Second main theorems and uniqueness problem of meromorphic mappings with moving hypersurfaces

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1 Bull Math Soc Sc Math Roumane Tome 5705 No 3, 04, Secon man theorems an unqueness problem of meromorphc mappngs wth movng hypersurfaces by S Duc Quang Abstract In ths artcle, we establsh some new secon man theorems for meromorphc mappngs of C m nto P n C an movng hypersurfaces wth truncate countng functons A unqueness theorem for these mappngs sharng few movng hypersurfaces wthout countng multplcty s also gven Ths result s an mprovement of the recent result of Dethloff - Tan [3] Moreover the meromorphc mappngs n our result may be algebracally egenerate The last purpose of ths artcle s to stuy unqueness problem n the case where the meromorphc mappngs agree on small entcal sets Key Wors: Secon man theorem, meromorphc mappng, movng hypersurface, unqueness problem, truncate multplcty 00 Mathematcs Subject Classfcaton: Prmary 3H30, Seconary 30D35, 3A Introucton In 004, Mn Ru [7] showe a secon man theorem for algebracally nonegenerate meromorphc mappngs an a famly of hypersurfaces n weakly general poston After that, wth the same assumptons, T T H An an H T Phuong [] mprove the result of Mn Ru by gvng an explct truncaton level for countng functons Recently, n [] Dethloff an Tan generalze an mprove the secon man theorems of Mn Ru an An - Phuong to the case of movng hypersurfaces They prove that Theorem A Dethloff - Tan [] Let f be a nonconstant meromorphc map of C m nto P n C Let {Q } q be a set of slow wth respect to f movng hypersurfaces n weakly general poston wth eg Q j = j q Assume that f s algebracally nonegenerate over K {Q} q Then for any ɛ > 0 there exst postve ntegers L j j =,, q, epenng only on n, ɛ an j j =,, q n an explct way such that q n ɛt f r N [Lj] Q f r + ot f r

2 80 S Duc Quang Here, the truncaton level L j s estmate by L j j n+m n tp0+ j +, where s the least common multple of the j s, = lcm,, q, an M = [n + n n + ɛ + n + ], p 0 =[ n+m n q n log n+m n q n ɛ log + + ], n+m n M n n+m n n q + M q an t p0+ < + p 0, n n where [x] = max{k Z ; k x} for a real number x By usng ths secon man theorem, Dethloff an Tan prove a unqueness theorem for meromorphc mappngs whch share slow movng hypersurfaces as follows Let f, g : C m P n C be two meromorphc mappngs Let {Q } q be q movng hypersurfaces of P n C n weakly general poston, eg Q =, an let,, be respectvely the least common multple, the maxmum number an the mnmum number of the j s Take M, p 0 be as above wth ɛ = an set n n+m n + M q t p0+ = + p 0 n n L =[ n+m n tp0+ + ] n q Wth the above notatons, n 0, Dethloff an Tan prove the followng Theorem B Theorem 3 [3] a Assume that f an g are algebracally nonegenerate over K {Qj} such that: D α f k f s = D α g k g s on for all α < p, p Z + an 0 k s n Then for q > n + nl p + 3, we have f g b Assume f an g as a satsfy an q ZeroQ f ZeroQ g, m n ZeroQ j f m 0 < < n q j=0,

3 Secon man theorems an unqueness problem of meromorphc mappngs 8 Then for q > n + L p + 3, we have f g However, the number of movng hypersurfaces n Theorem B s stll bg, snce the truncaton levels gven n Theorem A s far from the sharp We also woul lke to note that, n all mentone results on secon man theorem of Mn Ru, An - Phuong an Dethloff - Tan the algebracally nonegeneracy conton of the meromorphc mappngs can not be remove an t plays an essental role n ther proofs The frst purpose of the present paper s to show some new secon man theorems for meromorphc mappngs sharng slow movng hypersurfaces wth better truncaton levels for countng functons Moreover the mappngs may be algebracally egenerate Namely, we prove the followng theorems Theorem Let f be a meromorphc mappng of C m nto P n C Let Q =,, q be slow wth respect to f movng hypersurfaces of P n C n weakly general poston wth eg Q =, q nn+n+, where N = n+ n an = lcm,, q Assume that Q f 0 q Then we have q nn + n + T f r N [N] Q r + ot f f r Theorem Let f be a meromorphc mappng of C m nto P n C Let Q =,, q be slow wth respect to f movng hypersurfaces of P n C n weakly general poston wth eg Q =, q N +, where N = n+ n an = lcm,, q Assume that f s algebracally nonegenerate over K {Q} q Then we have q N + T f r N [N] Q f r + ot f r The secon purpose of ths paper s to show a unqueness theorem for meromorphc mappngs sharng slow movng hypersurfaces wthout countng multplcty We wll prove the followng Theorem 3 Let f an g be nonconstant meromorphc mappngs of C m nto P n C Let Q =,, q be a set of slow wth respect to f an g movng hypersurfaces n P n C n weakly general poston wth eg Q = Put = lcm,, n+ an N = n+ n Let k k n be an nteger Assume that m k j=0 ZeroQ j f m for every 0 < < k q, f = g on q ZeroQ f ZeroQ j g Then the followng assertons hol: knnn + n + a If q > then f = g b In aton to the assumptons -, we assume further that both f an g are algebracally nonegenerate over K knn + {Q} q If q >, then f = g We note that the numbers of hypersurfaces n our results are really reuce when compare to that n Theorem B of Dethloff - Tan Also by ntroucng some new technques, we smplfy ther proofs

