]

i )(r) + o(T (r, f ))

≤ n ·

2n+2

X

i=1

N(g,a[1]

i )(r) + o(T (r, f ))

≤ n(2n + 2)(T (r, g)) + o(T (r, f ))

Then we have || T (r, f ) = O(T (r, g)) Similarly, we also have || T (r, g) = O(T (r, f ))

10 MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES

By changing the homogeneous coordinates of Pn(C) if necessary, we may assume that

ai0 6≡ 0 for all 1 ≤ i ≤ 2n + 2 We set ˜aij = aij/ai0,

˜i = (ai0

ai0,

ai1

ai0, ,

ain

ai0), (f, ˜ai) =

n

X

j=0

˜ijfj and (g, ˜ai) =

n

X

j=0

˜ijgj

We suppose contrarily that the map f ×g is algebraically non-degenerate over R{ai}2n+2

i=1 Define hi = (f, ˜ai)/(g, ˜ai) (1 ≤ i ≤ 2n+2) Then hi/hj = (f, ˜ai) · (g, ˜aj)
/ (g, ˜ai) · (f, ˜aj)
does not depend on the choice of representations of f and g Since Pn

k=0˜ikfk− hi ·

Pn

k=0˜ikgk = 0 (1 ≤ i ≤ 2n + 2), it implies that

Φ := det (˜ai0, , ˜ain, ˜ai0hi, , ˜ainhi; 1 ≤ i ≤ 2n + 2) ≡ 0

(2.1)

For each subset I ⊂ {1, 2, , 2n + 2}, put hI = Q

i∈Ihi, ˜hI = Q

i∈I

h i

h 1 Denote by I the set

I = {(i1, , in+1) ; 1 ≤ i1 < · · · < in+1 ≤ 2n + 2}

For each I = (i1, , in+1) ∈ I, define

AI = (−1)(n+1)(n+2)/2+i1 +···+i n+1× det(˜ai r l; 1 ≤ r ≤ n + 1, 0 ≤ l ≤ n)

× det(˜ajsl; 1 ≤ s ≤ n + 1, 0 ≤ l ≤ n), where J = (j1, , jn+1) ∈ I such that I ∪ J = {1, 2, , 2n + 2}

We define the following:

Let M be the field of holomorphic function on Cm and let R be the field of all mero-morphic functions on Cm which is small with respect to f We define a pre-equivalence relation on I as follows: I = (i1, , in+1)∼ J = (jR 1, , js) if ˜hI/˜hJ is a rational function

in logarithmic derivatives of (hi/h1)’s with weight ≤ D( 2n+2n+1
) and with coefficients in R

Take real numbers p1, , p2n+2 satisfying the condition of Proposition 1.7 Without loss of generality, we may assume that

p1 ≤ p2 ≤ · · · ≤ p2n+2 Now the equality (2.1) implies that

X

I∈I

AIhI = 0

For each I = (i1, in+1) ∈ I, we set ˜hI =Qn+1

j=1(hij/h1) The above equality yields that X

I∈I

AIh˜I = 0

Applying Proposition 1.4 to meromorphic mappings AI˜hI (I ∈ I), we have a partition

I = I1 ∪ · · · ∪ Ik with Iα 6= ∅ and Iα∩ Iβ = ∅ for α 6= β such that, for each α,

(i) P

I∈I AI˜hI ≡ 0,

8MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES< /small>

Proposition 2.1 (see [3, Proposition 5.9]) The map Φ is a birational map of Pn(C)... class="page_container" data-page="10">

10 MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES< /small>

By changing the homogeneous coordinates of Pn(C) if necessary,... and the number k0 is taken so as to be bounded by a constant k(q) which depends only on q

2 Proof of Main Theorem

In order to prove the main theorem, we need the