NCKH Về định lý cơ bản thứ hai cho hàm đếm mới

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❚❍⑩■ ◆●❯❨➊◆

❑❍❖❆ ❚❖⑩◆

◆●❯❨➍◆ ❚❍➚ ◆●➴❈ ❉■➏P

❱➋ ✣➚◆❍ ▲Þ ❈❒ ❇❷◆ ❚❍Ù ❍❆■ ❈❍❖ ❍⑨▼ ✣➌▼
▼❰■

✣➋ ❚⑨■ ◆●❍■➊◆ ❈Ù❯ ❑❍❖❆ ❍➴❈

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿ P●❙✳❚❙ ❍⑨ ❚❘❺◆ P❍×❒◆●

❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✻


▼ö❝ ❧ö❝
▼Ð ✣❺❯
✶

✶

▼ð ✤➛✉ ✈➲ ✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤
❤➻♥❤
✶✳✶

✶✳✷

✷

✸

▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✶✳✶

▼ët sè ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✶✳✷

❈→❝ ❤➔♠ ✤➦❝ tr÷♥❣ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

❍❛✐ ✤à♥❤ ❧þ ❝ì ❜↔♥ tr♦♥❣ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ✳ ✳ ✳ ✳

✽

✶✳✷✳✶

✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ♥❤➜t

✽

✶✳✷✳✷

▼ët sè ❞↕♥❣ ❝õ❛ ✤à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❦✐➸✉ ❈❛rt❛♥ ❝❤♦ ❤➔♠ ✤➳♠ ♠î✐

✶✶

✷✳✶

✶✶

✷✳✷

❙è ♠ô ✤➦❝ tr÷♥❣

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✶✳✶

❙è ♠ô ✤➦❝ tr÷♥❣ t↕✐ ❝→❝ ✤✐➸♠ ❤ú✉ ❤↕♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✷✳✶✳✷

❙è ♠ô ✤➦❝ tr÷♥❣ t↕✐

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✷✳✶✳✸

❇ë✐ rót ❣å♥ ✈➔ ❜ë✐ ❞÷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽

✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❦✐➸✉ ❈❛rt❛♥ ❝❤♦ ❤➔♠ ✤➳♠ ♠î✐ ✳ ✳ ✳ ✳

✷✸

✷✳✷✳✶

❍➔♠ ✤➳♠ ♠î✐ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t

✷✸

✷✳✷✳✷

✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❦✐➸✉ ❈❛rt❛♥ ❝❤♦ ❤➔♠ ✤➳♠ ♠î✐
✈➔ ❝❤ù♥❣ ♠✐♥❤

∞

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✐

✸✶



t ữủ t ữợ sỹ ữợ t t ừ P
r Pữỡ tổ tọ ỏ t ỡ s s t t ũ
ổ ữ t tớ ữợ ồ
ởt ró r t tổ tr tổ ỳ
qỵ tr ự ồ t tổ ữủ
tr ờ tự ợ ồ ồ ũ t ổ tr
trữớ ồ ữ t tr ờ r ừ

t ữủ tỹ t t tở trữớ
ồ ữ ồ tổ ỷ ớ ỡ t
tợ t ổ trữớ ồ ữ ồ
t t ổ tr tờ t t ổ ổ t t
t tổ tr sốt q tr ồ t tổ õ
t ữủ t ự t ồ t t ừ


ổ t ỡ s ợ
ổ q t ú ù tổ tr sốt q tr ồ t tỹ
t

tớ ừ t ỏ t
ổ tr ọ ỳ t sõt tổ rt ữủ ỵ
õ õ ừ t ổ ồ t ữủ t ỡ

ổ t ỡ



ồ




▼Ð ✣❺❯
▲þ t❤✉②➳t ♣❤➙♥ ❜è ❣✐→ trà ◆❡✈❛♥❧✐♥♥❛ ✤÷ñ❝ ✤→♥❤ ❣✐→ ♥❤÷ ❧➔ ♠ët tr♦♥❣
♥❤ú♥❣ t❤➔♥❤ tü✉ s➙✉ s➢❝ ✈➔ ✤➭♣ ✤➩ ♥❤➜t ❝õ❛ t♦→♥ ❤å❝ tr♦♥❣ t❤➳ ❦✛ ❳❳✳
✣÷ñ❝ ❤➻♥❤ t❤➔♥❤ tø ♥❤ú♥❣ ♥➠♠ ✤➛✉ ❝õ❛ t❤➳ ❦✛ ❳❳✱ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛
❜➢t ✤➛✉ ❜➡♥❣ ♥❤ú♥❣ ❝æ♥❣ tr➻♥❤ ❝õ❛ ❍❛❞❛♠❛r❞✱ ❇♦r❡❧ ✈➔ ♥❣➔② ❝➔♥❣ ❝â ♥❤✐➲✉
ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ❝õ❛ t♦→♥ ❤å❝✳ ▲þ t❤✉②➳t ♣❤➙♥ ❜è
❣✐→ trà ❧➔ sü tê♥❣ q✉→t ❤♦→ ✤à♥❤ ❧þ ❝ì ❜↔♥ ❝õ❛ ✤↕✐ sè✱ ❝❤➼♥❤ ①→❝ ❤ì♥✳ ▲þ
t❤✉②➳t ♥❣❤✐➯♥ ❝ù✉ sü ♣❤➙♥ ❜è ❣✐→ trà ❝õ❛ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥

C✳ ❚r✉♥❣

t➙♠ ❝õ❛ ❧þ t❤✉②➳t ❧➔ ❤❛✐ ✤à♥❤ ❧þ ❝ì ❜↔♥✳ ✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ♥❤➜t✱ ♠ët ❝→❝❤
✈✐➳t ❦❤→❝ ❝õ❛ ❝æ♥❣ t❤ù❝ P♦✐ss♦♥✲❏❡♥s❡♥✱ ❝❤♦ t❤➜② q✉❛♥ ❤➺ ❣✐ú❛ ❤➔♠ ✤➦❝
tr÷♥❣

Tf (r) ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f

✈î✐ ❤➔♠ ✤➦❝ tr÷♥❣

Tf (r, a) ❝õ❛ ❤➔♠

1
f −a ✳

✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ t❤➸ ❤✐➺♥ ♥❤ú♥❣ ❦➳t q✉↔ ✤➭♣ ✈➔ s➙✉ s➢❝ ♥❤➜t ❝õ❛ ❧þ
t❤✉②➳t✱ ✤÷ñ❝ ♣❤→t ❜✐➸✉ ❞÷î✐ ♥❤✐➲✉ ❞↕♥❣ ❦❤→❝ ♥❤❛✉✿ q✉❛♥ ❤➺ ❣✐ú❛ ❤➔♠ ✤➦❝
tr÷♥❣ ✈î✐ ❝→❝ ❤➔♠ ✤➳♠✱ ❝→❝ ❤➔♠ ✤➳♠ ❜ë✐ ❝➢t ❝öt✱ ❝→❝ ❤➔♠ ①➜♣ ①➾✱ ✳✳✳✳

❑➼ ❤✐➺✉

Pn (C)

❧➔ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤

n

❝❤✐➲✉ tr➯♥ tr÷í♥❣ sè ♣❤ù❝

C✳

▼ët ✈➜♥ ✤➲ tü ♥❤✐➯♥ ✤÷ñ❝ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤➦t r❛ ❧➔✿ ♥❣❤✐➯♥ ❝ù✉ ❧þ t❤✉②➳t
◆❡✈❛♥❧✐♥♥❛ ❝❤✐➲✉ ❝❛♦✱ tù❝ ❧➔ ①➨t ♣❤➙♥ ❜è ❣✐→ trà ❝❤♦ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❣✐ú❛
❝→❝ ✤❛ t↕♣ tr➯♥

C✳

✣➛✉ t✐➯♥ ♣❤↔✐ ❦➸ tî✐ ♥❤ú♥❣ ❝æ♥❣ tr➻♥❤ ❝õ❛ ❍✳ ❈❛rt❛♥

✭❬✸❪✮ ❝æ♥❣ ❜è ✈➔♦ ♥➠♠ ✶✾✸✸✳ ❱➲ s❛✉✱ ✈✐➺❝ t✐➳♣ tö❝ ♣❤→t tr✐➸♥ ❧þ t❤✉②➳t ♣❤➙♥
❜è ❣✐→ trà ❝❤♦ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ù♥❣ ❞ö♥❣ ❝õ❛ ❧þ t❤✉②➳t
✤â tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ❝õ❛ t♦→♥ ❤å❝ ♣❤→t tr✐➸♥ ♠↕♥❤ ♠➩ ✈➔ t❤✉
❤ót ✤÷ñ❝ sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ tr➯♥ t❤➳ ❣✐î✐✳