4 8 S Duc Quang We woul lke to emphasze here that n all Theorem 3 an prevous results on the unqueness problem, the meromorphc mappngs always are assume to agree on the nverse mages of all movng hypersurfaces Our last purpose n ths paper s to show an algebrac relaton between meromorphc mappngs n the case where they agree on the nverse mages of only n + movng hypersurfaces Namely, we wll prove the followng Theorem 4 Let f an g be nonconstant meromorphc mappngs of C m nto P n C Let Q =,, q be a set of slow wth respect to f an g movng hypersurfaces n P n C n weakly general poston wth eg Q = Put = lcm,, q, L j = [ j n+m n tp0+ j + ], where M = 4n + n n + + n +, p 0 = [ n+m n q n log n+m n q n + ] log + 4 n+m n M n+m n q Assume that f an g are algebracally none- an t p0+ = n q n+m n n + p0 generate over K {Q} q an m k j=0 ZeroQ j f m for every 0 < < k n +, mn{νq 0 z, L f } = mn{νq 0 z, L g } for every n + 3 q, ZeroQ f ZeroQ j g f = g on n+ L If q n + + kl, where L = max n+ n + 3 < < [ q n ]+ such that Q f Q g = Q f Q g = = Q [ then there exst at least [ q n ] + nces q n f ]+ Q [ q n g ]+ Basc notons an auxlary results from Nevanlnna theory We set z = z + + z m / for z = z,, z m C m an efne Br := {z C m : z < r}, Sr := {z C m : z = r} 0 < r < Defne v m z := c z m an σ m z := c log z c log z m on C m \ {0} Let F be a nonzero holomorphc functon on a oman Ω n C m For a set α = α,, α m of nonnegatve ntegers, we set α = α + + α m an D α α F F = We efne the α z αm z m map ν F : Ω Z by ν F z := max {k : D α F z = 0 for all α wth α < k} z Ω

5 Secon man theorems an unqueness problem of meromorphc mappngs 83 We mean by a vsor on a oman Ω n C m a map ν : Ω Z such that, for each a Ω, there are nonzero holomorphc functons F an G on a connecte neghborhoo U Ω of a such that νz = ν F z ν G z for each z U outse an analytc set of menson m Two vsors are regare as the same f they are entcal outse an analytc set of menson m For a vsor ν on Ω we set ν := {z : νz 0}, whch s ether a purely m -mensonal analytc subset of Ω or an empty set Take a nonzero meromorphc functon ϕ on a oman Ω n C m For each a Ω, we choose nonzero holomorphc functons F an G on a neghborhoo U Ω such that ϕ = F on U an G mf 0 G 0 m, an we efne the vsors νϕ, 0 νϕ by νϕ 0 := ν F, νϕ := ν G, whch are nepenent of choces of F an G an so globally well-efne on Ω 3 For a vsor ν on C m an for a postve nteger M or M =, we efne the countng functon of ν by ν [M] z = mn {M, νz}, Smlarly, we efne Defne νzv m f m, ν Bt nt = νz f m = n [M] t Nr, ν = z t r nt t tm < r < Smlarly, we efne Nr, ν [M] an enote t by N [M] r, ν Let ϕ : C m C be a meromorphc functon Defne N ϕ r = Nr, νϕ, 0 N ϕ [M] r = N [M] r, νϕ 0 For brevty we wll omt the character [M] f M = 4 Let f : C m P n C be a meromorphc mappng For arbtrarly fxe homogeneous coornates w 0 : : w n on P n C, we take a reuce representaton f = f 0 : : f n, whch means that each f s a holomorphc functon on C m an fz = f 0 z : : f n z outse the analytc set {f 0 = = f n = 0} of comenson Set f = f f n / The characterstc functon of f s efne by T f r = log f σ m log f σ m Sr S 5 Let ϕ be a nonzero meromorphc functon on C m, whch are occasonally regare as a meromorphc map nto P C The proxmty functon of ϕ s efne by mr, ϕ := log max ϕ, σ m Sr

