Lí thuyết nevanlinna và phương trình vi phân p adic

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖

◆●❯❨➍◆ ❚❍➚ ▲■➊◆

▲➑ ❚❍❯❨➌❚ ◆❊❱❆◆▲■◆◆❆
❱⑨ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ P✲❆❉■❈

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✕ ✷✵✶✻


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖

◆●❯❨➍◆ ❚❍➚ ▲■➊◆

▲➑ ❚❍❯❨➌❚ ◆❊❱❆◆▲■◆◆❆
❱⑨ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ P✲❆❉■❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ●■❷■ ❚➑❈❍
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝

●❙✳❚❙❑❍ ❍⑨ ❍❯❨ ❑❍❖⑩■

❚❤→✐ ◆❣✉②➯♥ ✕ ✷✵✶✻


▲í✐ ❝❛♠ ✤♦❛♥
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣
t❤ü❝✱ ❦❤æ♥❣ trò♥❣ ❧➦♣ ✈î✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝ ✈➔ ❝→❝ t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣
❧✉➟♥ ✈➠♥ ✤➣ ✤÷ñ❝ ❝❤➾ rã ♥❣✉ç♥ ❣è❝✳
❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✻

◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥

◆❣✉②➵♥ ❚❤à ▲✐➯♥

✐


ớ ỡ
ữủ t ớ sỹ ữợ t t ừ
tớ ổ sự tổ tr
q tr tỹ t t ồ tổ t
tổ tọ ỏ t ỡ s s t
ổ tr trồ ỡ trữớ P
ồ ừ trữớ t ồ
t ủ tổ t tốt ử ồ t ừ
ố ũ tổ t s ử t
trữớ P t ỗ
ở t ú ù tổ ồ t tr sốt q
tr ồ t t

r q tr t ụ ữ tr ỷ
ổ tr ọ ỳ t sõt t ữủ sỹ
õ ỵ ừ t ổ ỗ ữủ t ỡ
t

ữớ t






▼ö❝ ❧ö❝
▲í✐ ❝❛♠ ✤♦❛♥

✐

▲í✐ ❝↔♠ ì♥

✐✐

▼ö❝ ❧ö❝

✐✐✐

▼ð ✤➛✉

✶

✶ ❈ì sð ❧➼ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛

✷

✶✳✶ ▲➼ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣✲❛❞✐❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

✶✳✷ ◗✉❛♥ ❤➺ sè ❦❤✉②➳t ❝❤♦ ♠ö❝ t✐➯✉ ❞✐ ✤ë♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✸ ❳→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✹ ×î❝ ❧÷ñ♥❣ ❝➜♣ t➠♥❣ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

✷ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ p✲❛❞✐❝

✶✾

✷✳✶ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè p✲❛❞✐❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

✷✳✷ ✣à♥❤ ❧➼ ▼❛❧♠q✉✐st ❦✐➸✉ ✭■✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

✷✳✸ ✣à♥❤ ❧þ ▼❛❧♠q✉✐st ❦✐➸✉ ✭■■✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✼

✷✳✹ ◆❣❤✐➺♠ ❝❤➜♣ ♥❤➟♥ ✤÷ñ❝ ❝õ❛ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✳

✷✾

❑➳t ❧✉➟♥ ❝❤✉♥❣

✸✻

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✼

✐✐✐


▼Ð ✣❺❯

●➛♥ ✤➙②✱ ❧➼ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ p✲❛❞✐❝ ✤➣ trð t❤➔♥❤ ♠ët ❧➼♥❤ ✈ü❝ ❚♦→♥ ❤å❝
♥➠♥❣ ✤ë♥❣✳ ❈❤➥♥❣ ❤↕♥✱ ❑❤♦→✐ ❬✻❪✱ ❑❤♦→✐✲◗✉❛♥❣ ❬✼❪ ✈➔ ❇♦✉t❛❜❛❛ ❬✷❪ ✤➣ ❝❤ù♥❣
♠✐♥❤ t÷ì♥❣ tü p✲❛❞✐❝ ❝õ❛ ❤❛✐ ✧✤à♥❤ ❧➼ ❝ì ❜↔♥✧ ✈➔ q✉❛♥ ❤➺ sè ❦❤✉②➳t ❝õ❛ ❧➼
t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝ê ✤✐➸♥✳ ❍➔ ❍✉② ❑❤♦→✐✱ ▼❛✐ ✈➠♥ ❚÷ ✈➔ ❈❤❡rr②✲❨❡ ✤➣
♥❣❤✐➯♥ ❝ù✉ ❧➼ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ p✲❛❞✐❝ ♥❤✐➲✉ ❜✐➳♥ ✈➔ ❝❤ù♥❣ ♠✐♥❤ q✉❛♥ ❤➺ sè
❦❤✉②➳t ❝õ❛ ❝→❝ s✐➯✉ ♣❤➥♥❣ tr♦♥❣ tr÷í♥❣ ❤ñ♣ tê♥❣ q✉→t✳ ❍✉✲❨❛♥❣ ✤➣ ❝❤ù♥❣
♠✐♥❤ t÷ì♥❣ tü p✲❛❞✐❝ ✈➲ q✉❛♥ ❤➺ sè ❦❤✉②➳t ❝❤♦ ♠ö❝ t✐➯✉ ❞✐ ✤ë♥❣✱ ✤à♥❤ ❧➼
❝ì ❜↔♥ t❤ù ❤❛✐ ❝❤♦ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ✈➔ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ✈î✐ sè ♣❤➛♥