◆➠♠ ✷✵✵✾✱ ❤❛✐ t→❝ ❣✐↔ ❏✳▼✳ ❆♥❞❡rs♦♥ ✈➔ ❆✳ ❍✐♥❦❦❛♥❡♥ ✤➣ ✤÷❛ r❛ ❦❤→✐
♥✐➺♠ ❤➔♠ ✤➳♠ ♠î✐ ❝❤♦ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ ♠ët ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤
tr♦♥❣

Pn (C)

✈➔ ✤à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❦✐➸✉ ❈❛rt❛♥ ✈î✐ ❤➔♠ ✤➳♠ ♠î✐ ♥➔②✳

❙ü ❧ü❛ ❝❤å♥ ✤➲ t➔✐

✏❱➲ ✤à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❝❤♦ ❤➔♠ ✤➳♠

♠î✐✧ ❝õ❛ ❝❤ó♥❣ tæ✐ ❝ô♥❣ ♥❤➡♠ t✐➳♣ tö❝ ♣❤→t tr✐➸♥ t❤➯♠ ♥❤ú♥❣ ✤✐➲✉ ❧þ t❤ó
❝õ❛ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥✱ ✤➦❝ ❜✐➺t ❧➔ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❞↕♥❣ ✤à♥❤ ❧þ

✶


❝ì ❜↔♥ t❤ù ❤❛✐ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤✳ ✣➲ t➔✐ ✤➣ t➟♣ ❤ñ♣ ♠ët ❝→❝❤ ❝â
❤➺ t❤è♥❣ ♠ët sè ❞↕♥❣ ✤à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤
✤÷ñ❝ ❝æ♥❣ ❜è ❜ð✐ ❝→❝ t→❝ ❣✐↔ tr♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙② ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ❜➡♥❣
t✐➳♥❣ ❱✐➺t ❦➳t q✉↔ ❝õ❛ ❤❛✐ t→❝ ❣✐↔ ❏✳▼✳ ❆♥❞❡rs♦♥ ✈➔ ❆✳ ❍✐♥❦❦❛♥❡♥ tr♦♥❣
❜➔✐ ❜→♦ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✣➲ t➔✐ ✤÷ñ❝ ❝❤✐❛ t❤➔♥❤ ✷ ❝❤÷ì♥❣✿

❈❤÷ì♥❣ ✶ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ tr♦♥❣ ❧þ t❤✉②➳t
◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ✈➔ ❣✐î✐ t❤✐➺✉ ♠ët sè ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ ❞↕♥❣
✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ✈î✐ ❜ë✐ ❝➢t ❝öt ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ✈➔♦
❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤ ❦➳t ❤ñ♣ ✈î✐ ❝→❝ s✐➯✉ ♣❤➥♥❣✱ s✐➯✉ ♠➦t✳

❈❤÷ì♥❣ ✷ ❧➔ ❝❤÷ì♥❣ ❝❤➼♥❤ ❝õ❛ ✤➲ t➔✐✱ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ❝→❝❤ ①➙②
❞ü♥❣ ❤➔♠ ✤➳♠ ♠î✐ ❞ü❛ tr➯♥ ❤➺ t❤è♥❣

sè ♠ô ✤➦❝ tr÷♥❣

✈➔ ❝❤ù♥❣ ♠✐♥❤ ❧↕✐

♠ët ❞↕♥❣ ✤à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❝❤♦ ❤➔♠ ✤➳♠ ♠î✐ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♠ö❝
t✐➯✉ ❧➔ ❝→❝ s✐➯✉ ♣❤➥♥❣ ❝è ✤à♥❤✳


❈❤÷ì♥❣ ✶

▼ð ✤➛✉ ✈➲ ✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐
❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tæ✐ ❣✐î✐ t❤✐➺✉ ♠ët sè ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ✈➲
❝→❝ ❞↕♥❣ ✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ✈î✐ ❜ë✐ ❝➢t ❝öt ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤
✈➔♦ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤ ❦➳t ❤ñ♣ ✈î✐ ❝→❝ s✐➯✉ ♣❤➥♥❣✱ s✐➯✉ ♠➦t✳ ❚r÷î❝ ❤➳t✱
❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ tr♦♥❣ ❧þ t❤✉②➳t ♣❤➙♥ ❜è ❣✐→
trà ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥✳

✶✳✶
✶✳✶✳✶

▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥

▼ët sè ❦❤→✐ ♥✐➺♠

z0 ∈ C ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣
✤✐➸♠ ❜ë✐ k ❝õ❛ f ♥➳✉ tç♥ t↕✐ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ h(z) ❦❤æ♥❣ tr✐➺t t✐➯✉
tr♦♥❣ ♠ët ❧➟♥ ❝➟♥ U ❝õ❛ z0 s❛♦ ❝❤♦ tr♦♥❣ ❧➟♥ ❝➟♥ U ✤â ❤➔♠ f ✤÷ñ❝ ❜✐➸✉
❈❤♦ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤

f : C −→ C✱

✤✐➸♠

❞✐➵♥ ❞÷î✐ ❞↕♥❣

f (z) = (z − z0 )k h(z).

✸




f (z0 ) = f (z0 ) = ã ã ã = f (k1) (z0 ) = 0



f (k) (z0 ) = 0

ợ

z C

t

ordf (z) =




k



z



0



f (z) = 0.

ổ ở

k

ừ

f,

f1

tr õ f1 , f2
f2
ổ õ ổ ố ự z0 ồ ổ
ở k ừ f z0 ổ ở k ừ f1 z0 ồ ỹ ở k ừ
f z0 ổ ở k ừ f2
sỷ

f

ởt õ

f=

ởt tứ

C

ữớ tr ổ

Pn (C) ỏ ồ
Pn (C) ữủ







f = (f0 : ã ã ã : fn ) : C Pn (C)
z (f0 (z) : ã ã ã : fn (z)),

tr õ

fj , 0

tự t

j n, tr C
f ữủ ồ ữớ số

ữớ

s t t



fj , j = 0, . . . , n,

f : C Pn (C)



ữủ ồ

f ự tr ởt t t t tỹ
n
sỹ õ ừ ổ P (C) ữớ f ữủ ồ s
số ừ f ự tr ởt t số tỹ sỹ õ ừ
Pn (C)
ừ

ữớ

f = (f0 : ã ã ã : fn ) : C Pn (C),
f0 , . . . , fn ổ õ ổ tr C
n+1
(f0 , . . . , fn ) : C C
\{0} ởt tố ừ

tr õ
ồ

f
s ữủ s r trỹ t tứ ú tổ sỷ
ử ự ữớ ỗ t




❈❤♦ f, g : C −→ Pn (C) ❧➔ ❤❛✐ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ❦❤→❝
❤➡♥❣ ✈➔ (f0 , . . . , fn )✱ (g0 , . . . , gn ) ❧➛♥ ❧÷ñt ❧➔ ❝→❝ ❜✐➸✉ ❞✐➵♥ tè✐ ❣✐↔♥ ❝õ❛ f ✱
g ✳ ❑❤✐ ✤â f ≡ g ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ fi gj ≡ fj gi ✈î✐ ♠å✐ ❝➦♣ ❝❤➾ sè ♣❤➙♥ ❜✐➺t
i, j ∈ {0, . . . , n}✳

▼➺♥❤ ✤➲ ✶✳✶✳✹✳

✶✳✶✳✷

❈→❝ ❤➔♠ ✤➦❝ tr÷♥❣ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥
❚✐➳♣ t❤❡♦ t❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ✤➦❝ tr÷♥❣✱ ❤➔♠ ①➜♣ ①➾✱ ❤➔♠ ✤➳♠ ❝õ❛

✤÷í♥❣ ❝♦♥❣ ❦➳t ❤ñ♣ ✈î✐ ❝→❝ s✐➯✉ ♠➦t ❝è ✤à♥❤✳ ❈❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤

f : C −→ Pn (C)

✈➔ ♠ët ❜✐➸✉ ❞✐➵♥ tè✐ ❣✐↔♥

(f0 , . . . , fn )