6 84 S Duc Quang The Nevanlnna s characterstc functon of ϕ s efne as follows Then T r, ϕ := N r + mr, ϕ ϕ T ϕ r = T r, ϕ + O The functon ϕ s sa to be small wth respect to f f T ϕ r = ot f r Here, by the notaton P we mean the asserton P hols for all r [0, exclung a Borel subset E of the nterval [0, wth r < E We enote by M resp K f the fel of all meromorphc functons resp small meromorphc functons on C m 6 Denote by H C m the rng of all holomorphc functons on C m Let Q be a homogeneous polynomal n H C m[x 0,, x n ] of egree Denote by Qz the homogeneous polynomal over C obtane by substtutng a specfc pont z C m nto the coeffcents of Q We also call a movng hypersurface n P n C each homogeneous polynomal Q H C m[x 0,, x n ] such that the common zero set of all coeffcents of Q has comenson at least two Let Q be a movng hypersurface n P n C of egree gven by Qz = a I ω I, I I where I = { 0,, n N n+ 0 ; n = }, a I H C m an ω I = ω 0 0 ωn n We conser the meromorphc mappng Q : C m P N C, where N = n+ n, gven by Q z = a I0 z : : a IN z I = {I 0,, I N } The movng hypersurfaces Q s sa to be slow wth respect to f f T Q r = ot f r Ths s equvalent to T a I a Ij r = ot f r for every a Ij 0 Let {Q } q be a famly of movng hypersurfaces n Pn C, eg Q = Assume that Q = I I a I ω I We enote by K {Q} q ai the smallest subfel of M whch contans C an all a J wth a J 0 We say that {Q } q are n weakly general poston f there exsts z Cm such that all a I q, I I are holomorphc at z an for any 0 < < n q the system of equatons { Qj zw 0,, w n = 0 0 j n has only the trval soluton w = 0,, 0 n C n+ 7 Let f be a nonconstant meromorphc map of C m nto P n C Denote by C f the set of all non-negatve functons h : C m \ A [0, + ] R, whch are of the form h = g + + g l g l+ + + g l+k,

7 Secon man theorems an unqueness problem of meromorphc mappngs 85 where k, l N, g,, g l+k K f \ {0} an A C m, whch may epen on g,, g l+k, s an analytc subset of comenson at least two Then, for h C f we have log hσ m = ot f r Sr 8 We have some followng theorems Lemma see [] Let {Q } n be a set of homogeneous polynomals of egree n K f [x 0,, x n ] Then there exsts a functon h C f such that, outse an analytc set of C m of comenson at least two, max {0,,n} Q f 0,, f n h f If, moreover, ths set of homogeneous polynomals s n weakly general poston, then there exsts a nonzero functon h C f such that, outse an analytc set of C m of comenson at least two, h f max Q f 0,, f n {0,,n} Lemma Lemma on logarthmc ervatve [8, Lemma 3] Let f be a nonzero meromorphc functon on C m Then m r, Dα f = Olog + T r, f α Z m + f 9 Assume that L s a subset of a vector space V over a fel R We say that the set L s mnmal over R f t s lnearly epenent over R an each proper subset of L s lnearly nepenent over R Repeatng the argument n Prop 45 [4], we have the followng Proposton see [4, Prop 45] Let Φ 0,, Φ k be meromorphc functons on C m such that {Φ 0,, Φ k } are lnearly nepenent over C Then there exsts an amssble set {α = α,, α m } k Z m + wth α = m j= α j k 0 k such that the followng are satsfe: {D α Φ 0,, D α Φ k } k s lnearly nepenent over M, e, et Dα Φ j 0 et D α hφ j = h k+ et D α Φ j for any nonzero meromorphc functon h on C m 3 Secon man theorems for movng hypersurfaces In orer to prove Theorem we nee the followng Lemma 3 Let f be as n Theorem Let {Q } nn+ be a set of homogeneous polynomals n K f [x 0,, x n ] of common egree n weakly general poston, where N = n+ n Assume that Q f 0 0 nn + Then there exst a subset B of {Q f ; 0 nn + } an subsets I,, I k of B such that the followng are satsfe:

