H a i n h l caR i t t v v nd u y nht ivi a
Mtsk h ini mvk tqub t r
Trc h t, ch ng t i nh c l i c c khi u vk h i n i m c b n c n g v i c c ktqub t r d ngtrongChn g 1(xem[1]).
Gis f lh m ph n h nh kh c h ng tr n C Vim i a∈C ,tanhnghahm ν a : C→Nxcnhbi ν a (z)=
Gis m lsnguyndn g Vimi a ∈ C ∪{∞}, t axcn h hm m) af, tC ∪{∞}n N chobi a 0 nu ν a (z)>m t ν f ∞ ,m) = ν 0 f , , ν af, m)
Ta cng ccc hmm N m) (r, 1 ) ,N m) (r,f),N m) (r,f),N m) (r, 1 ) xcn h bi
Tn g ttan h ngha ν a xcn h bi a 0 nu ν a (z)k v a,b t ng ng l c c i m v kh ng i mca f Khi,tac
2.V bl khngi mca fn ntac f= (z−b) m l,l (b) 0,f n = ( z−b) mn l n ,
B1.3.6.Cho f , (f) (k) l c chmphnhnhkhch ngtrn C v k lsnguyndn g Khi, tac
(f) (k) Chngminh.DoB1.3.4vc h rng m(r, )=S(r,f) tac f
B1.3.7.Cho f lm thmphnhnhkhchngtrn C v n , k l ccsnguyndn g , n≥k+1 Khi, tac
Tyvd o fl k h chngnn(f n ) (k) lk h chng.B1 3 7 c chngminh.
B1 3 8 C h o f l m th mp h nh nhk h ch ngt r n C v n , k l ccsng uy ndng, n>2k Khi ,tac
(k) KhidoB1.3.7tac A lkhchng, A=f n−k F LidoB1.3.4tac
B1.3.9.Cho f lhmphnhnhkhchngtrn C v n,k l ccsnguyndng, n>2k, v P (z) lathccb c d>0 Khi,tac
T(r,P(f))= dT(r,f)+S(r,f),T(r,P n (f))= ndT(r,f)+S(r,f) Biv C,C n kh c h ngsv S(r, f) =S(r,C)=S(r, C n ) DoB1.3.7 tac A=(C n )
(k)lk h chng.MtkhctheoB1.3.6vB 1.3.8tac n 2k)T(r,C)+kN(r,C)+N(r, A ) 1(
B1 3 9 c chngminh. nhlsauy lktquv t pxcn h duynhti via thcviphn. nh l 1.3.10.Cho f, g l hai h m ph n h nh kh c h ng v P(x) l ath cchp nh n c Gi s r ngcc i u ki n (B 1 ),(B 2 ),(B 3 ) cth am n v n(2q+3) 2 , q1 v d3k+5 N u (P d (f)) (k) v (P d (g))
FC d−k ,B= ( D d ) (k) = Q D d−k pdngB 1.1.5c h o( C n ) (k) ,(D n ) (k) tacm ttrongcctrn g hpsauxyra.
Chngtathyr ng,n u a lc ci mc a A th C (a)= ∞ vi ν A ∞ (a)≥ d+k≥2 Vt h
F) +S(r,f)≤2nT(r,f)+kN 1 (r,C) +knT(r,f)+S(r,f)=2nT(r,f)+knT(r,f)+kN 1 (r,f)+S(r,f)
Tathyrng,nu z 0lm tkhngi mca f e i vi1 i n ,th z 0lmtkhngimca(C d ) (k) v z 0 cnglmtccimca(D d ) (k) Khi c z 0 lmtccimca gv
Ty , suyra d(n 2 3n) 2k(1 n), i un ymuthunvi n 4,d,n,k lccsn g u y ndn g
C,D,H,G khc hngs.p d ngB1 1 5 cho H v iccgit r ∞ ,0 ,1t ac
Bygi,tachngminh S (r,H)=S (r,f) V Hp=C d t ac ndT(r,f)+S(r,f)=T(r,C d )≤T(r,H )+(k−1)logr
Dot a gpmuthunv d 3k+ 5> 2 n+ 2k− 1 n Vy p = 0. Khi(P(f)) d = ( P(g)) d Vth P(f)= eP(g),e d = 1 Ty , tac
Vd 1.3.11.Xttrn g hp n = 2 5 ,q= 1.Ly a = 1, b , c,e lcchng sk h ckhngthucC, vt h amncci ukin: b e( te
Chngminh.Trc ht,tachngminhrngi ukin( B 1 )c thamn.V n=25, q= 1, a=1, b,c,e lcchngskhckhng,tac
v 3 v 1 j c c c 25 det A α ′ det A β ′ det A α ′ det A β ′ lttcc c nghimca(1.38).