❝õ❛

f✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳ ✭❬✸❪✮ ❍➔♠

1
Tf (r) =
2π
✤÷ñ❝ ❣å✐ ❧➔
❝õ❛

f✱

2π

log f (reiθ ) dθ
0

❤➔♠ ✤➦❝ tr÷♥❣ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥

✭❤❛② ❤➔♠ ✤ë ❝❛♦ ❈❛rt❛♥✮

tr♦♥❣ ✤â

f (z) = max{|f0 (z)|, . . . , |fn (z)|}.
●✐↔ sû

D

❧➔ ♠ët s✐➯✉ ♠➦t ✭❝è ✤à♥❤✮ ❜➟❝

✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t

d

tr♦♥❣

Pn (C)✱

①→❝ ✤à♥❤ ❜ð✐

Q✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻✳ ✭❬✼❪✱ ❬✺❪✱ ❬✶❪✮ ❍➔♠

1
mf (r, D) = mf (r, Q) :=
2π
✤÷ñ❝ ❣å✐ ❧➔

❤➔♠ ①➜♣ ①➾

❝õ❛

f

2π
0

f (reiθ ) d
log
dθ
|Q(f )(reiθ )|

❦➳t ❤ñ♣ ✈î✐ s✐➯✉ ♠➦t

D✳

❑➼ ❤✐➺✉ nf (r, D) ❧➔ sè ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ Q ◦ f tr♦♥❣ ✤➽❛ |z| < r ✱ ❦➸ ❝↔
M
❜ë✐✱ nf (r, D) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ Q ◦ f tr♦♥❣ ✤➽❛ |z| < r ✱ ❜ë✐ ❝➢t ❝öt ❜ð✐
♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣

M✳

◆❣❤➽❛ ❧➔

nf (r, D) =

ordQ◦f (z);
z∈C,|z|

nM
f (r, D) =

min{M, ordQ◦f (z)}.
z∈C,|z|
✺


❚❛ ❦➼ ❤✐➺✉

nf (0, D) = ordQ◦f (0);
nM
f (0, D) = min{M, ordQ◦f (0)}.
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼✳ ✭❬✼❪✱ ❬✺❪✱ ❬✶❪✮ ❍➔♠
r

Nf (r, D) = Nf (r, Q) :=
0
✤÷ñ❝ ❣å✐ ❧➔

❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐

✈➔ ❤➔♠

r

NfM (r, D)

=

NfM (r, Q)

:=
0

✤÷ñ❝ ❣å✐ ❧➔
♠➦t

❤➔♠ ✤➳♠ ❜ë✐ ❝➢t ❝öt

D✳ ❙è M

nf (t, D) − nf (0, D)
dt + nf (0, D) log r
t
M
nM
f (t, D) − nf (0, D)
dt + nM
f (0, D) log r
t
❜ð✐

M

❝õ❛ ✤÷í♥❣ ❝♦♥❣

f

❦➳t ❤ñ♣ ✈î✐ s✐➯✉

❝❤➾ sè ❜ë✐ ❝➢t ❝öt✳ ❚r÷í♥❣
1
❝❤♦ Nf (r, D) ✈➔ ❣å✐ ❧➔ ❤➔♠

M
tr♦♥❣ ❦➼ ❤✐➺✉ Nf (r, D) ✤÷ñ❝ ❣å✐ ❧➔

M = 1✱ t❛ ✈✐➳t N f (r, D) t❤❛②

❤ñ♣ ✤➦❝ ❜✐➺t✱ ♥➳✉

✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐✳

f : C −→ Pn (C) ❧➔ ♠ët ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤✱ D ❧➔ ♠ët s✐➯✉
n
♠➦t ❜➟❝ d tr♦♥❣ P (C) ✈➔ Q ❧➔ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ d ①→❝ ✤à♥❤ D ✳ ❱î✐
♠é✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ k ✱ ❦➼ ❤✐➺✉ nf (r, D,
k) ❧➔ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝â ❜ë✐
♥❤ä ❤ì♥ ❤❛② ❜➡♥❣ k ❝õ❛ Q ◦ f tr♦♥❣ ✤➽❛ |z| < r ✈➔ nf (r, D, > k) ❧➔ sè ❦❤æ♥❣
✤✐➸♠ ❝â ❜ë✐ ➼t ♥❤➜t ❜➡♥❣ k + 1 ❝õ❛ Q ◦ f tr♦♥❣ ✤➽❛ |z| < r ✳ ❈→❝ ❤➔♠ ✤➳♠
●✐↔ sû

✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉

r

Nf, k (r, D) =

0
r

Nf,>k (r, D) =
0

k) − nf (0, D, k)
dt + nf (0, D, k) log r;
t
nf (t, D, > k) − nf (0, D, > k)
dt + nf (0, D, > k) log r,
t

nf (t, D,

nf (0, D, k) = ordQ◦f (0) ♥➳✉ ordQ◦f (0) k ✱ ❜➡♥❣ 0 tr♦♥❣ tr÷í♥❣
❤ñ♣ ♥❣÷ñ❝ ❧↕✐✱ nf (0, D, > k) = ordQ◦f (0) ♥➳✉ ordQ◦f (0) > k ✱ ❜➡♥❣ 0 tr♦♥❣
tr÷í♥❣ ❤ñ♣ ♥❣÷ñ❝ ❧↕✐✳ ❱î✐ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ ∆, k ✳ ❚❛ ❦➼ ❤✐➺✉
tr♦♥❣ ✤â

n∆
f (r, D,

k) =

min{ordQ◦f (z), ∆};
|z| r;ordQ◦f (z) k

n∆
f (r, D, > k) =

min{ordQ◦f (z), ∆}.
|z| r;ordQ◦f (z)>k

✻


❈→❝ ❤➔♠ ✤➳♠ ❜ë✐ ❝➢t ❝öt ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉

r

Nf,∆ k (r, D)

=
0
r

∆
Nf,>k
(r, D)

=
0

n∆
f (t, D,

k) − n∆
k)
f (0, D,

dt + n∆
k) log r;
f (0, D,
t
∆
n∆
f (t, D, > k) − nf (0, D, > k)
dt + n∆
f (0, D, > k) log r,
t

n∆
f (0, D,

k) = min{ordQ◦f (0), ∆} ♥➳✉ ordQ◦f (0)
k ✱ ❜➡♥❣
0 tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥❣÷ñ❝ ❧↕✐✱ n∆
f (0, D, > k) = min{ordQ◦f (0), ∆} ♥➳✉
ordQ◦f (0) > k ✱ ❜➡♥❣ 0 tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥❣÷ñ❝ ❧↕✐✳
tr♦♥❣ ✤â

❇ê ✤➲ s❛✉ s➩ ❝❤♦ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ ✤➳♠ ✈➔ ❤➔♠ ✤➳♠ ❜ë✐
❝➢t ❝öt✳

❇ê ✤➲ ✶✳✶✳✽✳

❱î✐ ❝→❝ ❣✐↔ t❤✐➳t ✈➔ ❦➼ ❤✐➺✉ ♥❤÷ tr➯♥✱ t❛ ❝â

1) Nf (r, D) = Nf, k (r, D) + Nf,>k (r, D);
∆

2) Nf∆ (r, D) = Nf,∆ k (r, D) + Nf,>k
(r, D);

3) Nf∆ (r, D)

Nf (r, D);

4) Nf1 (r, D)

Nf∆ (r, D)

∆Nf1 (r, D);

5) Nf,1 k (r, D)

Nf,∆ k (r, D)

∆Nf,1 k (r, D);

1
6) Nf,>k
(r, D)

∆
Nf,>k
(r, D)

1
∆Nf,>k
(r, D);

7)

1
∆
Nf,∆ k (r, D) + Nf,>k
(r, D)
k+1

∆
Nf (r, D).
k+1

●✐↔ sû f = (f0 : · · · : fn ) : C −→ Pn (C) ❧➔ ♠ët ✤÷í♥❣ ❝♦♥❣
❝❤➾♥❤ ❤➻♥❤ ❦❤→❝ ❤➡♥❣✱ ❦❤✐ ✤â
❇ê ✤➲ ✶✳✶✳✾✳

Nfj (r, 0)

Tf (r) + O(1)

✈î✐ ♠é✐ j = 0, . . . , n, tr♦♥❣ ✤â Nfj (r, 0) ❧➔ ❤➔♠ ✤➳♠ t↕✐ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛
❤➔♠ fj ✳

f : C −→ Pn (C) ❧➔ ♠ët ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ✈➔ D
n
❜➟❝ d tr♦♥❣ P (C) ①→❝ ✤à♥❤ ❜ð✐ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t Q✳

●✐↔ sû
s✐➯✉ ♠➦t

✼

❧➔ ♠ët


✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✵✳ ❙è

δf (D) = δf (Q) = 1 − lim sup
r−→+∞
✤÷ñ❝ ❣å✐ ❧➔

sè ❦❤✉②➳t

✈➔ sè

δfM (D)
✤÷ñ❝ ❣å✐ ❧➔
tr♦♥❣ ✤â

M

Nf (r, Q)
,
dTf (r)

=

δfM (Q)

sè ❦❤✉②➳t ❜ë✐ ❝➢t ❝öt

NfM (r, Q)
,
= 1 − lim sup
dTf (r)
r−→+∞

❝õ❛ ✤÷í♥❣ ❝♦♥❣

f

❦➳t ❤ñ♣ ✈î✐ s✐➯✉ ♠➦t

D✱

f : C −→ Pn (C)✱

t❛

❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✳

❉➵ t❤➜② r➡♥❣✱ ✈î✐ ♠é✐ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤
❧✉æ♥ ❝â