8 86 S Duc Quang I s mnmal, I s nepenent over K f k B = k I, I I j = j an B n + For each k, there exst meromorphc functons c α K f \ {0} such that c α Q α f I j Q αf I j= Kf Proof: Denote by Vf the vector space of all homogeneous polynomals of egree n K f [x 0,, x n ] It s seen that m Vf = n+ n = N + We set A 0 = {Q f ; 0 nn + } We are gong to construct the subset B 0 of A 0 as follows: Snce A 0 > N + = m Vf, the set A 0 s lnearly nepenent over K f Therefore, there exsts a mnmal subset I 0 over K f of A 0 If I 0 n + or I 0 A K 0 \ I 0 f = {0} then K f we stop the process an set B 0 = I 0, A = A 0 \ B 0 Otherwse, snce I 0 A K 0 \ I 0 f {0}, we now choose a subset I 0 K f of A 0 \I 0 such that I 0 s the mnmal subset of A 0 \I 0 satsfyng I 0 I 0 K f {0} By the mnmalty, the subset K f I 0 s lnearly nepenent over K f If I 0 I 0 n+ or I 0 I 0 A K 0 \ I 0 f I 0 = {0} K f then we stop the process an set B 0 = I 0 I 0, A = A 0 \ B 0 Otherwse, by repeatng the above argument, we have a subset I3 0 of A 0 \ I 0 I 0 Contnutng ths process, there exst subsets I 0,, Ik 0 such that: I0 s a subset of A 0 \ j= I0 j, I0 j s lnearly nepenent over K f j k, I 0 {0}, B 0 K f j= I0 j n + or B 0 Kf A 0 \ B 0 Kf = {0} Also, by the mnmalty of each subset I 0 k, there exst nonzero meromorphc functons c 0 α K f such that c 0 αq α f Kf Q αf I 0 If B 0 n +, by settng B = B 0, I = I 0 then the proof s fnshe Otherwse, we have B 0 Kf A 0 \ B 0 Kf = {0} We set A = A 0 \ B 0 Then ma Kf N + mb 0 Kf N an A nn + > N ma Kf Smlarly, we construct the subset B of A wth the same propertes as B 0 If B n + then the proof s fnshe Otherwse, by repeatng the same argument we have subsets A 3, B 3 an I 3 Contnutng ths process, we have the followng two cases: Case By ths way, we may construct subsets B,, B N wth B n N We set B N+ = A 0 \ N B Then B N+ nn ++ nn + = Then m B N+ Kf On the other han, t s easy to see that m B N+ Kf = m A 0 Kf j= I 0 j K f N m B Kf N + N + = 0

9 Secon man theorems an unqueness problem of meromorphc mappngs 87 Ths s a contracton Hence ths case s mpossble Case At the step k th k N, we get B k n + Then smlarly as above, the proof s fnshe Lemma 4 Let f be as n Theorem Let {Q } nn+ be a set of homogeneous polynomals n K f [x 0,, x n ] of common egree n weakly general poston, where N = n+ n Assume that Q f 0 0 nn + Then we have T f r nn+ N [N] Q f r + ot f r Proof: By Lemma 3, we may assume that there exst subsets I = {Q t+f,, Q t+ f} k an functons c K f \ {0} t + t k+, where t =, whch satsfy the assertons of Lemma 3 Snce I s mnmal over K f, there exst c j R \ {0} such that Z m + t j=0 c j Q j f = 0 Defne c j = 0 for all j > t Then t k+ j=0 c jq j f = 0 Snce {c j Q j f} t j= s lnearly nepenent over K f, there exsts an amssble set {α,, α t } α j t N such that A D α c Q f D α c t Q t f D α c Q f D α c t Q t f D αt c Q f D αt c t Q t f f t 0 c Q f D α Q 0 f c Q f D α Q 0 f D αt c Q f Q 0 f ct Q D α t f Q 0 f ct Q D α t f Q 0 f D αt ct Q t f Q 0 f Q 0 f t à 0 Now conser We set c j = c j 0 t + j t +, then t + j=t c + jq j f j= I j Therefore, there exst meromorphc functons c j K f 0 j t such that K f t+ j=0 c jq j f = 0