tû ❤ú✉ ❤↕♥✳ ❈❤❡rr②✲❨❛♥❣ ❬✹❪ ✤➣ ♠æ t↔ ♠ët sè t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ✈î✐ sè
♣❤➛♥ tû ❤ú✉ ❤↕♥ ❝õ❛ ❝→❝ ❤➔♠ ♥❣✉②➯♥ p✲❛❞✐❝✳✳✳
▲✉➟♥ ✈➠♥ ♥➔② ♥❤➡♠ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ♥❣➢♥ ❣å♥ ✈➲ ❧➼ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛
✈➔ ù♥❣ ❞ö♥❣ ❝õ❛ ♥â ✤è✐ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ p✲❛❞✐❝✳
◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ❣ç♠ ✷ ❝❤÷ì♥❣✿
❈❤÷ì♥❣ ■✿ ❚r➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❧➼ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ✈➔
♠ët sè ❦➳t q✉↔ ✈➲ q✉❛♥ ❤➺ sè ❦❤✉②➳t✱ ❜➔✐ t♦→♥ ①→❝ ✤à♥❤ t➟♣ ❞✉② ♥❤➜t ❝õ❛
❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ ✈➔ ÷î❝ ❧÷ñ♥❣ ❝➜♣ t➠♥❣ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝✳
❈❤÷ì♥❣ ■■✿ ●✐î✐ t❤✐➺✉ ✤à♥❤ ♥❣❤➽❛✱ ❝→❝ t➼♥❤ ❝❤➜t ✈➔ ♠ët sè ❦➳t q✉↔ ✈➲
♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ p✲❛❞✐❝✱ ❜❛♦ ❣ç♠ ✣à♥❤ ❧➼ ▼❛❧♠q✉✐st ❦✐➸✉ ✭■✮✱ ✣à♥❤ ❧➼
▼❛❧♠q✉✐st ❦✐➸✉ ✭■■✮ ✈➔ ❝❤➾ r❛ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè p✲❛❞✐❝
❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t ❝❤➜♣ ♥❤➟♥ ✤÷ñ❝✳

❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✻

✶


❈❤÷ì♥❣ ✶
❈ì sð ❧➼ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛
✶✳✶

▲➼ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣✲❛❞✐❝✳

❈❤♦ p ❧➔ sè ♥❣✉②➯♥ tè✱ Qp ❧➔ tr÷í♥❣ ❝→❝ sè p✲❛❞✐❝ ✈➔ Cp ❧➔ ❜ê s✉♥❣ ✤➛② ✤õ
p✲❛❞✐❝ ❝õ❛ ❜❛♦ ✤â♥❣ ✤↕✐ sè ❝õ❛ Qp ✳ ●✐→ trà t✉②➺t ✤è✐ |.|p tr♦♥❣ Cp ✤➣ ✤÷ñ❝

❝❤✉➞♥ ❤â❛ s❛♦ ❝❤♦ |p|p = p−1 ✳ ❚❛ t✐➳♣ tö❝ sû ❞ö♥❣ ❦➼ ❤✐➺✉ ordp ❧➔ ✤à♥❤ ❣✐→
❝ë♥❣ t➼♥❤ tr➯♥ Cp .
◆❤➢❝ ❧↕✐ r➡♥❣ tr♦♥❣ ♥❤ú♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ ♠➔ ♠❡tr✐❝ ❝↔♠ s✐♥❤

❜ð✐ ❝❤✉➞♥ ❦❤æ♥❣ ❆❝s✐♠❡t✱ tê♥❣ ✈æ ❤↕♥ ❤ë✐ tö ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ sè ❤↕♥❣ tê♥❣
q✉→t ❞➛♥ ✤➳♥ 0✳ ❑❤✐ ✤â ❜✐➸✉ t❤ù❝ ❝â ❞↕♥❣ ❞÷î✐ ✤➙②✿
∞

an z n

f (z) =
n=0

①→❝ ✤à♥❤ ✤ó♥❣ ✤➢♥ ❦❤✐ |an z n |p → 0.
✣à♥❤ ♥❣❤➽❛ ❜→♥ ❦➼♥❤ ❤ë✐ tö ρ ❜ð✐
1
1
= lim sup |an |pn .
ρ n→∞
❑❤✐ ✤â✱ ❝❤✉é✐ ❤ë✐ tö ♥➳✉ |z|p < ρ ✈➔ ♣❤➙♥ ❦➻ ♥➳✉ |z|p > ρ✳ ◆❣♦➔✐ r❛✱ ❤➔♠
f (z) ✤÷ñ❝ ❣å✐ ❧➔ ❣✐↔✐ t➼❝❤ p✲❛❞✐❝ tr➯♥ B (ρ) ♥➳✉ ❝❤✉é✐ ❤ë✐ tö tr➯♥
B (ρ) = {z ∈ Cp | |z|p < ρ}.

✷


= f (z) ữủ ồ p tr Cp
f t p tr B () (0 < ) .
t ừ ữỡ r t ừ
số ỹ

à (r, f ) = max |an |p rn
n0

(0 < r < ) ,

ũ ợ số tr t
(r, f ) = max{n| |an |p rn = à (r, f )}.
n0

(0, f ) = lim (r, f ) . ỡ ỳ ú t ú ỵ r h
r0

ởt t p tr B () t
à (r, f h) = à (r, f ) à (r, h) .