0
✈î✐ ♠å✐ sè ♥❣✉②➯♥ ❞÷ì♥❣

✶✳✷

δfM (D)

δf (D)

M

✈➔ s✐➯✉ ♠➦t

1

D✳

❍❛✐ ✤à♥❤ ❧þ ❝ì ❜↔♥ tr♦♥❣ ❧þ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥

✶✳✷✳✶

✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ♥❤➜t

(z0 : · · · : zn ) ❧➔ ❤➺ tå❛ ✤ë t❤✉➛♥ ♥❤➜t tr♦♥❣ Pn (C)✳ ❈❤♦ ✤❛
t↕♣ ✤↕✐ sè ①↕ ↔♥❤ X ❝â sè ❝❤✐➲✉ ❜➡♥❣ k ✱ k
n ✈➔ ♠ët ❤å ❣ç♠ q s✐➯✉ ♠➦t
n
D = {D1 , . . . , Dq } tr♦♥❣ P (C)✱ tr♦♥❣ ✤â Dj ①→❝ ✤à♥❤ ❜ð✐ ✤❛ t❤ù❝ t❤✉➛♥
♥❤➜t Qj tr♦♥❣ C[z0 , . . . , zn ], j = 1, . . . , q ✳ ❱î✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ N
k ✱ t❛
❑➼ ❤✐➺✉

✤à♥❤ ♥❣❤➽❛ ❦❤→✐ ♥✐➺♠ ❤å ❝→❝ s✐➯✉ ♠➦t ð ✈à tr➼ tê♥❣ q✉→t ♥❤÷ s❛✉✿

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❍å

D

❝→❝ s✐➯✉ ♠➦t tr♦♥❣

Pn (C)

N ✲tê♥❣ q✉→t ✤è✐ ✈î✐ ✤❛ t↕♣ X ♥➳✉ q N + 1 ✈➔
s✐➯✉ ♠➦t Dj1 , . . . , DjN +1 tr♦♥❣ ❤å D ✱ t❛ ❧✉æ♥ ❝â

✤÷ñ❝ ❣å✐ ❧➔

ð ✈à tr➼

✈î✐ ♠å✐ ❝→❝❤ ❝❤å♥

N +1

{x ∈ X|Qi1 (x) = · · · = QiN +1 (x) = 0} = ∅.
N = k ✱ t❛ ♥â✐ ❤å ❝→❝ s✐➯✉ ♠➦t D ð ✈à tr➼ tê♥❣ q✉→t ✤è✐
N = k = n✱ t❛ ♥â✐ ❤å D ð ✈à tr➼ tê♥❣ q✉→t ✭✤è✐ ✈î✐ Pn (C))✳

✣➦❝ ❜✐➺t✱ ♥➳✉

X✳

◆➳✉

✈î✐

❍å ❝→❝ s✐➯✉ ♣❤➥♥❣ {Hj , q = 1, . . . , q} ð ✈à tr➼ tê♥❣ q✉→t
✤è✐ ✈î✐ P (C) ♥➳✉ q > n ✈➔ n + 1 s✐➯✉ ♣❤➥♥❣ ❜➜t ❦ý tr♦♥❣ ❝❤ó♥❣ ✤➲✉ ❧➔ ✤ë❝
❧➟♣ t✉②➳♥ t➼♥❤✳
◆❤➟♥ ①➨t ✶✳✷✳✷✳
n

✽


sỷ f : C Pn (C)
ởt ữớ D ởt s t d tr Pn (C)
f (C) D t ợ ộ số tỹ ữỡ r t õ

ỵ ỵ ỡ tự t

mf (r, D) + Nf (r, D) = dTf (r) + O(1),
tr õ O(1) ởt số ở ợ r


ởt số ừ ỵ ỡ tự
rt ự

ữớ ổ s t t
f : C P (C) q s H1 , . . . , Hq tr tờ qt tr Pn (C)
õ ợ ộ > 0 ợ ộ r > 0 ừ ợ ởt t õ ở
s ỳ
ỵ
n

q

Nfn (r, Hj ) + O(1).

(q n 1 )Tf (r)
j=1

t q tr ừ rt ởt ỵ ỡ tự ợ
ở t ửt ữớ tứ

C



Pn (C)

ổ s

t t t ủ ợ s tr tờ qt ổ tr
ừ ổ ữủ t sự q trồ õ r ởt ữợ
ự ợ tr t tr ỵ tt ố tr ự sỹ
ố tr ừ ồ ỵ

tt

rt s õ t ổ ố ỳ ổ
tr ỵ ỡ tự rt
ự

ữớ ổ s t t
f = (f0 : ã ã ã : fn ) : C Pn (C) q s t H1 , . . . , Hq tr

Pn (C) ồ Lj , 1 j
q, t t s Hj
tữỡ ự W rs ừ f õ
ỵ

2

max log
0

K

jK

f (rei ) Lj d
+ NW (r, 0)
|Lj (f )(rei )| 2



(n + 1)Tf (r) + o(Tf (r)),


tr õ ữủ tr tt t K ừ {1, . . . , q} s
t t Lj , j K ở t t Lj tr
ợ t ừ ổ số tr Lj
ự

ởt ữớ ổ s số
f : C P (C) ởt ồ s t Dj , 1 j q, tr Pn (C) õ

dj tữỡ ự tr tờ qt õ ợ ộ > 0, t õ
ỵ
n

q

d1
j mf (r, Dj ) (n + 1 + )Tf (r).
j=1
ự

ởt ữớ ổ s số
f : C P (C) ởt ồ s t Dj , 1 j q, tr Pn (C) õ
dj tữỡ ự tr tờ qt õ ợ ồ số tỹ ữỡ > 0 tỗ
t ởt số ữỡ M s
ỵ
n

q
M
d1
j Nf (r, Dj ) + o(Tf (r)),

(q (n + 1) )Tf (r)
j=1

tr õ t tự tr ú ợ ồ r > 0 ừ ợ ởt t
t ủ õ ở s ỳ
Pữỡ ự

ởt ữớ ổ s số
f : C P (C) ởt ồ s t Dj , 1 j q, tr Pn (C) õ
dj tữỡ ự tr tờ qt ồ d ở số ọ t ừ dj
ỵ
n

ợ 0 < < 1 M

n

2d 2n (n + 1)n(d + 1)1 , t õ
q
M
d1
j Nf (r, Dj ),

(q (n + 1) )Tf (r)



j=1

tr õ t tự tr ú ợ ồ r > 0 ừ ợ ởt t
t ủ õ ở s ỳ




❈❤÷ì♥❣ ✷

✣à♥❤ ❧þ ❝ì ❜↔♥ t❤ù ❤❛✐ ❦✐➸✉ ❈❛rt❛♥
❝❤♦ ❤➔♠ ✤➳♠ ♠î✐
P❤➛♥ ❦✐➳♥ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷ñ❝ tr➻♥❤ ❜➔② ❧↕✐ t❤❡♦ þ ❤✐➸✉ ❝õ❛
t→❝ ❣✐↔✱ ❞ü❛ tr➯♥ ❜➔✐ ❜→♦✳✳✳✳✳✳✳✳✳✳✳✳

✷✳✶

❙è ♠ô ✤➦❝ tr÷♥❣
❚r÷î❝ t✐➯♥ t❛ ✤✐ ①➙② ❞ü♥❣ ❤➺ t❤è♥❣

❤ñ♣

z

✷✳✶✳✶

t❤✉ë❝

C

❤ú✉ ❤↕♥ ✈➔

z = ∞✱

sè ♠ô ✤➦❝ tr÷♥❣

❜➢t ✤➛✉ tø

z

❝❤♦ ❝→❝ tr÷í♥❣

❤ú✉ ❤↕♥✳

❙è ♠ô ✤➦❝ tr÷♥❣ t↕✐ ❝→❝ ✤✐➸♠ ❤ú✉ ❤↕♥
❈❤♦

f0 , f1 , ..., fn ✱ n ∈ N∗

❧➔ ❝→❝ ❤➔♠ ♥❣✉②➯♥✱ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✱

❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ tr➯♥
❲r♦♥s❦✐❛♥ ❝õ❛

C✳

❑➼ ❤✐➺✉

W (f0 , f1 , ..., fn )

f0 , f1 , ..., fn ✳

W = W (f0 , f1 , . . . , fn ) =

f0

f1

...

fn

f0

f1

...

fn

✳✳
✳

✳✳
✳

...

✳✳
✳

(n)

f0
✶✶

(n)

f1

(n)

. . . fn

.