10 88 S Duc Quang Defne c j = 0 for all j > t + Then t k+ j=0 c jq j f = 0 Snce {c j Q j f} t+ j=t s lnearly nepenent over K + f, there exsts {α j } t+ j=t + Zm + α j t + t N such that t+ t+ cs Q A = et D αj c s Q s f = Q 0 f t+ t et D αj s f j,s=t + Q 0 f j,s=t + =Q 0 f t+ t à 0 Conser an t k+ t k+ + mnor matrxes T an T gven by D α c 0Q 0f D α c tk+ Q tk+ f D α c 0Q 0f D α c tk+ Q tk+ f D α t c0q 0f D α t ctk+ Q tk+ f D α t + c 0Q 0f D α t + c tk+ Q tk+ f D α t + c 0Q 0f D α t + c tk+ Q tk+ f T = D α t 3 c0q 0f D α t 3 ctk+ Q tk+ f D α kt k + c k0 Q 0f D α kt k + c ktk+ Q tk+ f D α kt k + c k0 Q 0f D α kt k + c ktk+ Q tk+ f D α kt k+ c k0 Q 0f D α kt k+ c ktk+ Q tk+ f c0q 0f ctk+ Q tk+ f T = D α Q 0f D α t c0q 0f Q 0f D α t + c0q 0f Q 0f D α t c0q 3 0f Q 0f D α ck0 Q kt k + 0f Q 0f D α kt ck0 k+ Q 0f Q 0f D α Q 0f D α ctk+ Q tk+ f t Q 0f D α ctk+ Q tk+ f t + Q 0f D α ctk+ Q tk+ f t 3 D α kt k + Q 0f cktk+ Q tk+ f Q 0f D α kt k+ cktk+ Q tk+ f Q 0f

11 Secon man theorems an unqueness problem of meromorphc mappngs 89 Denote by D resp D the etermnant of the matrx obtane by eletng the + -th column of the mnor matrx T resp T It s clear that the sum of each row of T resp T s zero, then we have D = D 0 = k A = Q 0 f t k+ = Q 0 f t k+ D0 = Q 0 f t k+ D Snce k I n + an Q 0,, Q tk+ are n weakly general poston, by Lemma 8 there exsts a functon Ψ C f such that k fz Ψz max Q fz z C m 0 t k+ Fx z 0 C m Take 0 t k such that Q fz 0 = max 0 j tk Q j fz 0 Then D 0 z 0 fz 0 D z 0 tk+ j=0 Q = tk+ jfz 0 j=0 Q j fz 0 fz0 D z 0 Ψz 0 Q fz 0 tk+ j=0 Q j fz 0 j j Ths mples that log D 0z 0 fz 0 tk+ j=0 Q jfz 0 D z 0 log+ Ψz 0 tk+ j=0,j Q jfz 0 log + D z 0 tk + log + Ψz 0 j=0,j Q j fz 0 Thus, for each z C m, we have Note that D = et tk+ Q j f j=0,j Q 0 f log D 0z fz t k+ tk+ Q fz log + t k+ = log + c0 Q 0 f D α Q 0 f Q 0 f Q 0 f D α kt ck0 k+ Q 0 f Q 0 f Q 0 f Q 0 f D z tk j=0,j Q j fz D z Q j fz j=0,j Q 0 fz tk à + log + Ψz + log + Ψz 3 ctk+ Q D α tk+ f Q 0 f Q tk+ f Q 0 f D α cktk+ Q kt tk+ f k+ Q 0 f Q tk+ f Q 0 f

12 90 S Duc Quang The etermnant s counte after eletng the -th column n the above matrx By the lemma on logarthmc ervatve, for each an c K f we have D α cqj f D α cqj f Q 0 f Q 0 f m r, m r, +mr, c Q j f cq j f Q 0 f Q 0 f O log + T cqj f r Q 0 f +T c r = ot f r Therefore, we have m r, D tk+ Q j f j=0,j Q 0 f = ot f r 0 t k Integratng both ses of the nequalty 3, we get log f D 0 σ m + log tk+ Sr Sr Q σ m f t k+ log + D t k+ Sr m r, tk+ j=0,j Q jf Q 0 f D tk+ Q j f j=0,j Q 0 f By Jensen formula, the above nequalty mples that σ m + log + Ψzσ m Sr +ot f r = ot f r t k+ T f r + N D0 r N r N D 0 Qfr ot f r 3 We see that a pole of D 0 must be pole of some c s or pole of some nonzero coeffcents a I of Q an N r O N r + N r = ot D 0 c s a f r I,s a I 0 Therefore, the nequalty 3 mples that t k+ T f r N Qfr N D0 r + ot f r 33 Here we note that D = D 0, then ν 0 D = ν 0 D 0

