✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖ ▼❆ ❚❍➚ ◆❍❯◆● ❱➋ ❍⑨▼ P❍❹◆ ❍➐◆❍ f P (f ) ❱⑨ g P (g) ❈❍❯◆● ◆❍❆❯ ▼❐❚ ❍⑨▼ ◆❍➘ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖ ▼❆ ❚❍➚ ◆❍❯◆● ❱➋ ❍⑨▼ P❍❹◆ ❍➐◆❍ f P (f ) ❱⑨ g P (g) ❈❍❯◆● ◆❍❆❯ ▼❐❚ ❍⑨▼ ◆❍➘ ❈❤✉②➯♥ ♥❣➔♥❤✿ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ữớ ữợ P ❍⑨ ❚❘❺◆ P❍×❒◆● ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺ ✐ ▲í✐ ❝❛♠ ✤♦❛♥ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤æ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝ ✤➣ ❝ỉ♥❣ ❜è ð ❱✐➺t ◆❛♠✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷đ❝ ❝↔♠ ì♥ ✈➔ ❝→❝ t❤ỉ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr ữủ ró ỗ ố ♥❣✉②➯♥✱ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✺ ◆❣÷í✐ ✈✐➳t ▲✉➟♥ ✈➠♥ ▼❛ ❚❤à ◆❤✉♥❣ ❳→❝ ♥❤➟♥ ❳→❝ ♥❤➟♥ ❝õ❛ tr÷ð♥❣ ❦❤♦❛ ❝❤✉②➯♥ ổ ữớ ữợ P r P❤÷ì♥❣ ✐✐ ▲í✐ ❝↔♠ ì♥ ✣➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ tổ ổ ữủ sỹ ữợ ú ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ P●❙✳❚❙ ❍➔ ❚r➛♥ P❤÷ì♥❣ ✭❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❚❤→✐ ◆❣✉②➯♥✮✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛② ✈➔ ①✐♥ ❣û✐ ❧í✐ tr✐ ➙♥ ♥❤➜t ❝õ❛ tỉ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ✤✐➲✉ t❤➛② ✤➣ ❞➔♥❤ ❝❤♦ tæ✐✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ▲➣♥❤ ✤↕♦ ♣❤á♥❣ ✣➔♦ t↕♦✱ ✤➦❝ ❜✐➺t ❧➔ t ổ trỹ t q ỵ t s qỵ t ổ ợ ❑✷✶ ✭✷✵✶✸✲ ✷✵✶✺✮ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t ♥❤ú♥❣ ❦✐➳♥ tự qỵ ụ ữ t tổ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ t t tợ ỳ ữớ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❤é trđ ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦ ❚❤→✐ ♥❣✉②➯♥✱ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✺ ◆❣÷í✐ ✈✐➳t ▲✉➟♥ ✈➠♥ ▼❛ ❚❤à ◆❤✉♥❣ ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ✐ ▲í✐ ❝↔♠ ì♥ ✐✐ ▼ư❝ ❧ư❝ ✐✐✐ ▼ð ✤➛✉ ✶ ✶ ▼ët sè t➼♥❤ ❝❤➜t ✈➲ ♣❤➙♥ ❜è ❣✐→ trà ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✸ ✶✳✶ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✶ ❍➔♠ ✤➳♠ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈ỵ✐ ✤↕♦ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✶ ❚r÷í♥❣ ❤đ♣ ✤❛ t❤ù❝ ❝❤ù❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈ỵ✐ ✤↕♦ ❤➔♠ ✼ ✶✳✷✳✷ ❇ê ✤➲ ❝❤➻❛ ❦❤â❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ✷ ✶✶ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❦❤✐ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä ✷✶ ✷✳✶ ✷✶ ❳→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✶ ❚r÷í♥❣ ❤đ♣ P = b(x − a1 )n l (x − )ki i=2 ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✐✈ ✷✳✶✳✷ ❚r÷í♥❣ ❤đ♣ P = b(x − a1 ) n l (x − ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ i=2 ✷✳✷ ❳→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♥❣✉②➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ❚r÷í♥❣ ❤đ♣ P = b(x − a1 ) n l ✸✻ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ (x − ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ (x − )ki i=1 ✷✳✷✳✷ ❚r÷í♥❣ ❤đ♣ P = b(x − a1 )n l i=1 ❑➳t ❧✉➟♥ ✹✻ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✼ ✶ ▼ð ởt ự q trồ ỵ tt ❜è ❣✐→ trà ◆❡✈❛♥❧✐♥♥❛ ❧➔ ♥❣❤✐➯♥ ❝ù✉ sü ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝õ❛ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ t❤æ♥❣ q✉❛ ↔♥❤ ♥❣÷đ❝ ❝õ❛ ♠ët t➟♣ ❤ú✉ ❤↕♥✳ ◆➠♠ ✶✾✷✻✱ ❘✳ ◆❡✈❛♥❧✐♥♥❛ ✤÷đ❝ ❝❤ù♥❣ tä ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C ✤÷đ❝ ①→❝ ✤à♥❤ ♠ët ❝→❝❤ ❞✉② ♥❤➜t ❜ð✐ ↔♥❤ ♥❣÷đ❝ ❦❤ỉ♥❣ t➼♥❤ ❜ë✐ ❝õ❛ ✺ ♣❤➙♥ ❜✐➺t ❝→❝ ❣✐→ trà✳ ❈ỉ♥❣ tr➻♥❤ ♥➔② ❝õ❛ ➷♥❣ ✤÷đ❝ ①❡♠ ❧➔ ỗ ự t ①→❝ ✤à♥❤ ❞✉② ♥❤➜t✳ ❱➲ s❛✉✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ sü ①→❝ ✤à♥❤ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❜ð✐ ↔♥❤ ♥❣÷đ❝ ❝õ❛ ♠ët t➟♣ ❤ú✉ ❤↕♥ ♣❤➛♥ tû ✤➣ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ tr♦♥❣ ✈➔ ữợ rss t r ọ tỗ t ởt t ủ ỳ S ✤✐➲✉ ❦✐➺♥ E(S, f ) = E(S, f ) ❦➨♦ t❤❡♦ f ≡ g ❄✧✳ ◆➠♠ ✶✾✾✺✱ ❍✳❳✳ ❨✐ ✭❬✶✹❪✮ tr↔ ❧í✐ ❝➙✉ tr↔ ❧í✐ ❝➙✉ ❤ä✐ ❝õ❛ ●r♦ss tr♦♥❣ tr÷í♥❣ ❤đ♣ ❤➔♠ ♥❣✉②➯♥ ✈➔ ♥➠♠ ✶✾✾✽✱ ●✳ ❋r❛♥❦ ✈➔ ▼✳❘❡✐♥❞❡rs ✭❬✻❪✮ ✤➣ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ❚r♦♥❣ t❤ü❝ t➳✱ ❝➙✉ ❤ä✐ ❝õ❛ ●r♦ss ❝â t❤➸ ữủ t ữ s tỗ t ❦❤ỉ♥❣ ✤❛ t❤ù❝ P s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t ❝ù ❝➦♣ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ f ✈➔ g t❛ ❝â f ≡ g ♥➳✉ P (f ) ✈➔ P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà ♠ët ❣✐→ trà ❈▼❄ ▼ët tỹ t ữ r ọ s tỗ t↕✐ ❤❛② ❦❤æ♥❣ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ P s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t ❝ù ❝➦♣ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ f ✈➔ g t❛ ❝â f ≡ g ♥➳✉ P (f ) ✈➔ P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà ❈▼❄ ✣➣ ❝â ♠ët sè ❝æ♥❣ tr➻♥❤ ❝æ♥❣ ❜è t❤❡♦ ữợ ự ❋❛♥❣ ❛♥❞ ❲✳ ❍♦♥❣ ✭❬✼❪✮ ✤➣ ❝❤ù♥❣ ♠✐♥❤✿ ❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t✱ n ≥ 11 ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ◆➳✉ f n(f − 1)f ✈➔ gn(g − 1)g ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà ❦➸ ❝↔ ❜ë✐ t❤➻ f = g ◆➠♠ ✷✵✵✹✱ ❲✳ ❈✳ ▲✐♥ ✈➔ ❍✳ ❳✳ ❨✐ ✭❬✶✷❪✮ ❝❤ù♥❣ ♠✐♥❤✿ ❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t✱ n ≥ 13 ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ✷ ◆➳✉ f n(f − 1)2f ✈➔ gn(g − 1)2g ❝❤✉♥❣ ♥❤❛✉ z ❦➸ ❝↔ ❜ë✐ t❤➻ f = g.✳✳✳ ❱ỵ✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ ✈➜♥ ✤➲ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✤÷đ❝ ①→❝ ✤à♥❤ ♠ët ❝→❝❤ ❞✉② ♥❤➜t ❜ð✐ ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ❝â ❝❤ù❛ ✤↕♦ ❤➔♠ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t➔✐ ✏❱➲ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f P (f ) ✈➔ g P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä✑ ✳ ▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✤÷đ❝ ❝ỉ♥❣ ❜è ✈➔♦ ♥➠♠ ✷✵✶✸ ❜ð✐ ❑✳ ❇♦✉ss❛❢✱ ❆✳ ❊s❝❛ss✉t ✈➔ ❏✳ ❖❥❡❞❛ tr♦♥❣ ❬✷❪✳ ▲✉➟♥ ✈➠♥ ♥➔② ỗ õ ữỡ ữ s ữỡ ởt số tự ỡ tr ỵ tt r ữỡ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì tr ỵ tt ố tr ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜ê ✤➲ sû ❞ö♥❣ tr♦♥❣ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ tr♦♥❣ ❈❤÷ì♥❣ ✷✳ ❈❤÷ì♥❣ ✷✿ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❦❤✐ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä✳ ✣➙② ❧➔ ❝❤÷ì♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦➳t q✉↔ ♥❣✉②➯♥ ❝ù✉ ❝õ❛ ❑✳ ❇♦✉ss❛❢✱ ❆✳ ❊s❝❛ss✉t ✈➔ ❏✳ ❖❥❡❞❛ ✈➲ ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ❝õ❛ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ✤➸ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❧➔ ❜➡♥❣ ♥❤❛✉✳ ✸ ❈❤÷ì♥❣ ✶ ▼ët sè t➼♥❤ ❝❤➜t ✈➲ ♣❤➙♥ ❜è ❣✐→ trà ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✶✳✶ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✶✳✶✳✶ ❍➔♠ ✤➳♠ ✈➔ t➼♥❤ ❝❤➜t ✣➸ t❤✉➟♥ t✐➺♥ ❝❤♦ ✈✐➺❝ t❤❡♦ ❞ã✐ ❝→❝ ✈➜♥ tr tr trữợ t ú tổ ởt số tr ỵ tt ❜è ❣✐→ trà ❝õ❛ ◆❡✈❛♥❧✐♥♥❛✳ ❈→❝ ❦✐➳♥ t❤ù❝ ♥➔② ❝â t❤➸ t➻♠ t❤➜② tr♦♥❣ ♥❤✐➲✉ t➔✐ ❧✐➺✉✱ ❝❤➥♥❣ ❤↕♥ tr♦♥❣ ❬✷❪✳ ❈❤♦ f ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ DR = {z ∈ C : |z| ≤ R} ✈➔ ♠ët sè t❤ü❝ r > 0✱ tr♦♥❣ ✤â < R ≤ ∞ ✈➔ < r < R✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❍➔♠ m(r, f ) = ❧➔ 2π 2π log+ |f (reiϕ )|dϕ ❤➔♠ ①➜♣ ①➾ ❝õ❛ ❤➔♠ f ✳ ❚❛ ❦➼ ❤✐➺✉ n(r, f ) ❧➔ sè ❝ü❝ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐✱ n(r, f ) ❧➔ sè ❝ü❝ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛ ❤➔♠ f tr♦♥❣ Dr ✳ ợ ởt số ữỡ n[] (r, f ) ❧➔ sè ❝ü❝ ✤✐➸♠ ❜ë✐ ❝❤➦♥ ❜ð✐ ∆ ❝õ❛ ❤➔♠ f ✭tù❝ ❧➔ ❝ü❝ ✤✐➸♠ ❜ë✐ k > ∆ ❝❤➾ ✤÷đ❝ t➼♥❤ ∆ ❧➛♥ tr♦♥❣ tê♥❣ n[∆] (r, f )✮ tr♦♥❣ Dr ✹ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❍➔♠ r N (r, f ) = n(t, f ) − n(0, f ) dt + n(0, f ) log r, t ❤➔♠ ✤➳♠ ❦➸ ❝↔ ❜ë✐ ❝õ❛ ❤➔♠ f ❝→❝ ❝ü❝ ✤✐➸♠✮✳ ❍➔♠ ✤÷đ❝ ❣å✐ ❧➔ r N (r, f ) = ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ t↕✐ n(t, f ) − n(0, f ) dt + n(0, f ) log r t ❤➔♠ ✤➳♠ ❦❤æ♥❣ ❦➸ ❜ë✐✳ ❍➔♠ r N[∆] (r, f ) = ✤÷đ❝ ❣å✐ ❧➔ ✭❝á♥ ✤÷đ❝ ❣å✐ ❧➔ n[∆] (t, f ) − n[∆] (0, f ) dt + n[∆] (0, f ) log r t ❤➔♠ ✤➳♠ ❜ë✐ ❝❤➦♥ ❜ð✐ ∆✱ tr♦♥❣ ✤â n(0, f ) = lim n(t, f )❀ t→0 n(0, f ) = lim n(t, f )❀ n[∆] (0, f ) = lim n[∆] (t, f )✳ ❙è ∆ tr♦♥❣ N[∆] (r, f ) t→0 ✤÷đ❝ ❣å✐ ❧➔ t→0 ❝❤➾ sè ❜ë✐ ❝❤➦♥✳ ❚❛ ❦➼ ❤✐➺✉ Z(r, f ) = N (r, 1/f ); ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ Z(r, f ) = N (r, 1/f ); Z[∆] (r, f ) = N[∆] (r, 1/f ) ❍➔♠ T (r, f ) = m(r, f ) + N (r, f ) ❣å✐ ❧➔ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ f ✳ ❈→❝ ❤➔♠ N (r, f ), m(r, f ), T (r, f ) ✤÷đ❝ ❣å✐ ❝❤✉♥❣ ❧➔ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ❈→❝ ❜ê ✤➲ s❛✉ ✤➙② ❧➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✿ ✸✹ ◆➯♥ tø ✭✷✳✷✾✮ t❛ ❝â (n + k + 1)(T (r, f ) + T (r, g)) ≤ (9 + 2l)(T (r, f ) + T (r, g)) + Sf (r) + Sg (r) ❙✉② r❛ n+k+1 + 2l, ❦➨♦ t❤❡♦ n + l, ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ n ≥ 10 + l tr ỵ F,G = ố ữ tr ự ỵ t õ t❤➸ ✈✐➳t ΨF,G = F (F − 1)2 ✈ỵ✐ φ = φ φ (G − 1)2 ✳ ❱➻ ΨF,G = tỗ t A, B C s G ❝❤♦ A = +B G−1 F −1 ✭✷✳✸✵✮ ✈➔ A = ú ỵ r Z(r, f ) ≤ T (r, f ), N (r, f ) ≤ T (r, f ), Z(r, f − ) ≤ T (r, f − ) ≤ T (r, f ) + O(1), ✈ỵ✐ i = 2, , l ✈➔ Z(r, f ) ≤ T (r, f ) ≤ 2T (r, f ) + O(1) ❚÷ì♥❣ tü t❛ ❝ơ♥❣ ❝â t tự ữ t ố ợ g g ✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✱ t❛ ❝â T (r, F ) ≥ (n + k)T (r, f ) − m(r, 1/f ) + Sf (r) ✭✷✳✸✶✮ ❚❛ s➩ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ B = ✈➔ B = ❚r÷í♥❣ ❤đ♣ ✶✿ B = ●✐↔ sû A = 1✳ ▲➟♣ ❧✉➟♥ ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ỵ t õ (n + k 2l − 5)[T (r, f ) + T (r, g)] ≤ Sf (r) + Sg (r), ✭✷✳✸✷✮ ✸✺ ✤✐➲✉ ♥➔② s➩ ❦❤æ♥❣ t❤➸ ①↔② r❛ ❦❤✐ n + k ≥ 2l + ứ tt ỵ t ❝â k = l − ✈➔ n ≥ l + 10 ♥➯♥ n + k ≥ 2l + 9✳ ❉♦ ✤â ✤✐➲✉ ❦✐➺♥ ✭✷✳✸✷✮ ❦❤ỉ♥❣ t❤➸ ①↔② r❛✳ ◆❤÷ ✈➟② A = ❈ơ♥❣ ❧➟♣ ❧✉➟♥ ❣✐è♥❣ ♥❤÷ tr♦♥❣ ự ỵ t s r f = g ❚r÷í♥❣ ❤đ♣ ✷✿ B = ▲➟♣ ❧✉➟♥ ố ữ tr ự ỵ t ❝â Z(r, F ) + Z(r, G) + N (r, F ) + N (r, G) 2l + ≤ max(T (r, F ), T (r, G)) + SF (r) + SG (r) n+k ✭✷✳✸✸✮ ❚ø ❣✐↔ t❤✐➳t ❝õ❛ ✤à♥❤ ỵ t õ n + k 2l + ✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ lim sup r→∞ Z(r, F ) + Z(r, G) + N (r, F ) + N (r, G) max(T (r, F ), T (r, G)) < ❈ơ♥❣ ❧➟♣ ❧✉➟♥ ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ỵ t s r f = g q✉↔ ✷✳✶✳✽✳ ✭❬✷❪✮ ❈❤♦ P ∈ C[x] t❤ä❛ ♠➣♥ P ữủ t ữợ l n P (x) = b(x − a1 ) (x − ), i=2 ✈ỵ✐ Φ(P ) ≥ 4, b ∈ C∗✱ t❤ä❛ ♠➣♥ n ≥ l + 10✳ ❈❤♦ f, g ∈ M(C) ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t ✈➔ α ∈ Mf (C) ∩ Mg (C) ❦❤→❝ 0✳ ◆➳✉ f P (f ) ✈➔ g P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà α ❦➸ ❝↔ ❜ë✐✱ t❤➻ f = g x18 2x17 x16 2x15 ❱➼ ❞ö ✷✳✶✳✾✳ ✭❬✷❪✮ ❈❤♦ P (x) = − − + ✱ t❤➻ P (x) = 18 17 16 15 x14 (x − 1)(x + 1)(x − 2)✳ ❚❛ t❤➜② r➡♥❣ P (0), P (1), P (−1), P (2) ❧➔ ❝→❝ ♥❣❤✐➺♠ r✐➯♥❣ ❜✐➺t✳ ❉♦ ✤â✱ φ(P ) = 4✳ ❉♦ ✤â P ❧➔ ✤❛ t❤ù❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t tr➯♥ M(C)✳ ❍ì♥ ♥ú❛ ✈ỵ✐ n = 14, l = ✈➔ →♣ ❞ö♥❣ ❤➺ q✉↔ ✷✳✶✳✽✱ ♥➳✉ f, g ∈ M(C) ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t t❤ä❛ ♠➣♥ f P (f ) ✈➔ g P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà α ∈ Mf (C) ∩ Mg (C) ❦➸ ❝↔ ❜ë✐✱ t❤➻ f = g ❍➺ q✉↔ ✷✳✶✳✶✵✳ ✭❬✷❪✮ ❈❤♦ P C[x] tọ P ữủ t ữợ P (x) = b(x − a1 )n (x − a2 )(x − a3 ) ✸✻ ✈ỵ✐ Φ(P ) = 3, b ∈ C∗✱ t❤ä❛ ♠➣♥ n ≥ 13✳ ❈❤♦ f, g ∈ M(C) ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t ✈➔ α ∈ Mf (C) ∩ Mg (C) ❦❤→❝ 0✳ ◆➳✉ f P (f ) ✈➔ g P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà α ❦➸ ❝↔ ❜ë✐✱ t❤➻ f = g ❱➼ ❞ö ✷✳✶✳✶✶✳ ✭❬✷❪✮ ❳➨t ❞↕♥❣ tê♥❣ q✉→t ❝õ❛ ✤❛ t❤ù❝ P ♠➔ P ❝â ✸ ♥❣❤✐➺♠✱ ♠ët tr♦♥❣ sè ✤â ❧➔ ✵ ❝â ❜➟❝ n ✈➔ ❤❛✐ sè ❦❤→❝ ❧➔ a ✈➔ b ❝â ❜➟❝ ✶✳ ❚❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ P (0) = 0✱ t❛ ❝â P (x) = xn+3 (n + 2)(n + 1) − xn+2 (a1 + a2 )(n + 3)(n + 1) + xn+1 (a1 a2 )(n + 3)(n + 2) ❞♦ ✤â P (x) = (n + 1)(n + 2)(n + 3)xn (x − a1 )(x − a2 ) n+3 a1 = ✈➔ ❚✐➳♣ t❤❡♦ t❛ t❤➜② r➡♥❣ P (a1 ) = P (0) = ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ a2 n+1 P (a1 ) = P (a2 ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ n+1 [an+3 − an+3 − an+1 ](n + 1) = [a1 a2 (a1 )](n + 3) ✣➦❝ ❜✐➺t✱ ✤➥♥❣ t❤ù❝ s❛✉ ❧✉ỉ♥ ✤ó♥❣ ✈ỵ✐ n ❧➔ sè ❧➫ ✈➔ a2 = −a1 ❉♦ ✤â✱ ♥➳✉ a1 n + a1 n+1 = , = , a2 n + a2 n+3 ✈➔ n+1 [an+3 − an+3 − an+1 ](n + 1) = [a1 a2 (a1 )](n + 3), t❤➻ P ❧➔ ✤❛ t❤ù❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣❤ù❝✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sỷ ợ trữớ ủ n a2 = −a1 ✱ t❤➻ t❛ ❝â t❤➸ ❦✐➸♠ tr❛ ✤÷đ❝ r➡♥❣ P (f ) = P (−f ) ✈ỵ✐ ♠å✐ ❤➔♠ sè✳ ❉♦ ✤â P ❦❤æ♥❣ ❧➔ ✤❛ t❤ù❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t tr➯♥ M(C) ✷✳✷ ❳→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♥❣✉②➯♥ ✷✳✷✳✶ ❚r÷í♥❣ ❤đ♣ P = b(x − a1 )n ỵ P = b(x a1 )n ❈❤♦ l i=2 (x − )ki i=1 ❧➔ ✤❛ t❤ù❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t tr➯♥ A(C)✱ ✈ỵ✐ b ∈ C∗, l ≥ ✈➔ ki ≥ ki+1, ≤ i ≤ l − P (x − )ki l ✸✼ ✈ỵ✐ l > ✈➔ ❝❤♦ k = l ki i=2 ●✐↔ sû P t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ n ≥ k + 2, l n≥5+ max(0, − ki ) + max(0, − k2 ) i=3 ❈❤♦ f, g ∈ A(C) ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t ✈➔ α ∈ Af (C) ∩ Ag (C) ❦❤→❝ 0✳ ◆➳✉ f P (f ) ✈➔ g P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà α ❦➸ ❝↔ ❜ë✐ t❤➻ f = g ✳ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t a1 = ✣➦t g P (g) f P (f ) ✈➔ G = ✱ ❦❤✐ ✤â F ✈➔ G ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà ✶ ❦➸ ❝↔ F = α α ❜ë✐✳ ❱➻ f, g ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t ♥➯♥ ❝→❝ ❤➔♠ F ✈➔ G ❝ô♥❣ ❧➔ ❤➔♠ s✐➯✉ ✈✐➺t✳ ◆❤➢❝ ❧↕✐ F 2F G 2G ΨF,G = − − + F F G G1 rữợ t t ự F,G = 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t F = P (f ), G = P (g)✳ ▲➟♣ ❧✉➟♥ ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ỵ t õ (n + k + 1)(T (r, f ) + T (r, g)) ≤ 5(T (r, f ) + T (r, g)) + (5 − k2 )(Z(r, f − a2 ) + Z(r, g − a2 )) l (4 − ki )(Z(r, f − ) + Z(r, g − )) + i=3 + 5(N (r, f ) + N (r, g)) + k(T (r, f ) + T (r, g)) + Sf (r) + Sg (r) ✭✷✳✸✹✮ ❉♦ f, g, α t❤✉ë❝ A(C) ♥➯♥ N (r, f ) = N (r, g)✱ ❞♦ ✤â ✭✷✳✸✹✮ ✤÷đ❝ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉✿ (n + k + 1)(T (r, f ) + T (r, g)) l ≤ (5 + k)(T (r, f ) + T (r, g)) + (4 − ki )(Z(r, f − ) + Z(r, g − )) i=3 + (5 − k2 )(Z(r, f − a2 ) + Z(r, g − a2 )) + Sf (r) + Sg (r), ✸✽ ✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ n(T (r, f ) + T (r, g)) l ≤ 4(T (r, f ) + T (r, g)) + (4 − ki )(Z(r, f − ) + Z(r, g − )) i=3 + (5 − k2 )(Z(r, f − a2 ) + Z(r, g − a2 )) + Sf (r) + Sg (r) l ≤ 4(T (r, f ) + T (r, g)) + max(0, − ki )(T (r, f ) + T (r, g)) i=3 + max(0, − k2 )(T (r, f ) + T (r, g)) + O(1) ◆❤÷ ✈➟② l n≤4+ max(0, − ki ) + max(0, − k2 ) i=3 ✣✐➲✉ ♥➔② ♠➝✉ t❤✉➝♥ ✈ỵ✐ n ≥ + l max(0, − ki ) + max(0, − k2 ) tr♦♥❣ i=3 ❣✐↔ t❤✐➳t ❝õ❛ ✤à♥❤ ỵ õ F,G = ợ = ú ỵ r t õ t t F,G = F,G = tỗ t A, B C s❛♦ ❝❤♦ F (F − 1)2 (G − 1)2 ✳ G A = + B G−1 F −1 ợ A = ố ữ tr ự ỵ t õ T (r, F ) ≥ (n + k)T (r, f ) − m(r, 1/f ) + Sf (r) ✭✷✳✸✻✮ ❚❛ s➩ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ B = ✈➔ B = 0✳ ❚r÷í♥❣ ❤đ♣ ✶✿ B = ●✐↔ sû A = 1✳ ❚❤➻ t❤❡♦ ✭✷✳✸✺✮ t❛ ❝â F = AG + (1 − A)✳ ▲➟♣ ❧✉➟♥ ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ỵ t õ (n + k)T (r, f ) l l Z(r, f − ) + Z(r, g) + ≤ Z(r, f ) + i=2 Z(r, g − ) i=2 + N (r, f ) + Z(r, g ) + T (r, f ) + Sf (r) + Sg (r) ✭✷✳✸✼✮ ✸✾ ❉➵ t❤➜②✱ ✈ỵ✐ ♠é✐ sè ❞÷ì♥❣ c t❛ ❧✉ỉ♥ ❝â max(0, − c) + c ≥ 4✳ ❚ø ✭✷✳✸✼✮ ✈➔ ❇ê ✤➲ ✶✳✶✳✹✱ t❛ ❝â l l (n + k)T (r, f ) ≤ Z(r, f ) + Z(r, f − ) + Z(r, g) + i=2 Z(r, g − ) i=2 + Z(r, g ) + T (r, f ) + Sf (r) + Sg (r) l l ≤ Z(r, f ) + Z(r, f − ) + Z(r, g) + i=2 Z(r, g − ) i=2 + T (r, g) + T (r, f ) + Sf (r) + Sg (r) ữỡ tỹ ố ợ g t❛ ❝â l (n + k)T (r, g) ≤ Z(r, g) + l Z(r, g − ) + Z(r, f ) + i=2 Z(r, f − ) i=2 + T (r, g) + T (r, f ) + Sf (r) + Sg (r) ❑➳t ❤ñ♣ ❤❛✐ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ❝â (n + k)(T (r, f ) + T (r, g)) ≤ (2l + 2)(T (r, f ) + T (r, g)), ❦➨♦ t❤❡♦ n + k 2l + ứ tt ỵ t❛ ❝â l n+k ≥5+ l max(0, − ki ) + max(0, − k2 ) + i=3 l ≥5+ ki i=2 max(ki , 4) ≥ 10 + 4(l − 1), i=2 ❞♦ ✤â n + k ≥ 4l + ✭✷✳✸✽✮ ◆❤÷ ✈➟② ✤✐➲✉ ❦✐➺♥ n + k ≤ 2l + ❦❤ỉ♥❣ t❤➸ ①↔② r❛✳ ◆❤÷ ✈➟② A = 1, ❦➨♦ t❤❡♦ F = G✳ ❉➝♥ ✤➳♥ αF = αG tù❝ ❧➔ (F ) = (G) ✳ ❚ø ❇ê ✤➲ ✶✳✷✳✷✱ t❛ ❝â F = G tù❝ ❧➔ P (f ) = P (g) ✈➔ ❞♦ P ❧➔ ✤❛ t❤ù❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t tr➯♥ A(C) ♥➯♥ t❛ s✉② r❛ f = g ✳ ❚r÷í♥❣ ❤đ♣ ✷✿ B = ✹✵ ❚❛ ❝â l Z(r, F ) ≤ Z(r, f ) + Z(r, f − ) + Z(r, f ) + Sf (r) i=2 ❚÷ì♥❣ tü ✤è✐ ✈ỵ✐ G t❛ ❝â l Z(r, G) ≤ Z(r, g) + Z(r, g − ) + Z(r, g ) + Sg (r) i=2 ❍ì♥ ♥ú❛✱ ❞♦ f, g ❧➔ ❝→❝ ❤➔♠ ♥❣✉②➯♥ ♥➯♥ N (r, F ) ≤ N (r, f ) + Sf (r) = Sf (r); N (r, G) ≤ N (r, g) + Sg (r) = Sg (r) ❑➳t ❤ñ♣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ❝â Z(r, F ) + Z(r, G) + N (r, F ) + N (r, G) l ≤ Z(r, f ) + Z(r, f − ) + Z(r, f ) + Z(r, g) i=2 l Z(r, g − ) + Z(r, g ) + Sf (r) + Sg (r) + ✭✷✳✸✾✮ i=2 ❚❤❡♦ ❇ê ✤➲ ✶✳✶✳✹✱ ✭✷✳✸✾✮ ✤÷đ❝ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉ Z(r, F ) + Z(r, G) + N (r, F ) + N (r, G) l ≤ Z(r, f ) + Z(r, f − ) + Z(r, f ) + Z(r, g) i=2 l Z(r, g − ) + Z(r, g ) + Sf (r) + Sg (r) + i=2 l ≤ Z(r, f ) + Z(r, f − ) + T (r, f ) + T (r, g) − m(r, 1/f ) i=2 l − m(r, 1/g ) + Z(r, g) + Z(r, g − ) + Sf (r) + Sg (r) i=2 ❑➨♦ t❤❡♦ Z(r, F ) + Z(r, G) + N (r, F ) + N (r, G) ≤ (l + 1)(T (r, f ) + T (r, g)) − m(r, 1/f )) − m(r, 1/g ) + Sf (r) + Sg (r) ✭✷✳✹✵✮ ✹✶ ❚ø ✭✷✳✸✻✮ t❛ ❝â 2(n + k) T (r, f ) + T (r, g) − m(r, 1/f ) − m(r, 1/g ) ≤ T (r, F ) + T (r, G) + Sf (r) + Sg (r), ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✹✵✮ t❛ ❝â Z(r, F ) + Z(r, G) + N (r, F ) + N (r, G) 2l + T (r, F ) + T (r, G) ≤ n+k + Sf (r) + Sg (r) ❚❤❡♦ ❇ê ✤➲ ✶✳✷✳✶ t❛ s✉② r❛ Z(r, F ) + Z(r, G) + N (r, F ) + N (r, G) 2l + T (r, F ) + T (r, G) ≤ + SF (r) + SG (r) n+k 2l + ≤ max(T (r, F ), T (r, G)) + SF (r) + SG (r) n+k ❚ø ✭✷✳✸✽✮ t❛ ❝â t❛ ❝â n + k ≥ 4l + 1, s✉② r❛ lim sup r→∞ ✭✷✳✹✶✮ 2l + < 1✱ ✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ n+k Z(r, F ) + Z(r, G) + N (r, F ) + N (r, G) max(T (r, F ), T (r, G)) < ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✷✳✸ t❛ s✉② r❛ F = G ❤♦➦❝ F G = ❱➻ αF, αG ✈➔ α ❝ò♥❣ t❤✉ë❝ A(C)✱ ♥➯♥ ♥➳✉ F G = t❤➻ (αF )(αG) = (α)2 ∈ Af (C)✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❇ê ✤➲ ✶✳✷✳✻✳ ❉♦ ✤â F = G ✈➔ ❣✐è♥❣ ♥❤÷ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✶ t❛ s✉② r❛ f = g ❍➺ q✉↔ ✷✳✷✳✷✳ ✭❬✷❪✮ ❈❤♦ P ∈ C[x] t❤ä❛ ♠➣♥ P (x) = b(x−a1) n l l (x−ai )ki i=2 ✈ỵ✐ b ∈ C∗, Φ(P ) ≥ 4, ki ≥ ki+1, ≤ i ≤ l − ✈➔ k = ki✳ ●✐↔ sû P t❤ä❛ i=2 ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ • n ≥ k + 2✱ l • n≥5+ max(0, − ki ) + max(0, − k2 ) i=3 ✹✷ ❈❤♦ f, g ∈ A(C) ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t ✈➔ α ∈ Af (C) ∩ Ag (C) ❦❤→❝ 0✳ ◆➳✉ f P (f ) ✈➔ g P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà α ❦➸ ❝↔ ❜ë✐✱ t❤➻ f = g ❍➺ q✉↔ ✷✳✷✳✸✳ ✭❬✷❪✮ ❈❤♦ P ∈ C[x] t❤ä❛ ♠➣♥ P (x) = b(x − a1 )n (x − a2 )k (x − a3 )k ✈ỵ✐ b ∈ C∗ , Φ(P ) = 3, k2 ≥ k3 ✳ ●✐↔ sû P t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ • n ≥ k2 + k3 + 2✱ • n ≥ + max(0, − k3 ) + max(0, − k2 ) ❈❤♦ f, g ∈ A(C) ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t ✈➔ α ∈ Af (C) ∩ Ag (C) ❦❤→❝ 0✳ ◆➳✉ f P (f ) ✈➔ g P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà α ❦➸ ❝↔ ❜ë✐✱ t❤➻ f = g ✷✳✷✳✷ ❚r÷í♥❣ ❤đ♣ P = b(x − a1 )n l (x − ) i=1 ✭❬✷❪✮ ❈❤♦ P ❧➔ ✤❛ t❤ù❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t tr♦♥❣ A(C) s ữủ t ữợ ỵ P l n b(x − a1 ) (x − ), i=2 ✈ỵ✐ l ≥ 3, b ∈ C∗✱ t❤ä❛ ♠➣♥ n ≥ l + ❈❤♦ f, g ∈ A(C) ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t ✈➔ α ∈ Af (C) ∩ Ag (C) ❦❤→❝ 0✳ ◆➳✉ f P (f ) ✈➔ g P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà α ❦➸ ❝↔ ❜ë✐✱ t❤➻ f = g ❚❛ ❝ô♥❣ t ự ố ữ ỵ g P (g) f P (f ) ✈➔ G = ✱ ❦❤✐ ✤â F ✈➔ G ❝❤✉♥❣ ♥❤❛✉ ❣✐→ trà ✶ ❦➸ ✣➦t F = α α ❝↔ ❜ë✐✳ ❱➻ f, g ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t✳ ❚❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ ΨF,G = F 2F G 2G − − + F F −1 G G−1 ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ΨF,G = ●✐↔ sû ΨF,G = 0✱ ✤➦t F = P (f ), G = P (g)✱ ✹✸ ❧➟♣ ❧✉➟♥ ố ữ ự ỵ t õ l T (r, F ) + T (r, G) ≤ 5T (r, f ) + (T (r, f − ) + T (r, g − )) i=2 + 5T (r, g) + (l − 1)(T (r, f ) + T (r, g)) + T (r, f − a2 ) + T (r, g − a2 ) + 5(N (r, f ) + N (r, g)) + Sf (r) + Sg (r) ❉♦ f, g, α t❤✉ë❝ A(C) ♥➯♥ N (r, f ) = N (r, g) = 0, ♥➯♥ t❛ ❝â T (r, F ) + T (r, G) ≤ (4 + 2l)(T (r, f ) + T (r, g)) + Sf (r) + Sg (r) ✭✷✳✹✷✮ ❱➻ F ❧➔ ✤❛ t❤ù❝ ❝õ❛ f ❝â ❜➟❝ n + k + 1✳ ❚❛ ❝â T (r, F ) = (n + k + 1)T (r, f ) + O(1), t÷ì♥❣ tü T (r, G) = (n + k + 1)T (r, g) + O(1) ◆➯♥ tø ✭✷✳✹✷✮ t❛ ❝â (n + k + 1)(T (r, f ) + T (r, g)) ≤ (4 + 2l)(T (r, f ) + T (r, g)) + Sf (r) + Sg (r) ❙✉② r❛ n+k+1 + 2l, ❦➨♦ t❤❡♦ n + l, ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ n ≥ + l tr ỵ F,G = ố ữ tr ự ỵ t ❝â t❤➸ ✈✐➳t ΨF,G = ✈ỵ✐ φ = F (F − 1)2 φ φ (G − 1)2 ✳ ❱➻ ΨF,G = tỗ t A, B C s G ❝❤♦ A = + B, G−1 F −1 A = ú ỵ r Z(r, f ) ≤ T (r, f ), N (r, f ) ≤ T (r, f ), Z(r, f − ) ≤ T (r, f − ) ≤ T (r, f ) + O(1), ✈ỵ✐ i = 2, , l ✈➔ Z(r, f ) ≤ T (r, f ) ≤ 2T (r, f ) + O(1) ❚÷ì♥❣ tü t❛ ❝ơ♥❣ ❝â ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ♥❤÷ t❤➳ ✤è✐ ✈ỵ✐ g ✈➔ g ✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✱ t❛ ❝â T (r, F ) ≥ (n + k)T (r, f ) − m(r, 1/f ) + Sf (r) ✭✷✳✹✹✮ ❚❛ s➩ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ B = ✈➔ B = ❚r÷í♥❣ ❤đ♣ ✶✿ B = ●✐↔ sû A = 1✳ ▲➟♣ ❧✉➟♥ ❣✐è♥❣ ♥❤÷ tr ự ỵ t õ n + k ≤ 2l + ❚ø ❣✐↔ t❤✐➳t ỵ t õ n l + s✉② r❛ n + k ≥ 2l + 4✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ ✭✷✳✹✺✮ ❦❤ỉ♥❣ t❤➸ ①↔② r❛✳ ◆❤÷ ✈➟② A = ❈ơ♥❣ ❧➟♣ ❧✉➟♥ ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ỵ t s r f = g ❚r÷í♥❣ ❤đ♣ ✷✿ B = ▲➟♣ ❧✉➟♥ ❣✐è♥❣ ữ tr ự ỵ t õ Z(r, F ) + Z(r, G) + N (r, F ) + N (r, G) 2l + max(T (r, F ), T (r, G)) + SF (r) + SG (r) n+k ứ tt ỵ t❛ ❝â n ≥ l + 5✱ s✉② r❛ n + k ≥ 2l + 4✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ lim sup r→∞ Z(r, F ) + Z(r, G) + N (r, F ) + N (r, G) max(T (r, F ), T (r, G)) < ❈ô♥❣ ❧➟♣ ❧✉➟♥ ố ữ tr ự ỵ t s✉② r❛ f = g ✹✺ l ✭❬✷❪✮ ❈❤♦ P ∈ C[x], P = b(x − a1) (x − ai) ✈ỵ✐ b ∈ C∗✱ i=2 Φ(P ) ≥ ✈➔ t❤ä❛ ♠➣♥ n ≥ l + 5✳ ❈❤♦ f, g ∈ A(C) ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t ✈➔ α ∈ Af (C) ∩ Ag (C) ❦❤→❝ 0✳ ◆➳✉ f P (f ) ✈➔ g P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà α ❦➸ ❝↔ ❜ë✐✱ t❤➻ f = g ❍➺ q✉↔ ✷✳✷✳✻✳ ✭❬✷❪✮ ❈❤♦ P ∈ C[x], P = b(x − a1 )n (x − a2 )k (x − a3 )k ✈ỵ✐ b ∈ C∗ , Φ(P ) = ✈➔ t❤ä❛ ♠➣♥ n ≥ 8✳ ❈❤♦ f, g ∈ A(C) ❧➔ ❝→❝ ❤➔♠ s✐➯✉ ✈✐➺t ✈➔ α ∈ Af (C) ∩ Ag (C) ❦❤→❝ 0✳ ◆➳✉ f P (f ) ✈➔ g P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà α ❦➸ ❝↔ ❜ë✐✱ t❤➻ f = g✳ ❍➺ q✉↔ ✷✳✷✳✺✳ n ✹✻ ❑➳t ❧✉➟♥ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤✐ ❤❛✐ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❜➟❝ ♥❤➜t ❝õ❛ ❝❤ó♥❣ ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ tr ỵ tt ố tr t t ỵ ỡ r❛✱ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜ê ✤➲ ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✱ ❝➛♥ t❤✐➳t ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ tr♦♥❣ ữỡ ữỡ tr ố ỵ ✈➔ ♠ët sè ❤➺ q✉↔ ✈➲ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❤❛② ❤➔♠ ♥❣✉②➯♥ s✐➯✉ ✈✐➺t t❤æ♥❣ q✉❛ ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ ❜➟❝ ♥❤➜t ❝❤✉♥❣ ♥❤❛✉ ♠ët ❤➔♠ ♥❤ä✳ ❈ö t❤➸ ✣à♥❤ ❧➼ ✷✳✷✳✶ ✈➔ ✷✳✶✳✻ ❝❤♦ ❝❤ó♥❣ t❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤❛ t❤ù❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ✣à♥❤ ❧➼ ✷✳✷✳✶ ✈➔ ✷✳✷✳✹ ❝❤♦ ❝→❝ ❦➳t q✉↔ t÷ì♥❣ tü tr♦♥❣ tr÷í♥❣ ❤đ♣ ❤➔♠ ♥❣✉②➯♥✳ ◆❣♦➔✐ r❛✱ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ❝á♥ tr➻♥❤ ❜➔② ♠ët sè ✈➼ ỵ t❤❛♠ ❦❤↔♦ ❬✶❪ ✭✷✵✵✹✮✱ ❙tr♦♥❣ ❝❛s❡✱ ❈♦♠♣❧❡①✳ ❱❛r✱ ✹✾✭✶✮✱ ♣♣ ❆♥✱ ❚✳ ❚✳ ❍✳✱ ❲❛♥❣✱ ❏✳ ❚✳ ❨✳✱ ❲♦♥❣✱ P✳ ▼✳ ✉♥✐q✉❡♥❡ss ♣♦❧②♥♦♠✐❛❧s✿ t❤❡ ❝♦♠♣❧❡① ✷✺✲✺✹✳ ✭✷✵✶✸✮✱ ❈♦♠♣❧❡① ♠❡r♦♠♦r✲ ♣❤✐❝ ❢✉♥❝t✐♦♥s f P (f )✱ g P (g) s❤❛r✐♥❣ ❛ s♠❛❧❧ ❢✉♥❝t✐♦♥✱ ■♥❞❛❣❛t✐♦♥❡s ▼❛t❤❡♠❛t✐❝❛❡✱ ✷✹✱ ♣♣ ✶✺✲✹✶✳ ❬✷❪ ❇♦✉ss❛❢✱ ❑✳✱ ❊s❝❛ss✉t✱ ❆✳✱ ❖❥❡❞❛✱ ❏✳ ❬✸❪ ❉②❛✈❛♥❛❧✱ ❘✳ ❙✳ ❯♥✐q✉❡♥❡ss ❛♥❞ ✈❛❧✉❡✲s❤❛r✐♥❣ ♦❢ ❞✐❢❢❡r❡♥✲ t✐❛❧ ♣♦❧②♥♦♠✐❛❧s ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✱ ✸✼✹✱ ✭✷✵✶✶✮✱ ♣♣ ✸✸✺✲✸✹✺✳ ✭✷✵✵✼✮✱ ▼❡r♦♠♦r♣❤✐❝ ❙❝✐✳ ▼❛t❤✱ ✶✸✶✭✸✮✱ ♣♣ ✷✶✾✲✷✹✶✳ ❢✉♥❝t✐♦♥s ♦❢ ✉♥✐q✉❡♥❡ss✱ ❇✉❧❧✳ ❬✹❪ ❊s❝❛ss✉t✱ ❆✳ ❬✺❪ ❊s❝❛ss✉t✱ ❆✳✱ ❍❛❞❞❛❞✱ ▲✳✱ ❱✐❞❛❧✱ ❘✳ ❬✻❪ ❋r❛♥❦✱ ●✳✱ ❘❡✐♥❞❡rs✱ ▼✳ ❛♥❞ ♥♦♥ ✭✶✾✾✾✮✱ ❯❘❙ ❯❘❙✱ ❏✳ ◆✉♠❜❡r ❚❤❡♦r②✱ ✼✺✱ ♣♣ ✶✸✸✲✶✹✹✳ ♣❤✐❝ ❢✉♥❝t✐♦♥s ✇✐t❤ ✶✶ ❛♥❞ ❯❘❙■▼ ✭✶✾✾✽✮✱ ❆ ✉♥✐q✉❡ r❛♥❣❡ s❡t ❢♦r ♠❡r♦♠♦r✲ ❡❧❡♠❡♥ts✱ ❈♦♠♣❧❡①✳ ❱❛r✳ ❚❤❡♦r②✳ ❆♣♣❧✱ ✸✼✱ ♣♣ ✶✽✺✲✶✾✸✳ ❬✼❪ ❆ ❯♥✐❝✐t② ❚❤❡♦r❡♠ ❋♦r ❊♥t✐r❡ ❋✉♥❝t✐♦♥s ❈♦♥❝❡r♥✐♥❣ ❉✐❢❢❡r❡♥t✐❛❧ P♦❧②♥♣♠✐❛❧s✱ ■♥❞✐❛♥✳ ❏✳ P✉r❡ ❆♣♣❧✳ ❋❛♥❣✱ ▼✳ ▲✳✱ ❍♦♥❣✱ ❲✳ ✭✷✵✵✶✮✱ ▼❛t❤✳✱ ✸✷✱ ♣♣ ✶✸✹✸✲✶✸✹✽✳ ❬✽❪ ✭✶✾✼✼✮✱ ❋❛❝t♦r✐③❛t✐♦♥ ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ s♦♠❡ ♦♣❡♥ ♣r♦❜❧❡♠s✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳✱ ✺✾✾✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✱ ♣♣ ✺✶✕✻✼✳ ●r♦ss✱ ❋✳ ✹✽ ❬✾❪ ❬✶✵❪ ▲✐✱ P✳✱ ❨❛♥❣✱ ❈✳ ❈✳ s❡ts ♦❢ ♠❡r♦♠♦r♣❤✐❝ ✭✶✾✾✺✮✱ ❙♦♠❡ ❢✉rt❤❡r r❡s✉❧ts ♦♥ t❤❡ ✉♥✐q✉❡ r❛♥❣❡ ❢✉♥❝t✐♦♥s✱ ❑♦❞❛✐ ▼❛t❤✳ ❏✱ ✶✽✱ ♣♣ ✹✼✸✲✹✺✵✳ ❯♥✐q✉❡♥❡ss t❤❡♦r❡♠s ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝✲ t✐♦♥s ❝♦♥❝❡r♥✐♥❣ ❢✐①❡❞ ✲ ♣♦✐♥ts✱ ❈♦♠♣❧❡①✳ ❱❛r✳ ❚❤❡♦r② ❆♣♣❧✳✱ ✹✾✱ ♣♣ ▲✐♥✱ ❲✳✱ ❨✐✱ ❍✳ ✭✷✵✵✹✮✱ ✼✾✸✲✽✵✻✳ ❬✶✶❪ ❬✶✷❪ ✭✷✵✵✹✮✱ ❯♥✐q✉❡♥❡ss t❤❡♦r❡♠s ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝✲ t✐♦♥s✱ ■♥❞✐❛♥✳ ❏✳ P✉r❡ ❆♣♣❧✳ ▼❛t❤✳✱ ✸✺✱ ♣♣ ✶✷✶✲✶✸✷✳ ▲✐♥✱ ❲✳✱ ❨✐✱ ❍✳ ▲✐♥✱ ❲✳ ❈✳✱ ❨✐✱ ❍✳ ❳✳ t✐♦♥s ❈♦♥❝❡r♥✐♥❣ ❋✐①❡❞ ✭✷✵✵✹✮✱ ❆ ❯♥✐❝✐t② ❚❤❡♦r❡♠ ❋♦r ❊♥t✐r❡ ❋✉♥❝✲ P♦✐♥ts✱ ❈♦♠♣❧❡①✳ ❱❛r✳ ❚❤❡♦r②✳ ❆♣♣❧✱ ✹✾✱ ♣♣ ✼✾✸✲✽✵✻✳ ❬✶✸❪ ✭✶✾✾✼✮✱ ❯♥✐q✉❡♥❡ss ❛♥❞ ✈❛❧✉❡ s❤❛r✐♥❣ ❢✉♥❝t✐♦♥✱ ❆♥♥✳ ❆❝❛❞✳ ❙❝✐✳ ❋❡♥♥✳ ▼❛t❤✱ ✷✵✷✭✷✮✱ ♣♣ ❨❛♥❣✱ ❈✳ ❈✳✱ ❍✉❛✱ ❳✳ ❍✳ ♦❢ ♠❡r♦♠♦r♣❤✐❝ ✸✾✺✲✹✵✻✳ ❬✶✹❪ ✭✶✾✾✺✮✱ ❆ q✉❡st✐♦♥ ♦❢ ●r♦ss ❛♥❞ ❢✉♥❝t✐♦♥s✱ ◆❛❣♦②❛ ▼❛t❤✳ ❏✳✱ ✶✸✽✱ ✶✻✾✕✶✼✼✳ ❨✐✱ ❍✳ ❨✳ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ ❡♥t✐r❡ ✭✷✵✵✽✮✱ ❊♥t✐r❡ ♦r ♠❡r♦✲ ♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s s❤❛r✐♥❣ ♦♥❡ ✈❛❧✉❡✱ ❈♦♠♣✉t❡rs ❛♥❞ ▼❛t❤❡♠❛t✐❝s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ✺✻✱ ✶✽✼✻✲✶✽✽✸✳ ❬✶✺❪ ❩❤❛♥❣✱ ❳✳ ❨✳✱ ❈❤❡♥✱ ❏✳ ❋✳ ▲✐♥✱ ❲✳ ❈✳ ❬✶✻❪ ❲❛♥❣✱ ▲✳✱ ▲✉♦✱ ❳✳ ❝♦♥❝❡r♥✐♥❣ ❢✐①❡❞ ✭✷✵✶✷✮✱ ❯♥✐q✉❡♥❡ss ♦❢ ♠❡r♦♠♦r♣❤✐❝ ♣♦✐♥ts✱ ▼❛t❤✳ ❙❧♦✈❛❝✱ ✻✷✭✶✮✱ ✷✾✲✸✽✳ ❢✉♥❝t✐♦♥s ... + ✭t÷ì♥❣ ù♥❣ n ≥ deg(Q) + 2✮✳ ◆➳✉ P (f )f = P (g)g t❤➻ P (f ) = P (g) ❇ê ✤➲ ✶✳✷✳✷✳ ✣➦t k = deg(Q)✳ ❱➻ P (f )f = P (g)g tỗ t c C s P (f ) = P (g) + c✳ ●✐↔ sû r➡♥❣ c = t ỵ t õ ự ♠✐♥❤✳ T (r,... P (f )) ≤ (k + 1)T (r, f ) + O(1) ❚❛ ❝ô♥❣ ❝â Z(r, P (f ) − c) = Z(r, P (g)) ≤ Z(r, g) + Z(r, Q(g)) ≤ T (r, g) + T (r, Q(g)) ❚❤❡♦ ❇ê ✤➲ ✶✳✶✳✷✱ t❛ ❝â Z(r, P (f ) − c) ≤ (k + 1)T (r, g) + O(1) ... P (g) ❝❤✉♥❣ ♥❤❛✉ ♠ët ❣✐→ trà ♠ët ❣✐→ trà ❈▼❄ ▼ët ❝→❝❤ tỹ t ữ r ọ s tỗ t ❤❛② ❦❤æ♥❣ ✤❛ t❤ù❝ ❝❤ù❛ ✤↕♦ ❤➔♠ P s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t ❝ù ❝➦♣ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ f ✈➔ g t❛ ❝â f ≡ g ♥➳✉ P (f ) ✈➔ P (g)