(1)

ờ số tr t (r, f ) t r tọ ổ
tự
r

log à (r, f ) = log a(0,f ) p +

(t, f ) (0, f )
dt+ (0, f ) log r
t

(0 < r < ) .

0

ờ ỵ rstrss ỗ t t tự
P õ (r, f ) ởt t p g tr B [r] s f = gP

õ

B [r] = {z Cp | |z|p r}.
ỡ ỳ g ổ õ t ổ tr B [r] P õ ú (r, f )
ổ ở tr B [r] .

ồ n r, f1

số ổ ở ừ f ợ tr tt ố r

ừ f ố ợ
r

1
N r,
f

=

n t, f1 n 0, f1

dt + n 0,

t
0



1
f

logr

(0 < r < ) .


ờ r r
n r,

1
f

= (r, f ) .

ứ ờ s r ổ tự s
N r,

1
f

= log à (r, f ) log an(0, 1 ) .
f
p

(2)

ú t ụ số ổ t ừ f tr B [r]
n r, f1

r

1
N r,
f

=

n t, f1 n 0, f1

dt + n 0,

t

1
f

log r

(0 < r < ) .

0

ợ ộ n t ỗ t (t) t ordp (an z n ) ữ ừ t = ordp (z).
õ (t) ữớ t ợ ở n. ồ (t, f ) ừ
ừ tt ỷ t ữợ ữớ t n (t). ữớ
ữủ ồ t ừ f (z). t t õ (t, f )
õ ữủ ồ tợ ừ f (z). ỳ [, ] ự
ỳ tợ ó r r t tợ t ordp (an )+nt
t tợ tr ọ t t tr n. ú t õ

à(r, f ) = p(t,f )

tr õ r = pt . t ỡ ừ t t = ordp (z)
ổ tợ t
|f (z)|p = p(t,f )

t
|f (z)|p = à(r, f ).

f tr B() ữủ tữỡ

g
h

ừ t

p g h s g h ổ õ tỷ tr

t p tr B [] . à tọ ừ




t p tỗ t t õ t rở t à
f=

g
h

à(r, f ) =

à(r, g)
.
à(r, h)

ụ t
(t, f ) = (t, g) (t, h).

ó r r t = ordp (z) ổ tợ ừ f (z) õ ởt
t ổ tợ ừ g(z) h(z) t
|f (z)|p = p(t,f ) = à(r, f ).


|Cp | = {|z|p |z Cp }.

ú ỵ r {pw |w Q} |C|p }. |C|p trũ t tr R[0, +).
a : R[0, +) R b : |C|p R tr tỹ t
||a(r)|| b(z)

ợ t số ữỡ ỳ 0 < R < õ ởt t ỳ E
tr|Cp | [0, R] s
a(r) b(r),

r = |z|p |Cp | [0, R] E.

sỷ ử t õ
||à(r, f )|| = |f (z)|p

p f tr B().

n(r, f ) N (r, f ) ừ f ố ợ ỹ
1
1
n(r, f ) = n(r, ), N (r, f ) = N (r, ).
h
h
õ ử g h t t ữủ ổ tự s
1
N (r, ) N (r, f ) = log à(r, f ) Cf ,
f



(3)


ð ✤â Cf ❧➔ ❤➡♥❣ sè ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ f. ✣à♥❤ ♥❣❤➽❛
m(r, f ) = log+ µ(r, f ) = max {0, log µ(r, f )}

❈✉è✐ ❝ò♥❣ t❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ✤➦❝ tr÷♥❣✿
T (r, f ) = m(r, f ) + N (r, f ).

❇ê ✤➲ ✶✳✶✳✸✳

✭✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ♥❤➜t✮ ❈❤♦ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝

❤➡♥❣ tr♦♥❣ B(ρ). ❑❤✐ ✤â✱ ✈î✐ ♠é✐ a ∈ Cp t❛ ❝â

m(r,

❇ê ✤➲ ✶✳✶✳✹✳

1
1
) + N (r,
) = T (r, f ) + O(1)
f −a
f −a

(r → ρ).

✭❇ê ✤➲ ✤↕♦ ❤➔♠ ❧♦❣❛r✐t✮ ❈❤♦ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝

❤➡♥❣ tr♦♥❣ B(ρ). ❱î✐ ♠é✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ n ❜➜t ❦➻✱ t❛ ❝â

fn
m(r, ) = O(1)
f

❇ê ✤➲ ✶✳✶✳✺✳

(r → ρ).

✭✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐✮ ❈❤♦ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝

❤➡♥❣ tr♦♥❣ B(ρ) ✈➔ a1 , ..., aq ❧➔ ❝→❝ sè ♣❤➙♥ ❜✐➺t tr♦♥❣ Cp . ❑❤✐ ✤â
q

(q − 1)T (r, f ) ≤ N (r, f ) +

N (r,
j=1

1
) − N1 (r, f ) − log r + O(1)
f − aj

ð ✤â

N1 (r, f ) = 2N (r, f ) − N (r, f ) + N (r,

1
).
f

❍ì♥ ♥ú❛✱ t❛ ❝â
q

q

1
1
1
)−N1 (r, f ) ≤ N (r, f )+
N (r,
)−N0 (r, ),
N (r, f )+
N (r,
f − aj
f − aj

f
j=1
j=1
f (a)