13 Secon man theorems an unqueness problem of meromorphc mappngs 9 We now assume that z s a zero of some functons Q f Snce t k+ + n + an z can not be zero of more than n functons Q f, wthout loss of generalty we may assume that z s not zero of Q 0 f Then ν 0 D α st s +j c sq f z mn β Z m + wth αst s +j β Zm + mn β Z m + wth αst s +j β Zm + max{0, ν 0 Q f z N} N + ν c s z {ν 0 D β c sd α st s +j β Q f z} { max{0, ν 0 Qf z α st s +j β } β + ν c s z} for each t k+, j t s t s, s k +, where t 0 = 0 Put Iz = N + k tk s= t s t s νc s z Then t k+ ν D0 z max{0, νq 0 f z N} Iz 34 We note that f z s not zero of a functon Q f wth 0, replacng D 0 by D an repeatng the same above argument we agan get the nequalty 34 Hence 34 hols for all z C m It follows that t k+ νq 0 z ν f D 0 z t k Integratng both ses of the above nequalty, we get t k+ N Qfr N D0 r Combnng ths an 33, we get mn{n, νq 0 f z} + Iz t k+ N [N] Q f r + ot f r The lemma s prove T f r nn+ N [N] Q f r + ot f r Proof of Theorem We frst prove the theorem for the case where all Q =,, q have the same egree By changng the homogeneous coornates of P n C f necessary, we may assume that a I 0 for every =,, q We set Q = Q Then { a Q } q s a set of I homogeneous polynomals n K f [x 0,, x n ] n weakly general poston Conser nn + n + polynomals Q,, Q nn+n+ j q Applyng Lemma 4, we have T f r nn+n+ j= nn+n+ N [N] Q r + ot f r f N [N] Q r + ot f f r j=

14 9 S Duc Quang Takng summng-up of both ses of ths nequalty over all combnatons {,, nn+n+ } wth < < nn+n+ q, we have nn+n+ q nn + n + T f r N [N] Q r + ot f f r j= The theorem s prove n ths case We now prove the theorem for the general case where eg Q = Then, applyng the above case for f an the movng hypersurfaces Q The theorem s prove q nn + n + T f r = j= N [N] =,, q of common egree, we have Q / f r + ot f r N [N] Q r + ot f f r j= j= N [N] Q f r + ot f r Proof of Theorem By repeatng the argument as n the proof of Theorem, t suffces to prove the theorem for the case where all Q have the same egree By changng the homogeneous coornates of P n C f necessary, we may assume that a I 0 for every =,, q We set Q = Q Then { a Q } q s a set of homogeneous I polynomals n K f [x 0,, x n ] n weakly general poston Conser N + polynomals Q,, Q N+ j q We see that m Q j ; j N + K{Q N + < N + Then the set {Q } q,, Q N+ } s lnearly nepenent over K {Q} q Hence, there exsts a mnmal subset over K {Q} q, for nstance that s { Q,, Q t }, of { Q,, Q N+ } Then, there exst nonzero functons c j j t n K {Q} q c Q + + c t Qt = 0 such that Snce Q,, Q N+ are n weakly general poston, t n + Settng F j = c j Q j f, we have F + F t = F t Choose a meromorphc functons h so that F = hf : : hf t s a reuce representaton of a meromorphc mappng F from C m nto P n C It s seen that t N h r N r + N aj c I j r = ot f r j=

15 Secon man theorems an unqueness problem of meromorphc mappngs 93 On the other han, by the mnmalty of the set { Q,, Q t }, then F s lnearly nonegenerate over C Applyng the secon man theorem for fxe hyperplanes, we get It follows that T F r = t j= t N [t ] hf j r + ot F r N [t ] Q j f j= t j= r + N [t ] c j r + tn [t ] r + ot F r N+ N [t ] Q j f r + ot f r N [N] Q j f r + ot f r T f r = T F r + ot f r N+ j= j= h N [N] Q j f r + ot f r Takng summng-up of both ses of ths nequalty over all combnatons {,, N+ } wth < < N+ q, we have q N + T f r j= N [N] Q f r + ot f r The theorem s prove 4 Unqueness problem of meromorphc mappngs sharng movng hypersurfaces In orer to prove Theorem 3 an Theorem 4 we nee the followng Lemma 5 Let f an g be nonconstant meromorphc mappngs of C m nto P n C Let Q =,, q be slow wth respect to f an g movng hypersurfaces n P n C n weakly general poston wth eg Q = Assume that mn{νq 0 z, } = f mn{ν0 Q g z, } for all q Put = lcm,, q an N = n+ n Then the followng assertons hol: If q > NnN+n+ then T f r = OT g r an T g r = OT f r If both f an g are algebracally nonegenerate over K {Q} q an q n + then T f r = OT g r an T g r = OT f r Proof: It s clear that q > nn + n + Then applyng Theorem for f, we have