❧➔ ✤à♥❤ t❤ù❝


❱î✐

z0 ∈ C✱

❦➼ ❤✐➺✉

W (z0 ) =

f0

f1

...

fn

f0

f1

...

fn

✳✳
✳

✳✳
✳

...

✳✳
✳

(n)

f0

(n)

f1

❧➔ ✤à♥❤ t❤ù❝ ❲r♦♥s❦✐❛♥ ❝õ❛ ❝→❝ ❤➔♠

▼➺♥❤ ✤➲ ✷✳✶✳✶✳

(z0 ).

(n)

. . . fn

f0 , f1 , ..., fn

t↕✐

z0 ✳

◆➳✉ W (z0 ) = 0 t❤➻ ✈î✐ ♠é✐ k t❤✉ë❝ {1, ..., n} ❧✉æ♥ tç♥ t↕✐
n

♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ g =

aj fj ❝õ❛ ❝→❝ ❤➔♠ f0 , f1 , . . . , fn s❛♦ ❝❤♦ g ❝â

j=0

❦❤æ♥❣ ✤✐➸♠ ❜ë✐ k t↕✐ z0 ✳ ◆❣♦➔✐ r❛ ♠å✐ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ t➛♠ t❤÷í♥❣
g ❝õ❛ ❝→❝ ❤➔♠ f0 , f1 , . . . , fn ❝❤➾ ❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ ❦❤æ♥❣ ✈÷ñt q✉→ n t↕✐ z0 ✳
❈❤ù♥❣ ♠✐♥❤✳

❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤✿

n
(m)

aj f j

(z0 ) = 0, 0 ≤ m ≤ k − 1, k ∈ {1, 2, . . . , n}

✭✷✳✶✮

j=0
❚❛ t❤➜② ✭✷✳✶✮ ❧➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t ❝õ❛

n+1

➞♥

n

a0 , a1 , . . . , an ✳

✣➸ ❤➔♠

g=

aj f j

❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐

k

z0

t↕✐

t❤➻ ❤➺

(2.1)

j=0
♣❤↔✐ ❝â ♥❣❤✐➺♠ ❦❤æ♥❣ t➛♠ t❤÷í♥❣

(a0 , a1 , . . . , an )

t❤ä❛ ♠➣♥

n
(k)

aj fj (z0 ) = 0.

✭✷✳✷✮

j=0

W (z0 ) = 0 ♥➯♥ rankW (z0 ) = n+1✱ tø ✤â k ❞á♥❣ ❜➜t ❦➻ tr♦♥❣ W (z0 ) ✤➲✉
✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✱ ✤➦❝ ❜✐➺t k ❞á♥❣ ✤➛✉ t✐➯♥ ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ❙✉② r❛
✭✷✳✶✮ ❧✉æ♥ ❝â ♥❣❤✐➺♠ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ (a0 , a1 , . . . , an ) ♥➳✉ 1 ≤ k ≤ n✳
❉♦ ❞á♥❣ t❤ù k + 1 ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈î✐ k ❞á♥❣ ✤ù♥❣ tr÷î❝ ♥➯♥ ❝â
t❤➸ ❝❤å♥ ✤÷ñ❝ (a0 , a1 , . . . , an ) t❤ä❛ ♠➣♥ ✭✷✳✶✮ ✈➔ ✭✷✳✷✮✳❱➟② ❧✉æ♥ ❝❤å♥ ✤÷ñ❝
❉♦

n

(a0 , a1 , . . . , an )

s❛♦ ❝❤♦ ❤➔♠

g=

aj f j
j=0

t↕✐

z0 ✳
✶✷

❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐

k ✱ (1 ≤ k ≤ n)


n
◆❣♦➔✐ r❛ ✈î✐ ♠å✐ tê ❤ñ♣ t✉②➳♥ t➼♥❤

aj f j

g =

♠➔

g(z0 ) = 0✱

j=0

W (z0 ) = 0

t❤➻ r❛♥❦W (z0 )

=n+1

♥➯♥ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✿

n
(m)

aj f j

(z0 ), tr♦♥❣

✤â

0 ≤ m ≤ k,

j=0
s➩ ❝❤➾ ❝â ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣ ❦❤✐

k ≥ n + 1✳

❑❤✐ ✤â

g

❧➔ tê ❤ñ♣ t✉②➳♥ t➼♥❤

t➛♠ t❤÷í♥❣✳ ❱➟② ♠å✐ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ ❝❤➾ ❝â ♥❣❤✐➺♠
❜ë✐ ❦❤æ♥❣ ✈÷ñt q✉→

n

t↕✐

z0 ✳

❇➙② ❣✐í t❛ ①➨t tr÷í♥❣ ❤ñ♣

W (z0 ) = 0✳

f0 , f1 , . . . , fn ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ tr➯♥ C ♥➯♥ ✈î✐ z0 ∈ C
fj (z)
tç♥ t↕✐ ♠ët fj ✱ j ∈ {0, 1, . . . , n} s❛♦ ❝❤♦ fj (z0 ) = 0✳ ✣➦t g0 (z) =
✱ ❦❤✐
fj z0
✤â g0 (z0 ) = 1✳
●✐↔ sû tç♥ t↕✐ ❤➔♠ fi ✱ i ∈ {0, 1, ..., n} \ {j} s❛♦ ❝❤♦ fi (z0 ) = 0✱ t❛ ①➨t
❤➔♠ hi ♠î✐ ♥❤÷ s❛✉✿
❉♦ ❝→❝

hi = fi (z) − fi (z0 )g0 (z)
fj (z)
= fi (z) − fi (z0 )
fj (z0 )
t❤ä❛ ♠➣♥

hi (z0 ) = 0✳

●✐↔ sû

ordhi (z0 ) = m ≥ 1✱

❦❤✐ ✤â t❛ ❝â ❜✐➸✉ ❞✐➵♥ ❝õ❛

hi (z)✿
hi (z) = (z − z0 )m ti (z)
= (z − z0 )m (c0 + c1 (z − z0 ) + O((z − z0 )2 ))
= c0 (z − z0 )m + c1 (z − z0 )m+1 + O((z − z0 )m+2 ),
❚❛ t❤❛② t❤➳ ❤➔♠

z0

fi

❜ð✐ ❤➔♠

fi

♠î✐✿

fi =

1
hi ✱
c0

❧ó❝ ♥➔②

fi

✈î✐

c0 = 0.

s➩ tr✐➺t t✐➯✉ t↕✐

✈➔

fi (z) = (z − z0 )m + O((z − z0 )m+1 ),
O((z − z0 )m+1 )

z − z0 ❜➟❝ ❝❛♦ ❤ì♥ ♠✳
▲➦♣ ❧↕✐ q✉→ tr➻♥❤ ♥➔② ✈î✐ ♠å✐ ❤➔♠ fi (i = j) ♠➔ fi (z0 ) = 0✱ ❦❤✐ ✤â t❛
✤÷ñ❝ n ❤➔♠ ♠➔ ♠é✐ ❤➔♠ tr♦♥❣ ✤â ❤♦➦❝ ❧➔ ❤➔♠ fi ❜❛♥ ✤➛✉✱ ❤♦➦❝ ❧➔ ❤➔♠ fi
✤➣ t❤❛② t❤➳ ♥❤÷ tr➯♥✳ n ❤➔♠ ♥➔② ❝ò♥❣ ✈î✐ ❤➔♠ g0 s➩ t↕♦ t❤➔♥❤ n + 1 ❤➔♠

tr♦♥❣ ✤â

❧➔ ❤➔♠ ❝õ❛

✶✸


g0 (z0 ) = 1 fi (z0 ) = 0

ở t t tọ
rfi (z0 )

n

= 0

1

ỹ
tr

rg0 (z0 )



fi

số ụ trữ

tỗ t
rfk (z0 )

fk

ừ

(f0 , f1 , . . . , fn )

ữ s

s

= min rfk (z0 ) = d1 1,

i=j

õ

fk = ad1 (z z0 )d1 + ad1 +1 (z z0 )d1 +1 + O((z z0 )d1 +2 ), tr
t

g1 =

1
fk
ad1

ad1 = 0.