≤2

a∈Cp ∪{∞}

ð ✤â N0 (r, f1 ) ❧➔ ❤➔♠ ✤➳♠ sè ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f ✱ f ❦❤æ♥❣ ♥❤➟♥ ♠ët tr♦♥❣
❝→❝ ❣✐→ trà a1 , ..., aq ✈➔
f (a)

= 1 − lim sup
r→∞

✻

1
N (r, f −a
)
j

T (r, f )

.

số t ử t ở

ồ Pn (Cp ) ổ n tr Cp . ởt ữớ


f : Cp Pn (Cp ),

t ởt ợ tữỡ ữỡ ừ (n + 1) ở p
f = (f0 , ..., fn ) : Cp Cn+1
p

s f0 , ..., fn ổ õ tỷ tr p
tr Cp s ổ tt fi ỗ t é f ụ
ữủ ồ tự t ồ ừ f. t
||f(z)|| = max |fk (z)p |.
k

õ trữ
T (r, f ) = log ||f(z)|| (|z|p = r)

ữủ s ữủ O(1).
g : Cp Pn (Cp ) ởt ữớ ợ tự
t ồ g = (g0 , ..., gn ). (f, g) ữủ ồ tỹ
f, g = g0 f0 + ... + gn fn = 0

tt r (f, g) tỹ t
Nf (r, g) = N (r,

1
),

f, g

mf (r, g) = log

à(r, f, g )
,
||f(z)||||
g (z)||

õ |z|p = r. ứ ổ tự s t õ ỵ ỡ tự t
Nf (r, g) + mf (r, g) = T (r, f ) + T (r, g) + O(1).
ố t ừ f t g ữủ

f (g) = 1 lim sup
r



Nf (r, g)
T (r, f ) + T (r, g)


❱î✐ q ≥ n, ❝❤♦
gj : Cp −→ Pn (Cp ),

j = 0, ..., q

❧➔ q + 1 ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ✈î✐ ❜✐➸✉ t❤ù❝ t❤✉ ❣å♥
g˜ = (j0 , ..., fjn ) : Cp −→ Cn+1
p .

❍å {gj } ✤÷ñ❝ ❣å✐ ❧➔ ð ✈à tr➼ tê♥❣ q✉→t ♥➳✉ det(gjk l ) = 0 t↕✐ ❜➜t ❦➻ j0 , ..., jn
✈î✐
0 ≤ j0 < ... < jn ≤ q. ◆➳✉ ✈➟②✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣
gj0 = 0,

j = 0, ..., q,

❜➡♥❣ ❝→❝❤ t❤❛② ✤ê✐ ❤➺ t❤è♥❣ tå❛ ✤ë t❤✉➛♥ ♥❤➜t ❝õ❛ Pn (Cp ) ♥➳✉ ❝➛♥ t❤✐➳t✳
❙❛✉ ✤â ✤➦t
ζjk =

gjk
gj0

✈î✐ ζj0 = 1. ●å✐ G ❧➔ tr÷í♥❣ ❝♦♥ ♥❤ä ♥❤➜t ❝❤ù❛
{ζjk |0 ≤ j ≤ q, 0 ≤ k ≤ n} ∪ Cp

❝õ❛ tr÷í♥❣ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ Cp . ✣÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ f ✤÷ñ❝ ❣å✐
❧➔ ❦❤æ♥❣ s✉② ❜✐➳♥ tr➯♥ G ♥➳✉ f0 , ..., fn ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ tr➯♥ G ❈❤ó♥❣
t❛ ❝â q✉❛♥ ❤➺ sè ❦❤✉②➳t ❞÷î✐ ✤➙②✿

✣à♥❤ ❧þ ✶✳✷✳✶✳ ❈❤♦ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤
f, gj : Cp −→ Pn (Cp ),

j = 0, ..., q.

✈î✐ q ≥ n. ◆➳✉ ❤å {gj } ❧➔ ð ✈à tr➼ tê♥❣ q✉→t s❛♦ ❝❤♦

r → ∞,

T (r, gj ) = o(T (r, f )),
✈➔ ♥➳✉ f ❦❤æ♥❣ s✉② ❜✐➳♥ tr➯♥ G ✱ t❤➻
q

δf (gj ) ≤ n + 1.
j=0

✽

j = 0, ..., q


✣à♥❤ ❧þ ✶✳✷✳✷✳ ❈❤♦ V ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì n ❝❤✐➲✉ tr➯♥ Cp . ●å✐ G = {gj }qj=0
❧➔ ❤å ❤ú✉ ❤↕♥ ❝õ❛ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ p✲❛❞✐❝✱ gj : Cp −→ P(V ∗ ), ð
✈à tr➼ tê♥❣ q✉→t ✈î✐ q ≥ n. ▲➜② ♠ët sè ♥❣✉②➯♥ k ✈î✐ 1 ≤ k ≤ n. ●✐↔ sû

f : Cp −→ P(V ) ❧➔ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ p✲❛❞✐❝ k ✲♣❤➥♥❣ tr➯♥ R, s❛♦ ❝❤♦
♠é✐ ❝➦♣ (f, gj ) ❧➔ tü ❞♦ ✈î✐ j = 0, ..., q. ●✐↔ t❤✐➳t r➡♥❣ gj t➠♥❣ ❝❤➟♠ ❤ì♥ f
✈î✐ ♠é✐ j. ❑❤✐ ✤â t❛ ❝â✿
q

δf (gj ) ≤ 2n − k + 1.
j=0

✶✳✸

❳→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝

❈❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ f tr➯♥ C ✈➔ t➟♣ S ⊂ C ∪ {∞}. ❚❛ ✤à♥❤

♥❣❤➽❛
{mz|f (z) = a ✈î✐ ❜ë✐ m},

Ef (S) =
a∈S

✈➔
{z|f (z) = a ❦❤æ♥❣ ❦➸ ❜ë✐}.