) 24 vdo(1.38)tac ch −t 25 r 24 r 24 = −c(cr+ e) 24 ,− j ( cr+ e)=(cr+ e) 25 ,P(r) =r 25 +b+(cr+ e) 25 j c r 24 j r 24
Dd ngthyrngi ukin(B 2 )cthamn.Bygit a chngminh iukin(B 3 )cthamn.t v 1 =(1,0) , v 2 =(0,b), v 3 =(c,e) Taxt α=(1,2)β=(2,3) Khi
Taxtcctrn g hpcthxyranh sau.Trn g hp1 α ′ =(2,1)β ′ =(1,3) Khi
Do b 25 det A tanhnc det A α ′ det A det A β ′
Trong Ch ng 1, ch ng t i ch ng minh hai ktqut n g tnhlR i t t c h o h m p h n h n h , l n h l 1 2 2 , n h l 1 2 5 C h n g t i c ng ch ng minh c m t ktquv B i URSM ( nhl1.3.2),m t ktqu v URSM ( nhl1.3.3), m t ktqu vt p x c n h d u y nhtchoathcviphn(n h l1 3 1 0 )
Chng2 nh lthh a i c a R i t t v v n duy nh t c aath c vi ph n tr nmttrngkhng- Acsimet
Trong chn g n y c h n g t i khi uKlm t trn g ng iscs0 , y ivim t git r t u y t ikh ng-Acsimetkhi u bi T pxcn h cs p h ntn h n h t i vihmph nhnhtrn Kl t pc1 0ph n tc c h n g m i n h H u v Y a n g
[ 2 3 ] G n y , n h i u k t qu c ngnh ncivi ath c vi ph n, ch ng h ncd ng(f n ) (k) (An, Hoa, vKhoai [6]; Khoai, An, vLai [30]) vcd ng(f)
(′ ) P ′(f)(Boussaf- Escassut- Ojeda [11]) Trong [30] Khoai, An, vLaic h n g minhktqusau. nh l C.Cho f(z) v g(z) l hai h m ph n h nh kh c h ng tr n K v n , k l h a i s n g u y n d n g s a o c h o n 3k+ 8 K h i n u ( f n ) (k) v (g n ) (k) nh n 1 ct nh b i th f(z) =tg(z) v i t l h ng s th a m n t n =1
Trong[49]Yangt ravns a u : liungthc f −1 (S)=g −1 (S) vi S= 1,1 ivicca thccngbc f , g sk otheo f = g h a yl f= g ?Cuhinycngcgiiptrong[42],[43].
Thitlpmtsnhltn g tnhlRittthh a i chohmphnhnhvc chm nguyntrnK.
C c k t qun y g p ph n trl i C u h i c a C.C.Yang [49], c aF.Packovich [43]trongtrn g hp p -adic. Σ f
Tanhclimtskhinimsau. nhngha2.1.1.Hmchnhhnh f t r n K c gilm thmnguyn.
Kh i uloglh mlogaritcs ρ >1,ln lh mlogaritcc s e ,
A(K)lv nhcchmnguyntrn K , kh i u M (K) lt r n g cchm ph nh nhvkhi u K ^= K ∪{∞}
Gis fl hmnguyntrn Kvb ∈K Khi f v i tdidng f= ∞ n=q b n (z− b) n ,b n ∈ K,z∈ K, b q 0,v à 0 (b)= q lb ikhngi mca f ti b
Cn h st h c ρ 0vi0k v a l mtccimc a f Khi
B2.2.2.Cho f , ( f) (k) l cchmphnhnhkhchngtrn K v k lmtsnguyndn g Khi, tac
B2 2 3 [30]Cho f l mthmphnhnhkhchngtrn K v n,k lhaisnguyndn g thamn n≥k+1 Khi, tac
B2.2.4.[30]Cho f l mthmphnhnhkhchngtrn K v n,k lccsnguyndn g thamn n>2k Khi , tac
B2.2.5.Cho f lh mphnhnhkhchngtrn K v n , k l c csnguyndngtha mn n>2k, c h o P(z) l m ta thcbc d>0 Khi
P(z)=z d +a 1 z d−m +b 1 ,Q (z)=z d +a 2 z d−m +b 2 , (2.1) lc c a t h c b c dk i u F e r m a t - W a r i n g t r o n g K[z]k h n g cn g h i m b i Ktqusaulm r ngcan h l 1 2 2 choc chmphnhnhtr ntrn g khng-Acsimet. n
2 m nh l2.2.6.Cho d2m+3 v hoc m3 hoc (d,m) =1 v m2 , c=0 , P(z), Q(z) c x cnh nht r o n g ( 2 1 ) G i sr ngphngtrnh P(f)=cQ(g) cn g h i m (f,g) phnhnhkhch ng.Khi g=hf v i h ∈K saocho h d = 1
Chngminh.V P(f)=cQ(g)nntac f d +a 1 f d−m +b 1 = c (g d +a 2 g d−m +b 2 ),dT(r,f)+O(1)= dT(r,g),
Phngtrnh(2.2)ct h v i tlinhs a u f 1 +f 2 = cb 2 − b 1 ,f 1 = f d−m (f m +a 1 ),f 2 = −cg d−m (g m +a 2 )
1 +N 1 (r, g m + a )−logr+O(1), dT(r,f)≤ (2m+3)T(r,f)−logr+O(1), (d−2m−3)T(r,f)≤−logr+O(1), iunymuthunvigit h i t d 2m+3.Vvy cb 2 b 1 =0
Khi(2.2)trthnh f d +a 1 f d−m =cg d +ca 2 g d−m (2.3) ngin,tat h= g f v α = 1 c a 1
Gis h kh chngs.Xtc ctrn g hpct h x yranhs a u
(i).Nu h d − α v h d−m − β k hngck h ngi mchungthm ikhng imca h d −α ucbi≥ m Khi
(d−2− m )T(r,h)≤−logr+O(1), i u nyd nn d(m1)