ữủ

g1 (z) = (z z0 )d1 + O((z z0 )d1 +1 )
tr

õ

n1

ỏ trứ

ở ú

d1

t

z0

fk

ợ

d1 1

tỗ t

t t t t

fl

t ủ s ợ õ ổ ở t t

fl

õ õ ổ

fl g1 = 0
d1 + 1 t z0



ử t sỷ

fl (z) = bd1 (z z0 )d1 + bd1 +1 (z z0 )d1 +1 + O((z z0 )d1 +2 ), ợ bd1 = 0,

fl bd1 g1 t ữủ fl bd1 g1 = O((z z0 )d1 +1 )
õ ổ ở t t d1 + 1 t z0
q tr tr ợ n 1 trứ g0 fk ữủ t t
g1 t t ữủ n + 1 g0 , g1 , . . . , gn õ ổ ở di t z0 t
t ởt 0 = d0 < d1 < d2 < . . . < dn g0 , g1 , . . . , gn õ
t

fl



t t

gj

ộ

ởt tờ ủ t t ổ t tữớ ừ

(f0 , f1 , . . . , fn )


n+1



ợ ộ
ổ

{g0 , g1 , . . . , gn }

ở t t tr

j {0, 1, . . . , n} gj (z) = (z z0 )dj + O((z z0 )dj +1 ) gj
ở dj t z0

ồ số
ừ

C

(f0 , . . . , fn )

t

z0

dj

ữ tr

ró r t õ t t

õ ởt tữỡ tỹ ữ



õ

số ụ trữ

dj = dj (z0 )

W (z0 ) = 0


▼➺♥❤ ✤➲ ✷✳✶✳✸✳

●✐↔ sû g =

n

aj fj ❧➔ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ t➛♠

j=0

t❤÷í♥❣ ❝õ❛ ❝→❝ ❤➔♠ f0 , . . . , fn ✳ ❑❤✐ ✤â ♥➳✉ g ❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ m t↕✐ z0
t❤➻ m ✤ó♥❣ ❜➡♥❣ ♠ët tr♦♥❣ sè ❝→❝ dj ✱ j ∈ {0, 1, . . . , n}✱ tr♦♥❣ ✤â ❝→❝ sè dj
❧➔ sè ♠ô ✤➦❝ tr÷♥❣ ❝õ❛ (f0 , . . . , fn ) ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ tr➯♥✳
n

❈❤ù♥❣ ♠✐♥❤✳

❉♦

aj fj

g=

❧➔ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ t➛♠ t❤÷í♥❣

j=0
n
❝õ❛ ❝→❝ ❤➔♠

f0 , . . . , f n

♥➯♥ t❛ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥

kj gj ✱ ❝→❝ gj

g=

①→❝ ✤à♥❤

j=0

kj = 0✱ (0 ≤ j ≤ n), tù❝ ❧➔ g ❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ ♠
✤ó♥❣ ❜➡♥❣ ♠ët tr♦♥❣ sè ❝→❝ dj t↕✐ z0 ✳ Ð ✤➙② ♥➳✉ g(z0 ) = 0 t❤➻ m = 0 ✈➔
k0 = 0✱ tù❝ ❧➔ g ❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ d0 = 0 t↕✐ z0 ✳

♥❤÷ tr➯♥✱ ✈➔ tç♥ t↕✐

❚❤❡♦ ❦➳t qõ❛ tø ✷✳✶✳✶ ✈➔ ✷✳✶✳✸ t❛ ❝â ❜ê ✤➲ s❛✉✿

❈❤♦ n + 1 ❤➔♠ ♥❣✉②➯♥ f0 , . . . , fn ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣
tr➯♥ C✱ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ tr➯♥ C ✈➔ z0 ∈ C✳ ❑➼ ❤✐➺✉ £ = £(f0 , . . . , fn ) ❧➔
t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ f0 , . . . , fn ✳ ❑❤✐
✤â✿
❇ê ✤➲ ✷✳✶✳✹✳

✭✶✮ ◆➳✉ W (z0 ) = 0✱ ❝→❝ ❜➟❝ ❝â t❤➸ ❝â t❤✉ë❝ £ t↕✐ z0 t↕♦ ♥➯♥ ♠ët ❞➣②
0 = d0 < d1 < . . . < dn = n✳
✭✷✮ ◆➳✉ W (z0 ) = 0✱ ❝→❝ ❜➟❝ ❝â t❤➸ ❝â ❝õ❛ ❤➔♠ t❤✉ë❝ £(f0 , f1 , . . . , fn ) t↕✐
z0 t↕♦ ♥➯♥ ♠ët ❞➣② 0 = d0 < d1 < . . . < dn ✱ tr♦♥❣ ✤â dj = dj (z0 ) ❧➔ ❝→❝ sè
♠ô ✤➦❝ tr÷♥❣ ❝õ❛ f0 , . . . , fn t↕✐ z0 ✳
❇➙② ❣✐í t❛ ①➨t tr÷í♥❣ ❤ñ♣

✷✳✶✳✷

❙è ♠ô ✤➦❝ tr÷♥❣ t↕✐

z = ∞✳
∞

f0 , . . . , fn ❧➔ ❝→❝ ✤❛ t❤ù❝ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✱
✤✐➸♠ ❝❤✉♥❣ tr➯♥ C✳ ✣➦t d = max{deg fj | 0 ≤ j ≤ n}✳
tr♦♥❣ sè {0, . . . , n} s❛♦ ❝❤♦ deg fj0 = d✳ ❑❤✐ ✤â
●✐↔ sû

fj = ad z d + . . . + a0 , tr♦♥❣

✶✺

✤â

ad = 0.

❦❤æ♥❣ ❝â ❦❤æ♥❣
❈❤å♥

j0

❧➔ ♠ët


t

1
fj = fj
ad

g0 =

s r

g0 = z d + o(z d1 ).
i = j0 i {0, . . . , n}

tỗ t

(fj g1 )

deg fi = d t t t fj
ỡ d số ừ số

s

s ợ õ ọ

t
r

n

n+1

ỏ tữỡ tỹ t ữủ



{g0 , . . . , gn }

tọ


gi



n+1



gi = 0

tờ ủ t t ổ t tữớ ừ

gi

f0 , . . . , f n

ở t t

ợ ồ t

gj

õ

j

tự ợ ồ

j {0, . . . , n}

gj (z) = z j + o(z j 1 ), z .
t ụ õ

W (f0 , . . . , fn ) = cW (g0 , . . . , gn )
tr õ

c

số

ó r t õ

d = 0 > 1 > . . . > n 0.
s

W (f0 , . . . , fn )



số ụ trữ

f0 , . . . , f n

t



ổ tự t ừ

tự

fn (z)
f0 (z)
,
.
.
.
,
)
zm

zm

ữ ợ ồ m N ợ ồ z = 0 ợ tự f0 , . . . , fn t
d = max{deg fj , j = 0, n}. sỷ fs tở {f0 , . . . , fn } s deg fs = d
õ z t ữủ
t tr

(

Pn (C)



(f0 (z), . . . , fn (z))



(

fn (z)
f0 (z)
,
.
.
.
,
) (a0 , . . . , an ) Pn (C),
d
d
z

z

a0 , . . . , an C as 0 ồ (a0 , . . . , an ) ừ
(f0 (z), . . . , fn (z)) t z =
sỷ f0 , . . . , fn n + 1 tự ở t t gj , j =
0, n tờ ủ t t ữủ ỹ ữ tr d =
tr õ




g0
gn
,
.
.
.
,
) −−−→ (b0 , . . . , bn ) t❤➻
zd
z d z→∞
(b0 , . . . , bn ) ❧➔ ↔♥❤ ❝õ❛ (g0 , . . . , gn ) t↕✐ z = ∞✳ ❉➵ t❤➜②✱ ♣❤➛♥ tû t❤ù j tr♦♥❣
gn
g0
,
.
.
.
,
) ❝â ❞↕♥❣ z δj −δi + o(z δj −δi −1 ) ❦❤✐ z → ∞✳ ◆❤÷ ✈➟② ✈î✐ ♠å✐

❜ë (
d
d
z
z
j ∈ {1, . . . , n} t❤➔♥❤ ♣❤➛♥ t❤ù j ❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ δ0 − δj t↕✐ ∞✳
δ0 = max{deg f0 , . . . , deg fn }✳

●✐↔ sû

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✺✳ ❚❛ ❣å✐ dj (∞)
❝õ❛

(

= δ0 − δj ✱ (0 ≤ j ≤ n) ❧➔ sè ♠ô ✤➦❝ tr÷♥❣

(f0 , . . . , fn ) t↕✐ ∞✳
❉➵ t❤➜②✿ 0 = d0 (∞) < d1 (∞) < . . . < dn (∞)✳

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✻✳ ❈❤♦

g ❧➔ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛
g(z)
(f1 , . . . , fp )✳ ◆➳✉ d = az −m + o(a−m−1 ) ❦❤✐ z → ∞✱ a = 0✱ m ≥ 1 t❤➻ t❛
z
♥â✐ g ❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ m t↕✐ ∞ ✈î✐ d = δ0 = max{deg fj }✳
j=0,n

f0 , . . . , fn ❧➔ ❝→❝ ✤❛ t❤ù❝

✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✱ ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ tr➯♥ C✳ ●✐↔ sû g0 , . . . , gn ❧➔
❝→❝ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ f0 , . . . , fn ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷
❚❛ ✤✐ t➼♥❤ ❜➟❝ ❝õ❛ ❲r♦♥s❦✐❛♥ ✤❛ t❤ù❝✳ ❈❤♦

tr➯♥✳ ❑❤✐ ✤â✱

W (f0 , . . . , fn ) = cW (g0 , . . . , gn )
gj (z) = tδj + 0(z δj−1 ) s✉② r❛ W ❧➔ ✤❛ t❤ù❝ ❝õ❛ ♠ët
♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n + 1 × n + 1 ❞á♥❣ i✱ (0 ≤ i ≤ n)✱ ❝ët j ✱ (0 ≤ j ≤ n)
tr♦♥❣ ✤â ♣❤➛♥ tû t❤ù ij tr♦♥❣ ✤à♥❤ t❤ù❝ ❲r♦♥s❦✐❛♥ W ❧➔✿
✈î✐

c=0

❧➔ ❤➡♥❣ sè✳ ❚ø

δj (δj − 1) . . . (δj − i)z δj −i + 0(z δj −i−1 ),
❦❤✐ ✤â

W (z) = az b + o(z b−1 )