E¯f (S) =
a∈S

❚➟♣ S ⊂ C ∪ {∞} ✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝→❝ ❤➔♠ ♣❤➙♥
❤➻♥❤ ✭❯❘❙▼✮ ♥➳✉ ✈î✐ ♠é✐ ❝➦♣ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ ❜➜t ❦➻ f ✈➔ g tr➯♥

C✱ ✤✐➲✉ ❦✐➺♥ Ef (S) = Eg (S) ❦➨♦ t❤❡♦ f = g.

❚➟♣ S ⊂ C∪{∞} ✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝→❝ ❤➔♠ ♥❣✉②➯♥ ✭❯❘❙❊✮
♥➳✉ ✈î✐ ♠é✐ ❝➦♣ ❤➔♠ ♥❣✉②➯♥ ❦❤→❝ ❤➡♥❣ ❜➜t ❦➻ f ✈➔ g tr➯♥ C✱ ✤✐➲✉ ❦✐➺♥
Ef (S) = Eg (S) ❦➨♦ t❤❡♦ f = g.

✣à♥❤ ❧þ ❝ê ✤✐➸♥ ◆❡✈❛♥❧✐♥♥❛ ❝❤➾ r❛ r➡♥❣ f = g ♥➳✉ E¯f (aj ) = E¯g (aj ) ✈î✐
❝→❝ ❣✐→ trà ♣❤➙♥ ❜✐➺t a1 , ..., a5 ✱ ✈➔ r➡♥❣ f ❧➔ ♠ët ❜✐➳♥ ✤ê✐ ▼o¨❜✐✉s ❝õ❛ g ♥➳✉
Ef (aj ) = Eg (aj ) ✈î✐ ❝→❝ ❣✐→ trà ♣❤➙♥ ❜✐➺t a1 , ..., a4 . ●r♦ss ✈➔ ❨❛♥❣ ✤➣ ❝❤➾ r❛

r➡♥❣ t➟♣
S = {z ∈ C|z + ez = 0}

✾

ợ ợ ỳ tỷ ữủ t
t srs ữ
r
M = inf{#S|S },
E = inf{#S|S },

õ #S ỹ ữủ ừ t S. tr ữợ tốt t ữủ t

5 E 7,

6 M 11.

p f tr Cp t õ t
tữỡ tỹ Ef (S) Ef (S) ợ S Cp {} ữ r M E .
ú t ởt số sỹ ỳ ữợ

ờ f p tr Cp f ổ õ ổ
t f

ỵ f g p tr Cp .
a1 , a2 , a3 , a4 ố t tr Cp {}.

Ef (aj ) = Eg (aj ),

j = 1, ..., 4

t f g.

ỵ tt r f g p

tr Cp õ tỗ t tr t a1 , a2 , a3 Cp {} s

Ef (aj ) = Eg (aj ),

j = 1, 2, 3.

õ f g.

strss r r f g p tr
Cp õ tỗ t tr t a1 , a2 Cp s

Ef (aj ) = Eg (aj ),

t f g.



j = 1, 2,


ỵ f t p tr Cp t ổ
tỗ t a Cp s Ef (a) = Ef (a).

ỵ ởt số n 4 ồ a, b Cp {0} s
t

S = {z Cp |z n + az n1 + b = 0}
ự n tỷ t f g t p tr

Cp s Ef (S) = Eg (S), t f g.

ỵ ởt số n 12 ồ a, b Cp {0} s
t

S = {z Cp |z n + az n2 + b = 0}
ự n tỷ t f g p
tr Cp s Ef (S) = Eg (S), t f g.

t õ t t ợ a, b tỗ t ởt
õ h(z) = cz +d ợ c = 0 s h(a) = b h(b) = a t
t S = {a, b} t r ợ ộ f t õ Ef (S) = Ehf (S).
tữỡ tỹ õ t t ố ợ a, b, c
tỗ t ởt s t t h t S = {a, b, c} ởt ổ
t tữớ t ợ ộ f t õ Ef (S) = Ehf (S).
t ssst trữ ữủ s tự
tr trữớ số õ t ự r tr t ổ
st tỗ t s ừ n tỷ ợ n 3 t
n = 3 ồ trữ ở tỷ

ỵ ởt số n 10 b Cp {0, 1}. õ
tự P (z) ữủ

P (z) =

n(n 1) n2
(n 1)(n 2) n
z n(n 2)z n1 +
z
+b
2

2




❝❤➾ ❝â ❦❤æ♥❣ ✤✐➸♠ ✤ì♥✱ ✈➔ ♥➳✉ f ✱ g ❧➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ ❦❤→❝ ❤➡♥❣
tr➯♥ Cp s❛♦ ❝❤♦ Ef (S) = Eg (S) t❤➻ f ≡ g ✱ ð ✤â

S = {z ∈ Cp |P (z) = 0}.

