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
MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE 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 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. 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,Hi ) = 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 representations ai = (ai0 : · · · : ain ) (1 ≤ i ≤ q). We say that a1 , . . . , aq are located in general position if det(aik l ) ≡ 0 for any 1 ≤ i0 < i1 < · · · < in ≤ q. We denote by M the field of all meromorphic functions on Cm and R{ai }qi=1 the smallest subfield of M which contains C and all ajk /ajl with ajl ≡ 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 ) = aI (z)ω I , I∈Id where d is an integer, Id = {(i0 , . . . , iN ) ; 0 ≤ ij ≤ d, iN ω I = ω0i0 · · · ωN 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 ) ≡ 0 on V . N j=0 ij = d}, aI ∈ R and MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES3 Now let f and g be two meromorphic mappings of Cm into Pn (C) with representations f = (f0 : · · · fn ) and g = (g0 : · · · : gn ). 2 We consider Pn (C) × Pn (C) as a projective subvariety of P(n+1) −1 (C) by Segre embedding. 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 aIJ (z)ω I ω J , Q(z)(ω0 , . . . , ωn , ω0 , . . . , ωn ) = J=(j0 ,...,jn )∈Zn+1 I=(i0 ,...,in )∈Zn+1 + + i0 +···+in =d j0 +···+jn =d where d, d 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 2n+2 Pn (C) × Pn (C) is algebraically degenerate over R{ai }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 institute 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 B(r) := {z ∈ Cm ; ||z|| < r}, 1/2 for z = (z1 , . . . , zm ) ∈ Cm and define S(r) := {z ∈ Cm ; ||z|| = r} (0 < r < ∞). Define vm−1 (z) := ddc ||z||2 m−1 σm (z) := dc log||z||2 ∧ ddc log||z||2 m−1 and on 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 . ∂ α1 z1 · · · ∂ αm zm Dα F = We denote by νF0 (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 = νµ Xµ , where {Xµ } is a locally m finite family of analytic hypersurfaces in C and νµ ∈ Z. We may assume that Xµ are irreducible and distinct to each other, and νµ = 0 for all µ. Then the set Supp(ν) = µ Xµ 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 (ω 1 , . . . , ω m ) and two holomorphic functions f and g on U such that νf0 − νg0 = ν|U . We define ν(p) : = max{d; Dα f (p) = 0 for all α with |α| < d} − max{d; Dα g(p) = 0 for all α with |α| < d}, m i=1 where α = (α1 , . . . , αm ) ∈ Zm + , |α| = αi and ∂ |α| ϕ ∂ α1 ω 1 · · · ∂ αm ω m 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 Dα ϕ = ν [M ] (z) = min {M, ν(z)}, ν [M ] (z) if ν(z) > k, [M ] ν>k (z) = 0 if ν(z) ≤ k, ν(z)vm−1 if m ≥ 2, |ν| ∩B(t) n(t) = ν(z) if m = 1. |z|≤t [M ] Similarly, we define n[M ] (t) and n>k (t). Define r N (r, ν) = 1 n(t) dt (1 < r < ∞). t2m−1 MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES5 [M ] [M ] Similarly, we define N (r, ν [M ] ), N (r, ν>k ) and denote them by N [M ] (r, ν), N>k (r, ν), respectively. Let ϕ : Cm → C be a meromorphic function. Define [M ] [M ] Nϕ (r) = N (r, νϕ0 ), Nϕ[M ] (r) = N [M ] (r, νϕ0 ), Nϕ,>k (r) = N>k (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 1/2 f = |f0 |2 + · · · + |fn |2 . The characteristic function of f is defined by log f σn − T (r, f ) = log f σn . S(1) S(r) 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) := a0 f0 + · · · + an fn . If (f, a) ≡ 0, then we define mf,a (r) = log ||f || · ||a|| σm − |(f, a)| S(r) log ||f || · ||a|| σm , |(f, a)| S(1) 1/2 where a = |a0 |2 + · · · + |an |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) := log max (|ϕ|, 1)σm . S(r) (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 E dr < ∞. 