16 94 S Duc Quang q nn + n + T gr N [N] Q g r + ot gr N N [] Q g r + ot gr N N [] Q f r + ot gr qn T f r + ot g r Hence T g r = OT f r Smlarly, we get T f r = OT g r Applyng Theorem A wth ɛ =, then there exsts a postve nteger L such that Therefore, we have q n 3 T f r q n 3 T f r q n 3 T gr = N [L] Q f r + ot f r N [L] Q f r + ot f r, N [L] Q g r + ot gr L N [] Q f r + ot f r L N [] Q g r + ot f r ql T g r + ot g r Hence T f r = OT g r Smlarly, we get T g r = OT f r Proof of Theorem 3 We assume that f an g have reuce representatons f = f 0 : : f n an g = g 0 : : g n respectvely a By Lemma 5, we have T f r = OT g r an T g r = OT f r Suppose that f an g are two stnct maps Then there exst two nex s, t 0 s < t n satsfyng H := f s g t f t g s 0 Set S = { k j=0 ZeroQ j f ; 0 < < k q} Then S s ether an analytc subset of comenson at least two of C m or an empty set Assume that z s a zero of some Q f q an z S Then the conton yels that z s a zero of the functon H Also, snce z S, z can not be zero of more than k functons Q f Therefore, we have ν 0 Hz = k mn{, νq 0 z} f

17 Secon man theorems an unqueness problem of meromorphc mappngs 95 Ths nequalty hols for every z outse the analytc subset S of comenson at least two Then, t follows that N H r k N [] Q f r 4 On the other han, by the efnton of the characterstc functon an Jensen formula, we have N H r = log f s g t f t g s σ m Sr log f σ m + log f σ m Sr Sr Combnng ths an 4, we obtan Smlarly, we have = T f r + T g r T f r + T g r k T f r + T g r k N [] Q f r N [] Q g r Summng-up both ses of the above two nequaltes, we have T f r + T g r k = k N [] Q f r + k N [] Q / f r + k kn N [N] Q / f r + From 4 an applyng Theorem for f an g, we have T f r + T g r kn N [N] Q / f r + N [] Q g r N [] r Q / g kn N [N] r 4 Q / g kn N [N] r Q / g q kn nn + n + T f r + T g r + ot f r + T g r Lettng r +, we get q kn nn+n+ q knnn+n+ Ths s a contracton Hence f = g The asserton a s prove

18 96 S Duc Quang b By Lemma 5, we have T f r = OT g r an T g r = OT f r Suppose that f an g are two stnct maps Repeatng the same argument as n a, we get the followng nequalty, whch s smlar to 4, T f r + T g r kn N [N] Q / f r + From 43 an applyng Theorem for f an g, we have T f r + T g r kn N [N] Q / f r + kn N [N] r 43 Q / g kn N [N] r Q / g q kn N + T f r + T g r + ot f r + T g r Lettng r +, we get q kn N+ q knn+ Ths s a contracton Hence f = g The asserton b s prove Proof of Theorem 4 By Lemma 5, we have T f r = OT g r an T g r = OT f r By changng nces f necessary, we may assume that where k s = q n+3 n+3 f n+3 k k Q Q Qn+3 g Q k k }{{ } group k + k f + k + k3 k 3 Q Q Qk g + Q k3 k }{{ 3 } group 3 k + f Q k f + Q g Q k + k g + Q k k k k f } {{ g } group ks + f Q k f s + Q g ks + Qk g Q s + ks k s f, ks k s g }{{} group s If there exst a group contanng more than [ q n ] elements then we have the esre concluson of the theorem We now suppose that the number of elements of each group s at most [ q n ] For each n + 3 q, we set + [ q n ] f + [ q n ] q, σ = + [ q n ] q + n + f + [ q n ] > q,