✶✳✹

×î❝ ❧÷ñ♥❣ ❝➜♣ t➠♥❣ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝

❈❤♦ M(Cp ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝õ❛ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ tr➯♥ Cp . ✣à♥❤
♥❣❤➽❛

k

aj (z)wj ,

A(z, w) =

(4)

j=0

ð ✤â aj ∈ M(Cp ) ✈î✐ ak ≡ 0

❇ê ✤➲ ✶✳✹✳✶✳ ◆➳✉ w ∈ M(Cp ) t❤➻

k

N (r, A) = kN (r, w) + O

N (r, aj ) + N (r,
j=0

1
aj

.

(5)

❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ a ∈ Cp ∪ {∞}✱ µaw (z) ❧➔ a✲❣✐→ trà ❜ë✐ ❝õ❛ w t↕✐ z. ❘ã r➔♥❣✱

t❛ ❝â

k

µ∞
A

≤

kµ∞
w

µ∞
aj ,

+
j=0

✈➔ ❞♦ ✤â

k

N (r, A) ≤ kN (r, w) +

N (r, aj ).

(6)

j=0

❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ❞÷î✐ ✤➙②✿
k

µ∞
A

≥

kµ∞
w

0
(µ∞
aj + µaj ).

−k

(7)

j=0

✣à♥❤ ♥❣❤➽❛ bj (z) = aj (z)w(z)j ,

j = 0, ..., k. ❇➙② ❣✐í ❝è ✤à♥❤ z ∈ Cp . ◆➳✉

∞
µ∞
w (z) = 0, ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ rã r➔♥❣ ✤ó♥❣✳ ●✐↔ sû r➡♥❣ µw (z) > 0. ◆➳✉
∞
µ∞
bj (z) < µbk (z) (j < k),

✶✷


t



0

0
à
A (z) = àbk (z) kàw (z) + àak (z) àak (z) kàw (z) àak (z).

tỗ t l < k s

à
bj (z) < àbl (z) (j = l),

t ợ j = k t õ

0


0
kà
w (z) + àak (z) àak (z) < làw (z) + àal (z) àal (z),

t


0
à
w (z) (k l)àw (z) àal (z) + àak (z).

à
bj (z) = àbl (z) ố ợ ởt số j > l, t


0
à
w (z) (j l)àw (z) àal (z) + àaj (z).

õ ữủ s r ữ ởt q t õ
k

N (r, A) kN (r, w) k

N (r, aj ) + N r,

1
aj

m(r, aj ) + m r,

1
aj

j=0

.

(8)

ó r ữủ s r tứ

ờ w M(Cp ), t
k

m(r, A) = km(r, w) + O
j=0

.

(9)

ự ú ỵ r

||à(r, A)|| = |A(z, w(z))|p max {|aj (z)|p |w(z)|jp } = max {à(r, aj )à(r, w)j }
0jk

0jk

õ
à(r, A) max {à(r, aj )à(r, w)j }
0jk




✤ó♥❣ ❝❤♦ t➜t ❝↔ r > 0 ❜ð✐ t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ ❝→❝ ❤➔♠ µ. ❉♦ ✤â t❛ ❝â
m(r, A) ≤ km(r, w) + max m(r, aj ).

(10)

0≤j≤k

▲➜② z ∈ Cp ✈î✐
w(z) = 0, ∞;

✈➔ ✤à♥❤ ♥❣❤➽❛

aj (z) = 0, ∞ (0 ≤ j ≤ k),

|aj (z)|p
A(z) = max {1,
0≤j≤k
|ak (z)|p

1
k−j

}

◆➳✉ |w(z)|p > A(z), t❛ t❤➜②
|aj (z)|p |w(z)|jp ≤ |ak (z)|p A(z)k−j |w(z)|jp < |ak (z)|p |w(z)|kp .

❉♦ ✤â
|A(z, w(z))|p = |ak (z)|p |w(z)|kp .

✣➦t r = |z|p , t❛ ✤÷ñ❝
µ(r, w)k =

µ(r, A)
.
µ(r, ak )

◆➳✉ |w(z)|p ≤ A(z), t❛ ❝â
µ(r, aj )
µ(r, w) ≤ max {1,
0≤jµ(r, ak )
k

k
k−j

}.

❱➻ t❤➳ t❛ ✤÷ñ❝
µ(r, A) µ(r, aj )
||µ(r, w) || ≤ max {1,
,
0≤jµ(r, ak ) µ(r, ak )
k

k
k−j

}.

◆❤÷ ✈➟②✱ ❞♦ t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ ❤➔♠ µ, t❛ ❝â
1
km(r, w) ≤ m(r, A) + km r,
+ k max m(r, aj ).
0≤jak
◆❤÷ ✈➟② ✭✾✮ ✤÷ñ❝ s✉② r❛ tø ✭✶✵✮ ✈➔ ✭✶✶✮✳
❇ê ✤➲ ✶✳✹✳✶ ✈➔ ✶✳✹✳✷ ♠❛♥❣ ❧↕✐ ❦➳t q✉↔ ♥❤÷ s❛✉✿

✶✹

(11)


✣à♥❤ ❧þ ✶✳✹✳✸✳ ◆➳✉ w ∈ M(Cp ), t❤➻
k

T (r, A) = kT (r, w) + O

T (r, aj ) .

(12)

j=0

▲➜② {b0 , ..., bq } ⊂ M(Cp ) ✈î✐ bq ≡ 0 ✈➔ ✤à♥❤ ♥❣❤➽❛
q

bj (z)wj .