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 i∈I fi = 0 for all 6MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES {0, . . . , n + 1}, then I n+1 [n] || T (r, f ) ≤ Nfi (r) + o(T (r, f )). i=0 Theorem 1.2 ([5] and [3, Theorem 5.5]). Let f be a nonzero meromorphic function on C . Then Dα (f ) = O(log+ T (r, f )) (α ∈ Zm m r, + ). f m 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 ≡ 0. Then T (r, fj ) ≤ T (r, f ) ≤ fk n T (r, j=0 fj ) + O(1). fk (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(. . . , Xjα , . . .) in variables . . . , Xjα , . . . , where j = 1, 2, . . . and α = (α1 , . . . , αn ) with nonnegative integers αl , is said to be of weight ≤ d if ˜ 1 , t2 , . . .) := Q(. . . , t|α| , . . .) Q(t j is of degeree ≤ d as a polynomial in t1 , t2 , . . .. Let h1 , h2 , . . . , hp be 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 (· · · , Dα hj /hj , · · · ) Q(· · · , Dα hj /hj , · · · ) with polynomials P (· · · , X α , · · · ) and Q(· · · , X α , · · · ) in variables . . . , Xjα , . . . of weight ≤ d. Proposition 1.4 ([3, Proposition 3.4]). Let h1 , h2 , . . . , hp (p ≥ 2) be nonzero meromorphic 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 = β such that, for each α, (i) i∈Jα hi = 0, (ii) hi /hi (i, i ∈ 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}. MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES7 R Definition 1.5. A relation ∼ on Iq,s is said to be a pre-quivalence relation if it satisfies; R (i) I ∼ I for all I ∈ Iq,s , R R (ii) I ∼ J, then J ∼ I. R We now consider a pre-quivalence relation ∼ on Iq,s . For I = (i1 , . . . , is ) and J = (j1 , . . . , js ) in Iq,s , we set RI,J = δi1 + · · · + δis − δj1 − · · · − δjs ∈ Zq , i−th where δi := (0, . . . , 0, 1 , 0 . . . , 0) ∈ Zq (1 ≤ i ≤ q). We denote by R the submodule of R Zq generated by all elements RI,J with I ∼ J. 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 j−th i−th (0, . . . , 0, m0 , 0, . . . , 0, −m0 , 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 R and I1 , J1 , . . . , Ik0 , Jk0 ∈ Iq,s with Il ∼ Jl (1 ≤ l ≤ k0 ) such that k0 m0 (gi /gj ) = GIl /GJl , l=1 where GI := gi1 · · · gis 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 : xi0 ω0 + xi1 ω1 + · · · + xin ωn = 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 xi0 v0 + xi1 v1 + · · · + xin vn . ui = xi0 w0 + xi1 w1 + · · · + xin wn 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 Cm into Pn (C) with reduced representations f = (f0 : · · · : fn ) and g = (g0 : · · · : gn ). Define hi = (f, bi )/(g, bi ) (1 ≤ i ≤ 2n + 1) and hI = i∈I hi for each subset I of 2n+1 {1, . . . , 2n + 1}. Set I = {I = (i1 , . . . , in ) ; 1 ≤ i1 < · · · < in ≤ 2n + 1}. Let R{bi }i=1 be the smallest subfield of M which contains C and {bil /bis ; bis ≡ 0, 1 ≤ i ≤ 2n + 1, 0 ≤ l, s ≤ n}. We have the following proposition. 