❦❤✐

z → ∞✱ α = 0✱
n

δj −

deg W = b =
j=0

(i > 0),

s✉② r❛

n(n + 1)
.
2

◆➳✉ t➜t ❝↔ ❝→❝ ❤➔♠ fj 0 ≤ j ≤ n ❧➔ ❝→❝ ✤❛ t❤ù❝ ✤ë❝ ❧➟♣ t✉②➳♥
t➼♥❤ ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ t❤➻ ✤à♥❤ t❤ù❝ ❲r♦♥s❦✐❛♥ ❝õ❛ ❝❤ó♥❣ ❧➔ ✤❛
t❤ù❝ ❜➟❝✿
❇ê ✤➲ ✷✳✶✳✼✳

n

δj −
j=0

n(n + 1)
.
2

❚r♦♥❣ ✤â δ0 , . . . , δn ❧➔ ❝→❝ sè ♠ô ✤➦❝ tr÷♥❣ ❝õ❛ fj t↕✐ ∞ ①→❝ ✤à♥❤ ♥❤÷ tr➯♥✳
✶✼


t t z0 = tr t t trữớ
ủ f0 , . . . , fn tự t õ t rở
f0 , . . . , fn s ffji ỳ t ợ ồ ở (i, j)

t

r trữớ ủ t ổ t ữủ tự F0 , . . . , Fn s

fj (z) = eh(z) Fj (z)

tr C.

ỹ tữỡ tỹ ữ tr tự Fj ồ õ
tữỡ ự ố ợ fj


ở rút ồ ở ữ

f0 , . . . , fn n+1 ở t t ổ õ ổ
tr C sỷ z0 C tũ ỵ W (z0 ) = 0 t ỳ
f0 , . . . , fn W (z0 ) = 0 t t f0 , . . . , fn g0 , . . . , gn


ữủ ữ tr ử t t t ừ

W (f0 , . . . , fn ) = cW (g0 , . . . , gn ) tr õ c
ổ ữ rW (z0 ) ổ tt

tự rs t õ
số õ

W (f0 , f1 , . . . , fn ) =

c.det|gji |

=

g0

g1

. . . gn

g0

g1

. . . gn







(n)

g0
ử t

W (f0 , . . . , fn )

(n)

g1

t ụ tứ ừ




...

.

(n)

. . . gn

z z0

t s õ

W (f0 , . . . , fn ) = (z z0 )b + O(z z0 )b+1
O(z z0 )b+1

z z0 ỡ b b 0
t tr C = 0 s b
ộ tỷ tr ởt ỏ ởt ởt ừ W t
tr ộ số ừ W
tr õ

ừ

n

n

dj

b=
j=0

n

dj

i=
i=1

j=0



n(n + 1)
2

ọ

ởt


✈➔

α = det |dj (dj−1 ) . . . (dj−i )|ni,j=0
✭✤✐➲✉ ♥➔② ✤ó♥❣ ♥❣❛② ❝↔ ❦❤✐ ❝â ♠ët sè ✈à tr➼ tr➯♥ ✤à♥❤ t❤ù❝ ❜➡♥❣ ✵✮✳

❚❛ t❤➜②✿

α=

1

...

1

d0

...

dn

d0 (d0 − 1)

...

dn (dn − 1)

...

...

...

.

d0 (d0 − 1) . . . (d0 − n) . . . dn (dn − 1) . . . (dn − n)
❇✐➳♥ ✤ê✐ ✤à♥❤ t❤ù❝ tr➯♥ ❜➡♥❣ ❝→❝ ❝ë♥❣ ❞á♥❣ ✷ ✈➔♦ ❞á♥❣ ✸✱ t÷ì♥❣ tü ✈î✐ ❝→❝
❞á♥❣ ❝á♥ ❧↕✐✱ t❛ ✤÷ñ❝✿

1

1

...

d0

d1

. . . dn

α = d20

d21

. . . d2n .

...
(n)

d0

... ... ...
(n)

d1

(n)

. . . dn

d0 , . . . , dn ✈➔ ♥❤÷ ✈➟②
n(n + 1)
n(n + 1)
=
dj −
.
dj −
α = 0 ✈➔ ♦r❞W (z0 ) = b =
2
2
j=1
j=0
❉♦ 1 ≤ d1 < d2 < . . . < dn ♥➯♥ dj ≥ j ✈î✐ ♠å✐ j ∈ {0, . . . , n}✳
❍✐➸♥ ♥❤✐➯♥ W (z0 ) = 0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ordW (z0 ) = b > 0✱ tù❝ ❧➔ b > 0✳ ❍ì♥
♥ú❛ b = 0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ dj = j ✈î✐ ♠å✐ j ∈ {1, . . . , n} ♥➯♥ b > 0 ❦❤✐ ✈➔ ❝❤➾
❉♦ ✤â

α

1

❝❤➼♥❤ ❧➔ ✤à♥❤ t❤ù❝ ❱❛♥❞❡♠♦♥❞❡ ❝õ❛ ❝→❝ sè

n

n

✶✾


j0 ∈ {1, . . . , n} s❛♦ ❝❤♦ dj0 > j0 ✳ ❱➻ ❞➣② {dj − j} ❧➔ ❞➣② ❦❤æ♥❣
❣✐↔♠ ♥➯♥ dn −n ≥ dj0 −j0 > 0✱ tø ✤â dn −(n+1) > −1 ❤❛② dn −(n+1) ≥ 0✳
❦❤✐ tç♥ t↕✐

❇ê ✤➲ ✷✳✶✳✾✳

✭✶✮ ◆➳✉ W (z0 ) = 0 t❤➻ ❜➟❝ ❝õ❛ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ ❲ t↕✐ z0 ❧➔
n

n(n + 1)
✳ ❍ì♥ ♥ú❛✱ tç♥ t↕✐ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ t➛♠
2
j=1
t❤÷í♥❣ ❝õ❛ f0 , ..., fn s❛♦ ❝❤♦ ♦r❞g (z0 ) ≥ n + 1✳
b=

dj (z0 ) −

✭✷✮ ◆➳✉ W (z0 ) = 0 s✉② r❛ ♦r❞W (z0 ) = 0 ✈➔ dj = j ✈î✐ ♠å✐ j ∈ {1, . . . , n}
♥❣♦➔✐ r❛ ❦❤æ♥❣ tç♥ t↕✐ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ g ❝õ❛ f0 , . . . , fn
s❛♦ ❝❤♦ ♦r❞g (z0 ) ≥ n + 1✳

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✵✳ ❈❤♦ ❣ ❧➔ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ t➛♠ t❤÷í♥❣

f0 , . . . , fn ❝â ❜➟❝ m ≥ 1 t↕✐ z0 ✳ ❑❤✐ ✤â
t❤✉ë❝ 1, . . . , n s❛♦ ❝❤♦ m = dj (z0 )✳ ❚❛ ❣å✐✿
❝õ❛

tç♥ t↕✐ ❞✉② ♥❤➜t ♠ët ❝❤➾ sè

j

νn (g, z0 ) = j ∈ {1, . . . , n}
❧➔ ❜ë✐ rót ❣å♥ ❝õ❛ ✵✲✤✐➸♠ ❝õ❛

g

t↕✐

z0 ✳

ε(g, z0 ) = dj − j
❧➔ ♣❤➛♥ ❞÷ ✭❜ë✐ ❞÷✮ ❝õ❛
◆➳✉
❞✉② ♥❤➜t✱

g

z0 ✳ ❚❛ ❝ô♥❣ ✈✐➳t εj (z0 ) = dj (z0 ) − j ✳
m ≥ 1 t↕✐ ∞ t❤➻ m = δ0 − δj ✈î✐ ♠ët ❣✐→

t↕✐

g ❝â ✵✲✤✐➸♠ ❜ë✐
1 ≤ j ≤ n✳ ❚❛ ❣å✐✿

trà

j

νn (g, ∞) = j ∈ {1, . . . , n}
❧➔ ❜ë✐ rót ❣å♥ ❝õ❛ ✵✲✤✐➸♠ ❝õ❛

g

t↕✐

∞✳

ε(g, ∞) = δi − δj − j)
❧➔ ♣❤➛♥ ❞÷ ✭❜ë✐ ❞÷✮ ❝õ❛
◆➳✉

g

g

t↕✐

∞✳

εj (∞) = dj (∞) − j ✳

νn (g, z) = ε(g, z) = 0.