B(z, w) =

(13)

j=0

●✐↔ sû r➡♥❣ A(z, w) ✈➔ B(z, w) ❧➔ ❝→❝ ✤❛ t❤ù❝ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ tr♦♥❣
w. ✣à♥❤ ♥❣❤➽❛
R(z, w) =

A(z, w)

.
B(z, w)

(14)

✣à♥❤ ❧þ ✶✳✹✳✹✳ ◆➳✉ w ∈ M(Cp ), t❤➻
q

k

T (r, R) = max{k, q}T (r, w) + O

T (r, aj ) +
j=0

T (r, bj ) .

(15)

j=0

❈❤ù♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ sû deg(A) = k ≥ q = deg(B). ❇➡♥❣ ❝→❝❤

sû ❞ö♥❣ t❤✉➟t t♦→♥ ❝❤✐❛✱ t❛ ❝â
A(z, w) = P1 (z, w)B(z, w) + Q1 (z, w)
deg(P1 ) = k − q,

deg(Q1 ) = t1 < q,

B(z, w) = P2 (z, w)Q1 (z, w) + Q2 (z, w)

deg(P2 ) = q − t1 , deg(Q2 ) = t2 < t1 ,
...
Qm−2 (z, w) = Pm (z, w)Qm−1 (z, w) + Qm (z)
deg(Pm ) = tm−2 − tm−1 ,

deg(Qm ) = tm = 0.

❉♦ A(z, w) ✈➔ B(z, w) ❧➔ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉✱ ♥➯♥ Qm (z) ≡ 0 ✈➔
A(z, w)Q(z, w) + B(z, w)P (z, w) = 1

ð ✤â P (z, w) ✈➔ Q(z, w) ❧➔ ❝→❝ ✤❛ t❤ù❝ ❝õ❛ w s❛♦ ❝❤♦
deg(P ) ≤ k − 1,

deg(Q) ≤ q − 1,

✶✺

(16)


✈➔ s❛♦ ❝❤♦ ❝→❝ ❤➺ sè ❧➔ ❝→❝ ❤➔♠ ❤ú✉ t✛ ❝õ❛ {aj (z)} ✈➔ {bj (z)}. ❈❤ó þ r➡♥❣
k ≥ q.

k + deg(Q) = q + deg(p),

❚❛ ❝ô♥❣ ❝â deg(Q) ≤ deg(P ). ❚ø ✤à♥❤ ❧➼ ✶✳✹✳✶ ✈➔ ✤à♥❤ ❧➼ ❝ì ❜↔♥ t❤ù ♥❤➜t t❛
t❤➜②
Q1
A
≤ T (r, P1 ) + T r,

B
B
B
= T (r, P1 ) + T r,
+ O(1)
Q1

T (r, R) = T r,

≤ T (r, P1 ) + ... + T (r, Pm ) + T r,

Qm−1
+ O(1)
Qm

=(k − q)T (r, w) + (q − t1 )T (r, w) + ... + (tm−1 − tm )T (r, w)
q

k

+O

(17)

T (r, aj ) +
j=0

T (r, bj )
j=0
q

k

=kT (r, w) + O

T (r, aj ) +
j=0

T (r, bj ) .
j=0

❇➙② ❣✐í t❛ sû ❞ö♥❣ ♣❤➨♣ q✉② ♥↕♣✳ ◆➳✉ q = 0, ✤à♥❤ ❧➼ ✶✳✹✳✹ ✤÷ñ❝ s✉② r❛ tø
✤à♥❤ ❧➼ ✶✳✹✳✸✳ ●✐↔ sû ✤à♥❤ ❧➼ ✶✳✹✳✹ ✤ó♥❣ ❝❤♦ ❤➔♠ ❤ú✉ t✛ ❝õ❛ w ✈î✐ ❜➟❝ ❝õ❛
♠➝✉ ≤ q − 1. ❚ø ✭✶✻✮✱ t❛ ❝â
T r,

1
Q B
+
= T r,
P
A
AP

= T (r, AP ) + O(1)
q

k

= (k + deg(P ))T (r, w) + O

T (r, aj ) +
j=0

T (r, bj ) .
j=0

❚ø ❣✐↔ t❤✐➳t q✉② ♥↕♣✱ t❛ ❝â
T r,

P
Q B
+
≤ T r,
+ T (r, R) + O(1)
P
A
Q
q

k

≤ deg(P )T (r, w) + T (r, R) + O

T (r, aj ) +
j=0

✶✻

T (r, bj ) .

j=0


t t ữủ
q

k

kT (r, w) T (r, R) + O

T (r, aj ) +
j=0

T (r, bj )
j=0

t ủ ợ s r ữủ

ỵ w p f M(Cp )
Cp (z), t

T (r, f w)
= +
r T (r, w)
lim

ự f M(Cp ) Cp (z), tỗ t ởt số c Cp s f c = 0

õ ổ số ổ a1 , a2 , ... ợ |aj al |p > 1(j = l). t
f (z) c = (z aj )gj (z),

j = 1, 2, ...

t ợ ộ số ữỡ tỗ t số ữỡ K (< 21 ) s
|gj (z)|p K,

|z aj |p ,

j = 1, ..., .