2n+1 (I ∈ I), not all zero, such Proposition 2.2. If there exist functions AI ∈ R{bi }i=1 that AI hI ≡ 0, I∈I then the map f × g into P (C) × P (C) is algebraically degenerate over R{bi }2n+1 i=1 . n n Proof. By changing the homogeneous coordinates of Pn (C), we may assume that bi0 ≡ 0 (1 ≤ i ≤ 2n + 1). Since b1 , . . . , b2n+1 are in general position, we have det(bij k )0≤j,k≤n ≡ 0 for 1 ≤ i0 < · · · in ≤ 2n + 1. Therefore, the set {z ∈ Cm ; AI (z) = 0} ∪ S= {z ∈ Cm ; det(bij k (z))0≤j,k≤n = 0} 1≤i0 i0 , then pi0 < pi0 +1 pji0 . This is a contradiction. Therefore ji0 = i0 , and hence j1 = 1, . . . , ji0 −1 = i0 − 1. We conclude that J = (1, . . . , i0 , ji0 +1 , . . . , jn+1 ) and i0 ≤ n + 1 for each J ∈ I1 . By (2.1), we have ˜ I\{i } ≡ 0. ˜ I = hi0 AI h AI h 0 h 1 I∈I I∈I 1 1 Thus ˜ I\{i } ≡ 0. AI h 0 I∈I1 It implies that AI hI\{i0 } ≡ 0. I∈I1 Then Proposition 2.2 shows that f × g is algebraically degenerate over R{ai }2n+2 i=1 . It contradicts to the supposition. Now the proof is reduced to the case where p1 = · · · = p2n+2 . We consider the torsion free Abelian group M of all meromorphic functions on C and consider meromorphic function (hi /h1 ), 1 ≤ i ≤ 2n + 2. Then by Lemma 1.8, there exists a positive constants m0 and kij , where kij ≤ k(2n + 2), such that hi /h1 m0 ) = ( hj /h1 kij l=1 ˜I h l , ˜hJ l R where Il , Jl ∈ I2n+2,n+1 and Il ∼ Jl . This implies that, for every 1 ≤ i, j ≤ 2n + 2, hi /h1 m0 ( ) is a rational funtion in logarithmic derivatives of (hs /h1 )’s with coefficients in hj /h1 R with weight bounded by d(n). Assume that ( hi /h1 m0 Pij (. . . , Dα (hs /h1 ), . . .) ) = , hj /h1 Qij (. . . , Dα (hs /h1 ), . . .) 12 MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES where Pij and Qij are polynomials in logarithmic derivatives of (hs /h1 )’s with coefficients in R and with weight bounded by a constant d(n). It is easy to see that 2n+2 νQ∞ij (z) d(n) min{1, νh∞s /h1 (z) + νh0s /h1 (z)} + Sij (z) ≤ i=1 2n+2 min{1, νh∞s (z) + νh0s (z)} + Sij (z), ≤ d(n) i=1 where Sij (z) is the sum of all pole orders of all coefficients of Qij at the point z. Hence, if we regard Sij as a divisor on Cm , then || N (r, Sij ) = o(T (r, g)). Thus 2n+2 || N (r, νQ∞ij ) (N [1] (r, νh∞s ) + N [1] (r, νh0s )) + o(T (r, g)) ≤ d(n) i=1 2n+2 [1] N(g,ai ),>l0 (r) ≤ d(n) i=1 ≤ d(n) ≤ l0 + 1 2n+2 N(g,ai ) (r) + o(T (r, g)) i=1 d(n)(2n + 2) T (r, g) + o(T (r, g)). l0 + 1 Therefore, by the lemma on logarithmic derivatives (Theorem 1.2), we have hi 1 m r, = m r, hj m0 (2.2) hi /h1 hj /h1 ≤ m(r, Pij ) + m(r, m0 +O(1) 1 ) ≤ m(r, Pij ) + T (r, Qij ) + O(1) Qij = m(r, Pij ) + m(r, Qij ) + N (r, νQ∞ij ) + O(1) ≤ d(n)(2n + 2) T (r, g) + o(T (r, g)). l0 + 1 Claim 2.4. For each i0 ∈ {1, . . . , 2n + 2}, we have q 2 (q − 1)(q − 2)(n + 1) || Nhi0 (r) ≤ T (r, g) + o(T (r, g)) 2(l0 + 1) q 2 (q − 1)(q − 2)(n + 1) and || N1/hi0 (r) ≤ T (r, g) + o(T (r, g)). 