19 Secon man theorems an unqueness problem of meromorphc mappngs 97 an P = Q fq σ σ g Q gq σ σ f Q Snce the number of elements of each group s at most [ q n σ σ f ], then Q f an Q g belong to two stnct groups, hence P 0 for every n + 3 q Then we Q σ σ g have q P := P 0 We set S = =n+3 < < k+ n+ k+ j= ZeroQ j f Then S s an analytc set of comenson at least of C m Clam: N P r q N L Q f Inee, fx a pont z If Ig S We assume that z s a zero of some functons Q f q We set I = { : n +, f, H z = 0} an t = I, J = { : n + 3 q, f, H z = 0} an l = J Here we note that 0 t, l k an t + l k For each nex, t s easy to see that ν P z mn{νq 0, L f } f J, σ J ν P z mn{νq 0 σ f, L σ} f J, σ J σ ν P z mn{νq 0, L f } + mn{νq 0 σ f, L σ} f, σ J σ ν P z 0 f, σ J an t = 0 ν P z f, σ J an t > 0 We set vz = {j : j, σj J} It easy to see that Then, we have the followng two cases: vz q n l tq n k

20 98 S Duc Quang Case t = 0 Then ν P z = Case 0 < t k Then ν P z = =n+3 J =n+3 =n+3 J =n+3 mn{ν 0 Q f, L } mn{νq 0, L f } + q n n+ mn{νq 0 k, } f mn{ν 0 Q f, L } + vz =n+3 mn{νq 0, L f } + q n n+ mn{νq 0 k, } f Therefore, from the above two cases t follows that ν P z =n+3 mn{νq 0, L tq n f } + k mn{νq 0, L f } + q n n+ mn{νq 0 k, } f for all z outse the analytc set If Ig S Integratng both ses of the above nequalty, we get N P r =n+3 =n+3 Here we note that q n kl Smlarly, we have N [L] Q r + q n n+ f N [] k Q r f n+ N [L] Q r + f kl kl q n N [L] kl Q r f n + N P r N [L] Q f r 44 N [L] Q g r 45 Then by 44 an 45 an by Theorem A wth ɛ =, we have N P r q n 3 T f r + T g r + ot f r 46

21 Secon man theorems an unqueness problem of meromorphc mappngs 99 Repeatng the same argument as n the proof of Theorem 3, by Jensen s formula an by the efnton of the characterstc functon, we have N P r = =n+3 From 46 an 47, we have N P r =n+3 T f r + T g r = q n T f r + T g r + ot f r 47 q n 3 T f r + T g r q n T f r + T g r + ot f r Lettng r +, we get q n 3 q n Ths s a contracton Therefore the supposton s mpossble Hence there must exst a group contanng more than [ q n ] elements, then we have the esre concluson of the theorem Acknowlegement Ths work was supporte n part by a NAFOSTED grant of Vetnam References [] T T H An an H T Phuong, An explct estmate on multplcty truncaton n the Secon Man Theorem for holomorphc curves encounterng hypersurfaces n general poston n projectve space, Houston J Math , [] G Dethloff an T V Tan, A secon man theorem for movng hypersurface targets Houston J Math 37 0, 79- [3] G Dethloff an T V Tan, A unqueness theorem for meromorphc maps wth movng hypersurfaces, Publ Math Debrecen 78 0, [4] H Fujmoto, Non-ntegrate efect relaton for meromorphc maps of complete Kähler manfols nto P N C P N k C, Japanese J Math 985, [5] J Noguch an T Ocha, Introucton to Geometrc Functon Theory n Several Complex Varables, Trans Math Monogr 80, Amer Math Soc, Provence, Rhoe Islan, 990 [6] S D Quang, Secon man theorem an uncty theorem for meromorphc mappngs sharng movng hypersurfaces regarless of multplcty, Bull Sc Math 36 0, [7] M Ru, A efect relaton for holomorphc curves ntersectng hypersurfaces, Amer J Math 6 004, 5-6 [8] B Shffman, Introucton to the Carlson - Grffths equstrbuton theory, Lecture Notes n Math , 44-89

22 300 S Duc Quang [9] D D Tha an S D Quang, Unqueness problem wth truncate multplctes of meromorphc mappngs n several compex varables for movng targets, Internat J Math 6 005, [0] D D Tha an S D Quang, Secon man theorem wth truncate countng functon n several complex varables for movng targets, Forum Mathematcum 0 008, Receve: 00 Accepte: Department of Mathematcs, Hano Natonal Unversty of Eucaton E-mal: quangs@hnueeuvn