õ t õ
1
log

|f (z) c|p



+

log+


log+ (K),
|z aj |p

m r,

1
1

log+ log+ (K).
w aj


j=1

z Cp ,

s r
1
m r,

f wc

ú ỵ r



j=1



N r,
j=1

1
w aj

N r,

1
.
f wc

ở t tự tr sỷ ử ỡ tự t t õ
T (r, w) T (r, f w) + O(1)

ứ T (r, w) r ữủ ự




❍➺ q✉↔ ✶✳✹✳✻✳ ❍➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ f tr➯♥ Cp ❧➔ ❤➔♠ ❤ú✉ t✛ ❜➟❝ d ♥➳✉ ✈➔
❝❤➾ ♥➳✉ ✈î✐ ❜➜t ❦➻ ❤➔♠ ♥❣✉②➯♥ p✲❛❞✐❝ ❦❤→❝ ❤➡♥❣ w tr➯♥ Cp ✱ t❛ ❝â

T (r, f ◦ w)
=d
r→∞ T (r, w)
lim

❍➺ q✉↔ ✶✳✹✳✼✳ ❍➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ f tr➯♥ Cp ❧➔ ❤➔♠ ❤ú✉ t✛ ❜➟❝ d ♥➳✉ ✈➔
❝❤➾ ♥➳✉

T (r, f )
=d
r→∞ log r
lim

❍➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ tr♦♥❣ M(Cp ) − Cp (z) ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ s✐➯✉ ✈✐➺t✳ ❘ã
r➔♥❣✱ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ tr♦♥❣ Cp ❧➔ ❤➔♠ s✐➯✉ ✈✐➺t ♥➳✉ ✈➔ ❝❤➾ ♥➳✉

T (r, f )
= +∞.
r→∞ log r
lim

✶✽


❈❤÷ì♥❣ ✷
P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ p✲❛❞✐❝
✷✳✶

P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè p✲❛❞✐❝✳

P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè p✲❛❞✐❝ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣
Ω(z, w, w , ...w(n) ) = R(z, w),

(18)

ð ✤â
Ω(z, w, w , ...wn ) =

ci wi0 (w )i1 ...(w(n) )in

(19)

i∈I

✈➔ i = (i0 , i1 , ..., in ) ❧➔ ❝→❝ ❝❤➾ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠✱ I ❧➔ t➟♣ ❤ú✉ ❤↕♥✱ ci ∈
M(Cp ), ✈➔ R(z, w) ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ tr➯♥ C2p . ✣à♥❤ ♥❣❤➽❛

n

deg(Ω) = max
i∈I

n

iα ,

Γ(Ω) = max
i∈I

α=0

(α + 1)iα ,
α=0

n

γ(Ω) = max
i∈I

αiα .
α=1

❚r÷î❝ t✐➯♥ ❝❤ó♥❣ t❛ ✤÷❛ r❛ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ t♦→♥ tû ✈✐ ♣❤➙♥ Ω. ❈❤♦
w ∈ M(Cp ), t❛ ✈✐➳t t➢t
Ω(z) = Ω(z, w(z), w (z), ..., wn (z)).

❈❤ó þ r➡♥❣

N (r, w(α) ) = N (r, w) + αN (r, w) ≤ (α + 1)N (r, w).

✶✾


❚❛ ❝â
N (r, Ω) ≤ deg(Ω)N (r, w) + γ(Ω)N (r, w) +

N (r, ci )

(20)

i∈I

✈➔
N (r, Ω) ≤ Γ(Ω)N (r, w) +

N (r, ci )

(21).

i∈I

❘ã r➔♥❣✱ t❛ ❝â
n

w(α)
iα m r,
m(r, Ω) ≤ deg(Ω)m(r, w) + max m(r, ci ) +
i∈I

w
α=1

.

(22)

❉♦ ✤â t❛ t❤✉ ✤÷ñ❝ tø ❜ê ✤➲ ✤↕♦ ❤➔♠ ▲♦❣❛r✐t
T (r, Ω) ≤ deg(Ω)T (r, w) + γ(Ω)N (r, w) +

T (r, ci ) + O(1)

(23)

i∈I

✈➔
T (r, Ω) ≤ Γ(Ω)T (r, w) +

T (r, ci ) + O(1)

(24)

i∈I

❚✐➳♣ t❤❡♦✱ t❛ s➩ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ✭✶✽✮ ✈î✐ R(z, w) =

A(z,w)
B(z,w) .

✣à♥❤ ❧➼ ❦✐➸✉

❈❧✉♥✐❡ ❞÷î✐ ✤➙② s➩ ✤â♥❣ ♠ët ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤
❧➼ ❦✐➸✉ ▼❛❧♠q✉✐st✳

❇ê ✤➲ ✷✳✶✳✶✳ ❈❤♦ w ∈ M(Cp ) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✽✮✳ ◆➳✉ q ≥ k, t❤➻
k

m(r, Ω) ≤
i∈I

1
+
m(r, ci ) +
m(r, aj ) + O m r,
b
q
j=0
N (r, ci ) +
i∈I

m(r, bj ) , (25)
j=0

q

k

N (r, Ω) ≤

q

N (r, aj ) + O
j=0

N r,
j=0

1
bj

❈❤ù♥❣ ♠✐♥❤✳ ▲➜② z ∈ Cp ✈î✐

w(z) = 0, ∞;

aj (z) = 0, ∞ (0 ≤ j ≤ k);

ci (z) = 0, ∞ (i ∈ I);

bj (z) = 0, ∞ (0 ≤ j ≤ q),

✷✵

.

(26)