2(l0 + 1) For the proof of this claim, we fix an index α. If Iα = 2, it is clear that we have a 2n+2 nontrivial algebraic relation (over R{ai }i=1 ) among f0 , . . . , fn , g0 , . . . , gn . It follows that 2n+2 f × g is algebraically degenerate over R{ai }i=1 . This is a contradiction. Then Iα > 2. Assume that Iα = {I0 , . . . , It+1 }, t ≥ 1. We consider the meromorphic mapping Fα of Cm into Pt (C) with the reduced representation Fα = (uAI0 hI0 : · · · : uAIt hIt ), where u is a meromorphic function. Then we see that each zero point of uAIi hIi (0 ≤ i ≤ t + 1) must be either zero or pole of some AIj hIj (0 ≤ j ≤ t). Then by Theorem 1.1, we MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES 13 have t+1 [t] || T (r, Fα ) ≤ NuAI (r) + o(T (r, Fα )) h i Ii i=0 t+1 [t] ≤ (NAI [t] h i Ii (r) + N1/AI h i Ii (r)) + o(T (r, Fα )) i=0 t+1 [t] ≤ [t] (NhI (r) + N1/hI (r)) + o(T (r, g)) i i i=0 t+1 [1] ≤t [1] (NhI (r) + N1/hI (r)) + o(T (r, g)) i i i=0 [t] ≤t [t] (NhI (r) + N1/hI (r)) + o(T (r, g)) I∈I [1] ≤t (2.3) [1] (Nhi (r) + N1/hi (r)) + o(T (r, g)) I∈I i∈I = ≤ tq 2 tq 2 2n+2 [1] [1] (Nhi (r) + N1/hi (r)) + o(T (r, g)) i=1 2n+2 [1] N(g,ai ),>l0 (r) + o(T (r, g)) i=1 tq ≤ 2(l0 + 1) 2n+2 N(g,ai ),>l0 (r) + o(T (r, g)) i=1 tq(2n + 2) T (r, g) + o(T (r, g)) ≤ 2(l0 + 1) q(q − 2)(n + 1) ≤ T (r, g) + o(T (r, g)). l0 + 1 Take a regular point z0 of the analytic subset 2n+2 {z ; (f, ai ) = 0} with z0 ∈ I(f ) ∪ I(g). i=1 We may assume that νh01 (z0 ) ≥ νh02 (z0 ) ≥ · · · ≥ νh02n+2 (z0 ). Set I = (1, . . . , n + 1). Then I ∈ Iα with an index α, 1 ≤ α ≤ k. It also is easy to see that if 1 ∈ I for all I ∈ Iα , then AI hI \{1} ≡ 0. I ∈Iα Therefore, Proposition 2.2 shows that f × g is algebraically degenerate (over R{ai }2n+2 i=1 ). This is a contradiction. Hence, there exists I ∈ Iα such that 1 ∈ I . Assume that I = (j1 , . . . , jn+1 ) with 1 < j1 < · · · < jn+1 . By Proposition 2.3, it yields that νh0j (z0 ) ≤ n+1 14 MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES 0 (z0 ). Then we have νdet(a jl ;j∈I ,0≤l≤n) νh0i0 (z0 ) ≤ νh01 (z0 ) ≤ νA0 I hI /AI ≤ νA0 I hI /AI hI hI (z0 ) + νA0 I (z0 ) + νA0 I /Ai (z0 ) + νh0jn+1 (z0 ) 0 + νdet(a (z0 ). jl ;j∈I ,0≤l≤n) /AI (z0 ) Thus we have k νh0i0 (z0 ) (νA0 I hI /AI ≤ hI (z0 ) + νA0 I /AI (z0 )) α=1 I,I ∈Iα 0 νdet(a (z0 ). j l ;1≤i≤n+1,0≤l≤n) + i 1≤j1 [...]... We complete the proof of Claim 2.4 We now continue the proof of Main Theorem By changing the homogeneous coordinates of Pn (C) if necessary, we may assume that ai0 ≡ 0 for all 1 ≤ i ≤ 2n + 2 Set n n aij a ˜ij = , a ˜i = (˜ ai0 , , a ˜in ), (f, a ˜i ) = a ˜ij fj and (g, a ˜i ) = a ˜ij gj ai0 j=0 j=0 MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES 15 2n+2 Then there exist.. .MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES 11 (ii) ˜ AI h I ˜I AI h ˜ J )’s (I, I ∈ Iα ) are rational functions in logarithmic derivatives of (AJ h with weight ≤ D( 2n+2 n+1 ) Moreover, we may assume that Iα is minimal, i.e., there is no proper subset Jα Iα with ˜ I∈Jα AI hI ≡ 0 We distinguish the following two cases: First we prove the theorem in the case where there... variables sharing small identical sets for moving target, Internat J Math 21 (2010), 1095–1120 [7] S D Quang and L N Quynh, Two meromorphic mappings sharing 2n + 2 hyperplanes regardless of multiplicity, J Math Anal Appl 410 (2014), 771–782 MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES 17 [8] M Ru, A uniqueness theorem with moving targets without counting