❚❛ ❝ô♥❣ ✈✐➳t

❦❤æ♥❣ ❜à tr✐➺t t✐➯✉ t↕✐

¯
z∈C

t❤➻

❚↕✐ ♠å✐ z0 ∈ C ❜ë✐ ❞÷ ❝õ❛ ❝→❝ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝
❤➔♠ f0 , . . . , fn t↕♦ ♥➯♥ ♠ët ❞➣② ❤ú✉ ❤↕♥ ❦❤æ♥❣ ❣✐↔♠✿
❇ê ✤➲ ✷✳✶✳✶✶✳

εj (z0 ) = dj − j ≤ dj+1 − (j + 1) ✈î✐ ♠å✐ j = 0, 1, 2, . . . , n − 1✳
❈❤ù♥❣ ♠✐♥❤✳

❍✐➸♥ ♥❤✐➯♥ ✈➻

dj+1 − dj ≥ 1✳
✷✵


❇➙② ❣✐í t❛ ❧➜②

Yj = Yj (z0 )

dj ≥ 1

sè ♠ô ✤➦❝ tr÷♥❣

❧➔

❝õ❛

f0 , . . . , f n ✳

●å✐

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝❤ù❛ ❤➔♠ ❦❤æ♥❣ ✈➔ ❝→❝ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❦❤æ♥❣

t➛♠ t❤÷í♥❣ ❝õ❛

f0 , ..., fn

♠➔ ❝â ❜ë✐ ➼t ♥❤➜t

dj

t↕✐

z0

n

aj fj : ordg (z0 ) ≥ dj }✳

Yj = {0, g =
j=0

Yj ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ♣❤ù❝ ✈➔ sè
❝❤✐➲✉ dimYj = n + 1 − j (1 ≤ n + 1 − j ≤ n)✳ ❚❤❡♦ ❜ê ✤➲ ✷✳✶✳✶✶ ✈î✐ ♠å✐
g ∈ Yj (z0 ) \ {0} s✉② r❛ ε(g, z0 ) ≥ dj − j ✳
❑❤✐ ✤â ❞➵ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝

◆➳✉ W (z0 ) = 0✱ t❤➻ ε(g, z0 ) = 0 ✈î✐ ♠å✐ tê ❤ñ♣ t✉②➳♥ t➼♥❤
❦❤æ♥❣ t➛♠ t❤÷í♥❣ g ❝õ❛ ❝→❝ ❤➔♠ f0 , ..., fn ✳ ❱➔ ♥➳✉ ordW (z0 ) = m ≥ 1 t❤➻

❇ê ✤➲ ✷✳✶✳✶✷✳
n

m=

εj (z0 )✳

j=1

❈❤ù♥❣ ♠✐♥❤✳
♠å✐

❍✐➸♥ ♥❤✐➯♥ ♥➳✉

W (z0 ) = 0

t❤➻

dj = j

s✉② r❛

ε(g, z0 ) = 0

✈î✐

g✳
◆➳✉

ordW (z0 ) = m ≥ 1
n

t❤➻

n

n

dj (z0 ) −

m=
j=1

(dj − j)(z0 ) =

j=
j=1

n

j=1

εj (z0 ).
j=1

✣➸ ❝→❝ ❦❤→✐ ♥✐➺♠ tr➯♥ ✤÷ñ❝ rã r➔♥❣✱ t❛ ①➨t ❤❛✐ ✈➼ ❞ö s❛✉✳

f : C −→ P2 (C) ❧➔ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ✈î✐
f (z) = (z : cosz : sinz)✳ ✣➦t f0 = z, f1 = cosz, f2 = sinz✱ ❦❤✐ ✤â ✤à♥❤
t❤ù❝ ❲r♦♥s❦✐❛♥ ❝õ❛ (f0 , f1 , f2 ) ❧➔

❱➼ ❞ö ✷✳✶✳✶✸✳ ❈❤♦

z

cosz

W (f0 , f1 , f2 ) = 1 −sinz

sinz
2
2
cosz = zsinz z + 2coszsinzz − xcosz z.

0 −cosz −sinz
❚❛ ①➙② ❞ü♥❣

sè ♠ô ✤➦❝ tr÷♥❣

❝❤♦ ❝→❝ ❤➔♠

z = 0✳

✷✶

f0 , f1 , f2

t↕✐

z=

π
2

✈➔ t↕✐


❱î✐

f0 = z

z=

π
✱
2

π
W ( ) = 0 ✈➔
2
π

t✐➯✉ t↕✐ z =
,
2

❞➵ t❤➜②

❦❤æ♥❣ ❜à tr✐➺t

π
π
f1 = cosz = −(z − ) + O((z − )3 )
2
2

❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ ✶ t↕✐

1
π
π
f2 = sinz = 1 − (z − )2 + O((z − )3 )
2
2
2

z=

❦❤æ♥❣ ❜à tr✐➺t t✐➯✉ t↕✐

π
,

2

z=

π
.
2

π
f0 ✱ t❤❛② t❤➳ ❤➔♠ f2 ❜ð✐ ❤➔♠ − f2 + f0 ✱ t❛ ✤÷ñ❝
2
π
π
π 2
π 3
❤➔♠ f2 ♠î✐✱ f2 = (z −
) + (z − ) + O(z − ) ✳ ❚❛ t❤➜② ❤➔♠ f2 ♠î✐
2
4
2
2
π
♥➔② ❝â ❦❤æ♥❣ ✤✐➸♠ ❜➡♥❣ ✈î✐ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ f1 t↕✐ z =
✳ ❚❤❛② t❤➳ ❤➔♠
2
π
π
π
f1 ❜ð✐ ❤➔♠ f1 + f2 ✱ t❛ ✤÷ñ❝ ❤➔♠ f1 ♠î✐✱ f1 = (z − )2 + O(z − )3 ✱ ❦❤✐
4

2
2
π
✤â f1 ❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ 2 t↕✐ z =
✳ ❉♦ ✤â sè ♠ô ✤➦❝ tr÷♥❣ ❝õ❛ ❝→❝ ❤➔♠
2
π
π
(f0 , f1 , f2 ) t↕✐ z =
❧➔ d0 = 0, d1 = 1, d2 = 2. ❚❛ ❝ô♥❣ ❝â ν(f1 , ) = 0,
2
π
π2
π
π
π
ν(f2 , ) = 2, ν(f3 , ) = 1, ε(f0 , ) = ε(f1 , ) = ε(f2 , ) = 0.
2
2
2
2
2
❱î✐ z = 0✱ ❞➵ t❤➜② W (0) = 0✱ ❲ ❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ m = 1 t↕✐ z = 0
●✐ú ♥❣✉②➯♥ ❤➔♠

✈➔

f0 = z

❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ ✶ t↕✐

z = 0✱

x2
f1 = cosz = 1 −
+ O(x3 )
2

❦❤æ♥❣ ❜à tr✐➺t t✐➯✉ t↕✐

x3
+ O(x4 )
f2 = sinz = z −
6

❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ ✶ t↕✐

z = 0✱
z = 0✳

●✐ú ♥❣✉②➯♥ f0 , f1 ✱ t❤❛② t❤➳ ❤➔♠ f2 ❜ð✐ f0 − f2 t❛ ✤÷ñ❝ ❤➔♠ f2 ♠î✐✱
x3
f2 =
+ O(x4 )✱ ❦❤✐ ✤â f2 ❝â ❦❤æ♥❣ ✤✐➸♠ ❜ë✐ ✸ t↕✐ ③❂✵✳ ❚❛ ✤÷ñ❝ sè ♠ô ✤➦❝
6
tr÷♥❣ ❝õ❛ ❝→❝ ❤➔♠ (f0 , f1 , f2 ) t↕✐ z = 0 ❧➔ d0 = 0, d1 = 1, d2 = 3. ❑❤✐ ✤â
ν(f0 , 0) = 1, ν(f1 , 0) = 0, ν(f2 , 0) = 2, ε(f0 , 0) = ε(f1 , 0) = 0, ε(f2 , 0) = 1.

✷✷