multiplicity, Proc Amer... The theorem is proved References [1] G Dethloff and T V Tan, Uniqueness problem for meromorphic mappings with truncated multiplicities and moving targets, Nagoya J Math 181 (2006), 75–101 [2] H Fujimoto, The uniqueness problem of meromorphic maps into the complex projective space, Nagoya Math J 58 (1975), 1–23 [3] H Fujimoto, Uniqueness problem with truncated multiplicities in value distribution theory,... Pt (C) with the reduced representation Fα = (uAI0 hI0 : · · · : uAIt hIt ), where u is a meromorphic function Then we see that each zero point of uAIi hIi (0 ≤ i ≤ t + 1) must be either zero or pole of some AIj hIj (0 ≤ j ≤ t) Then by Theorem 1.1, we MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES 13 have t+1 [t] || T (r, Fα ) ≤ NuAI (r) + o(T (r, Fα )) h i Ii i=0 t+1 [t] ≤... in logarithmic derivatives of (hs /h1 )’s with coefficients in hj /h1 R with weight bounded by d(n) Assume that ( hi /h1 m0 Pij ( , Dα (hs /h1 ), ) ) = , hj /h1 Qij ( , Dα (hs /h1 ), ) 12 MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES where Pij and Qij are polynomials in logarithmic derivatives of (hs /h1 )’s with coefficients in R and with weight bounded by a constant... equations in unkown variables g0 /gn , , gn−1 /gn and solve these to obtain that gi /gn (0 ≤ i ≤ n − 1) has the form gi Pi = , gn Qi 16 MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES where Pi and Qi are homogeneous polynomials in hj /h1 (1 ≤ j ≤ 2n + 1) of degree n with coefficients in R{aj }2n+2 j=1 Then by Theorem 1.2 we have n−1 (2.4) T (r, g) ≤ i=0 gi T (r, ) = gn n−1... Wang, Truncated second main theorem with moving targets, Trans Amer Math Soc 356 (2004), 557–571 [10] D D Thai and S D Quang, Uniqueness problem with truncated multiplicities of meromorphic mappings in several complex variables for moving target, Internat J Math 16 (2005), 903–939 [11] D D Thai and S D Quang, Second main theorem with truncated counting function in several complex variables for moving. .. is algebraically degenerate (over R{ai }2n+2 i=1 ) This is a contradiction Hence, there exists I ∈ Iα such that 1 ∈ I Assume that I = (j1 , , jn+1 ) with 1 < j1 < · · · < jn+1 By Proposition 2.3, it yields that νh0j (z0 ) ≤ n+1 14 MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES 0 (z0 ) Then we have νdet(a jl ;j∈I ,0≤l≤n) νh0i0 (z0 ) ≤ νh01 (z0 ) ≤ νA0 I hI /AI ≤ νA0 I... S D Quang, Uniqueness theorems for meromorphic mappings sharing few hyperplanes, J Math Anal Appl 393 (2012), 445–456 [5] J Noguchi and T Ochiai, Geometric function theory in several complex variables, Transl Math Monogr 80, American Mathematical Society, Providence, RI, 1990 [6] H H Pham and S D Quang and D T Do, Unicity theorems with truncated multiplicities of meromorphic mappings in several complex ... 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... i∈I c MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES9 Proposition 2.3 Let f be a meromorphic mapping of Cm into Pn (C) and let b1 , , bn+1 be moving hyperplanes of. .. + hyperplanes regardless of multiplicity, J Math Anal Appl 410 (2014), 771–782 MEROMORPHIC MAPPINGS HAVING THE SAME INVERSE IMAGES OF MOVING HYPERPLANES 17 [8] M Ru, A uniqueness theorem with