✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ Pề ì ị P P ệ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ Pề ì ị P P ệ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ữớ ữợ ❱Ơ ❍❖⑨■ ❆◆ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺ ✐ ▲í✐ ❝❛♠ ✤♦❛♥ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr➻♥❤ ự r tổ ữợ sỹ ữợ ❚❙✳ ❱ơ ❍♦➔✐ ❆♥✳ ▲✉➟♥ ✈➠♥ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❝ỉ♥❣ ❜è tr♦♥❣ ❜➜t ❦➻ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ♥➔♦ ✈➔ ♠å✐ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ tr✉♥❣ t❤ü❝✳ ❍å❝ ✈✐➯♥ P❤ị♥❣ ❚❤à ❍÷ì♥❣ ✐✐ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❑❤♦❛ s❛✉ ✤↕✐ ❤å❝✱ ữ ữợ sỹ ữợ s ụ ❞à♣ ♥➔②✱ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ♥❤➜t ✤➳♥ ❚✐➳♥ s➽ ❱ô ❍♦➔✐ ❆♥✱ ữớ tớ t t ữợ ❣✐ó♣ ✤ï ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ tèt ❧✉➟♥ ✈➠♥ ♥➔②✳ ▼ët ❧➛♥ ♥ú❛ t→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ✤➳♥ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ❝õ❛ ❑❤♦❛ ❚♦→♥✱ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✈➔ ❱✐➺♥ ❚♦→♥ ❤å❝ ❱✐➺t ◆❛♠✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✈➔ ❝→❝ t❤➔♥❤ ✈✐➯♥ tr♦♥❣ ❧ỵ♣ ❝❛♦ ❤å❝ ❑✷✶❇ ✤➣ ❧✉ỉ♥ õ♥❣ ❤ë ✈➔ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚✉② ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ s♦♥❣ t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝ ❝õ❛ ❜↔♥ t❤➙♥ ❝â ❤↕♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sõt t ữủ ỳ ỵ õ õ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈➔ ❜↕♥ ✤å❝✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✵✸ ♥➠♠ ✷✵✶✺ ❚→❝ ❣✐↔ P❤ị♥❣ ❚❤à ❍÷ì♥❣ ✐✐✐ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ỵ tt ✶✳✶ ✶✳✷ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✶ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ♣ ✲❛❞✐❝ ✶✳✶✳✶ ❚r÷í♥❣ ❝→❝ sè ✶✳✶✳✷ ❍➔♠ s✐♥❤ ❜ð✐ ❝❤✉é✐ ❧ô② t❤ø❛ ✶✳✶✳✸ ❍➔♠ ♣❤➙♥ ❤➻♥❤ ♣ ✲❛❞✐❝ ♣ ✲❛❞✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ❍➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣ ✲❛❞✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✶ ❍➔♠ ✤➦❝ tr÷♥❣ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✷ ❍❛✐ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✷✳✸ ❇ê ✤➲ q✉❛♥ ❤➺ sè ❦❤✉②➳t ✷✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ rt ỵ rt ỵ rt t t❤❛♠ ❦❤↔♦ ✷✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣ ✲❛❞✐❝ ✳ ✳ ✳ ✷✹ ✷✾ ✹✼ ✹✽ ✐✈ ❈→❝ ❦➼ ❤✐➺✉ • Cp ✿ ❚r÷í♥❣ sè ♣❤ù❝ ♣ ✲❛❞✐❝ ❢ ✿ ❍➔♠ ♣❤➙♥ ❤➻♥❤ ♣ ✲❛❞✐❝ • Nf (a, r)✿ ❍➔♠ ✤➳♠ ❝õ❛ ❢ t↕✐ ❛ • mf (∞, r) ✿ ❍➔♠ ã Tf (r) trữ ã ã O(1) ữủ ợ ã Nf (r), Nk (f, r)✿ • W (f ) • Hj ✿ ❍➔♠ ✤➳♠✱ ❤➔♠ ✤➳♠ ♠ù❝ ❲r♦♥s❦✐❛♥ ❝õ❛ ❤➔♠ f ã Fj (z) = Pữỡ tr ❝õ❛ s✐➯✉ ♣❤➥♥❣ k ✶ ▼ð ✤➛✉ ❚♦→♥ ❤å❝ ✤÷đ❝ ❝♦✐ ❧➔ ✤➾♥❤ ❝❛♦ tr➼ t✉➺ ❝õ❛ ❝♦♥ ♥❣÷í✐✱ ♥â ①➙♠ ♥❤➟♣ ✈➔♦ ❤➛✉ ❤➳t ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ❧➔ t ỵ tt q trồ ❚♦→♥ ❤å❝ ♥❣➔② ❝➔♥❣ ♣❤→t tr✐➸♥ ♠↕♥❤ ♠➩ q✉❛ tø♥❣ t❤í✐ ❦➻✳ ✣➦❝ ❜✐➺t tr♦♥❣ ✤➛✉ t❤➳ ❦✛ ❳❳ ❧➔ sỹ r ỵ tt ữủ ♠ët tr♦♥❣ ♥❤ú♥❣ t❤➔♥❤ tü✉ ♥ê✐ ❜➟t ✈➔ s➙✉ s➢❝ t rồ t ỵ tt ỵ rt rở ỵ tt trữớ ủ ữớ ✈➔ ✤÷❛ r❛ ♥❤✐➲✉ ù♥❣ ❞ư♥❣ q✉❛♥ trå♥❣✳ ❱➻ ✈➟② ỵ tt ố ợ ữớ ✤÷đ❝ ♠❛♥❣ t➯♥ ❤❛✐ ♥❤➔ t♦→♥ ❤å❝ ①✉➜t s➢❝ ❝õ❛ t õ ỵ tt rt ổ q ữợ ự t q tr♦♥❣ ❣✐↔✐ t➼❝❤ ❤➔♠✱ tr♦♥❣ ✤↕✐ sè ❝ơ♥❣ ♥❤÷ tr♦♥❣ ỵ tt số ữủ r ợ t➯♥ t✉ê✐ ❝õ❛ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ tr➯♥ t❤➳ ❣✐ỵ✐ ♥❤÷ P❤✳ ●r✐❢❢✐t❤s✱ ❍✳❲❡②❧✱ P✳❱♦❥t❛✱ ●✳❋❛❧t✐♥❣s✱✳✳✳ ♣ ✲❛❞✐❝✮✱ ❧➛♥ ✤➛✉ t✐➯♥ ❍❛ ❍✉② ❑❤♦❛✐ ✈➔ ▼② ❱✐♥❤ ◗✉❛♥❣ ✤➣ ①➙② ❞ü♥❣ t÷ì♥❣ tü ♣ ✲❛❞✐❝ ❝õ❛ ❚r➯♥ tr÷í♥❣ ❝ì sð ổ st trữớ số ỵ tt tổ q ỵ t q ỳ t tr t t ỵ tt ❝â t❤➸ t➻♠ t❤➜② tr♦♥❣ ♥❤ú♥❣ ❝æ♥❣ tr➻♥❤ ❬✷❪✱ ❬✸❪✱ ỵ tt rt ữớ ữ ✤➣ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝✳ ❚✉② ♥❤✐➯♥ sỹ t ỵ tt tr trữớ ỡ sð ❦❤ỉ♥❣ ❆❝s✐♠❡t ♠ỵ✐ ❝❤➾ ❜➢t ✤➛✉ ✈➔ ❝á♥ ❧➙✉ ợ ữủ t ự ữủ ỵ ỵ tt tr trữớ ủ ✷ ♠ët ❝❤✐➲✉✳ ◆➠♠ ✶✾✾✸✱ ❲✳❈❤❡rr② ✤➣ ①➙② ❞ü♥❣ ♠ët ❜↔♥ s❛♦ ♣ ✲❛❞✐❝ ❤➛✉ ❤➳t ❝→❝ ❦➳t q✉↔ ❝õ❛ ỵ tt ố ợ ✤à♥❤ tr➯♥ ✤➽❛ t❤õ♥❣ ❝õ❛ ♠➦t ♣❤➥♥❣ ♣ ✲❛❞✐❝ Cp✳ õ ú t ỵ tt rt ợ tr trữớ ủ ✶✾✾✺ ❍❛ ❍✉② ❑❤♦❛✐ ✈➔ ▼❛✐ ❱❛♥ ❚✉ ❬✺❪ ✤➣ t ự ỵ rt ữợ ự tổ ự t ỵ rt r ❧✉➟♥ ✈➠♥ ♥➔② tæ✐ s➩ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ỵ õ r ởt số ự q trồ ỵ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ♣ ✲❛❞✐❝ ✈➔♦ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ sü s✉② ❜✐➳♥ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ♣ ✲❛❞✐❝✳ ▼➦t ❦❤→❝✱ ỵ s rt q ✭①❡♠ ❬✶✲✷✲✸❪✮ ❧➔ ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ sæ✐ ✤ë♥❣ ✈➔ t❤í✐ sü✳ ❱➻ ✈➟② ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ tr➻♥❤ ❜➔② ❧↕✐ tữỡ tỹ ỵ s ✲❛❞✐❝✳ ✣➙② ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ✈➜♥ ✤➲ ♠❛♥❣ t➼♥❤ t❤í✐ sü ✈➔ ❝➜♣ t❤✐➳t ❝õ❛ ❣✐↔✐ t➼❝❤ ♣ ✲❛❞✐❝✱ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ♥❣✉②➯♥ ❝ù✉✳ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ✷ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✳ ❚r➻♥❤ ❜➔② ởt số tự ỵ tt ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣ ✲❛❞✐❝✳ ❈❤÷ì♥❣ ✷✳ ❚r➻♥❤ ❜➔② ❧↕✐ ỵ rt ự ỵ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ sü s✉② ❜✐➳♥ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ tữỡ tỹ ỵ ▼❛s♦♥ ❝❤♦ ❝→❝ ❤➔♠ ♥❣✉②➯♥ ♣ ✲❛❞✐❝✳ ✸ ❈❤÷ì♥❣ ✶ ỵ tt ♥❛② tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ❝❛♦ ❤å❝ ❝❤✉②➯♥ ♥❣➔♥❤ t♦→♥ ❣✐↔✐ t➼❝❤ t↕✐ ❑❤♦❛ ❚♦→♥ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❣✐→♦ tr➻♥❤ ❣✐↔✐ t➼❝❤ ♣ ✲❛❞✐❝ ❬✶❪ ✤➣ ✤÷đ❝ ✤÷❛ ✈➔♦ ❣✐↔♥❣ ❞↕②✳ ◆❣♦➔✐ r❛✱ ❝ơ♥❣ ❝â ♠ët sè t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❜➡♥❣ t✐➳♥❣ ❆♥❤ ❬✷❪✱ ❬✸✲✹❪ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ♥➔②✳ ❚ø ✤â ❝→❝ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝✱ ♥❣❤✐➯♥ ❝ù✉ s✐♥❤ ✈➔ ♥❤ú♥❣ ♥❣÷í✐ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✱ ❝â t❤➸ t❤❛♠ ❦❤↔♦ ❜ê s✉♥❣ ✈➔ ♠ð rë♥❣ t❤➯♠ ❦✐➳♥ t❤ù❝ ỵ tt ổ q t ❧✐➺✉ ♥➔②✱ tr➯♥ ❝ì sð ❝→❝ ❦✐➳♥ t❤ù❝ ✤➣ ❜✐➳t✱ tr♦♥❣ ❈❤÷ì♥❣ ✶ tỉ✐ ①✐♥ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ tự ỵ tt ✲❛❞✐❝ ✤➸ ❞ị♥❣ ❝❤♦ ❈❤÷ì♥❣ ✷✳ ✶✳✶ ▼ët sè ❦✐➳♥ tự ỡ rữớ số ợ p ❧➔ ♠ët sè ♥❣✉②➯♥ tè ❝è ✤à♥❤✱ ❖str♦✇s❦✐ ✤➣ ❦❤➥♥❣ ✤à♥❤✿ ❈❤➾ ❝â ❤❛✐ ❝→❝❤ tr❛♥❣ ❜à ❝❤✉➞♥ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ ❝❤♦ tr÷í♥❣ ❤ú✉ t➾ ▼ð rë♥❣ t❤❡♦ ❝❤✉➞♥ t❤ỉ♥❣ t❤÷í♥❣ t❛ ❝â tr÷í♥❣ sè t❤ü❝ ❝❤✉➞♥ ♣ ✲❛❞✐❝ t❛ ❝â tr÷í♥❣ sè Qp✳ ❑➼ ❤✐➺✉ Cp = Qp ❧➔ ❜ê s✉♥❣ ❝õ❛ ❜❛♦ ✤â♥❣ ✤↕✐ sè ❝õ❛ R✱ Qp ✳ Q✳ ♠ð rë♥❣ t❤❡♦ ❚❛ ❣å✐ Cp ❧➔ ✹ tr÷í♥❣ sè ♣❤ù❝ ♣ ✲❛❞✐❝✳ ❈❤✉➞♥ tr➯♥ ✤÷đ❝ ♠ð rë♥❣ tü ♥❤✐➯♥ ❝õ❛ ❝❤✉➞♥ Cp ♣ ✲❛❞✐❝ tr➯♥ Qp✳ ❑➼ ❤✐➺✉✿ Dr = {z ∈ Cp : |z| ≤ r} , D = {z ∈ Cp : |z| = r} ●✐↔ sû f (z) Dr ❧➔ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐ f (z) = an z n ✳ n≥0 ❉♦ lim |an | |z n | = n tỗ t n N ✤➸ |an | |z n | ✤↕t ❣✐→ trà ❧ỵ♥ ♥❤➜t✳ ❑❤✐ ✤â t❛ ✤➦t✿ |f |r = max {|an | |z n |} n≥0 ❚r♦♥❣ s✉èt ❧✉➟♥ ✈➠♥ t q ữợ log logp s ❜ð✐ ❝❤✉é✐ ❧ô② t❤ø❛ ♣ ✲❛❞✐❝ ❍➔♠ s✐♥❤ ❜ð✐ ❝❤✉é✐ ❧ô② t❤ø❛ ♣ ✲❛❞✐❝ ❧➔ ❤➔♠ ❝â ❞↕♥❣ ∞ an z n , f (z) = (an ∈ Cp ) ✭✶✳✶✮ n=0 ∞ ❚❛ ❝â t❤➸ ❣→♥ ❝❤♦ f (z) ❣✐→ trà ❝õ❛ tê♥❣ ❝❤✉é✐ an z n ✈ỵ✐ ♠é✐ z ∈ Cp n=0 ♠➔ ρ |an z n | −→ ❦❤✐ n −→ ∞ ✭✈➻ ❦❤✐ ✤â ❝❤✉é✐ ❤ë✐ tư✮✳ ❇→♥ ❦➼♥❤ ❤ë✐ tư ❝õ❛ ❝❤✉é✐ ✭✶✳✶✮ ✤÷đ❝ t➼♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ 1 = lim sup |an | n ρ n→∞ ∞ ●✐↔ sû ❝❤✉é✐ ❧ô② t❤ø❛ < ρ ≤ +∞✳ f (z) = ❱ỵ✐ ♠é✐ an z n ❝â ❜→♥ ❦➼♥❤ ❤ë✐ tö ❧➔ n=0 + r ∈ R : < r < số ợ t à(r, f ) = max |an |rn n≥0 ✈➔ ❝❤➾ sè tr✉♥❣ t➙♠ ν(r, f ) = max {n : |an |rn = µ(r, f )} n≥0 ρ✿ ✸✹ ❈❤ù♥❣ ♠✐♥❤✳ f = (f0 , f1 , f2 ) : C −→ P2 (Cp ) ❧➔ ♠ët ✤÷í♥❣ ❝♦♥❣ ↔♥❤ ♥➡♠ tr♦♥❣ ✤÷í♥❣ ❝♦♥❣ ①↕ ↔♥❤ ❋❡r♠❛t X ✳ ❑❤✐ ●✐↔ sû ❝❤➾♥❤ ❤➻♥❤ ❜➜t ❦➻ ❝â ✤â✱ t❛ ❝â f0d + f1d = f2d ❤❛② f0d + f1d − f2d = ❉♦ d ≥ 3✱ →♣ ❞ö♥❣ ❇ê ✤➲ ❇♦r❡❧ ♣ ✲❛❞✐❝ t❛ s✉② r❛ f0, f1, f2 t✉②➳♥ t➼♥❤ ✈ỵ✐ ♥❤❛✉ tø♥❣ ✤ỉ✐ ♠ët✳ ◆â✐ ❝→❝❤ ❦❤→❝ ✤✐ ♣❤ö t❤✉ë❝ f0 , f1 , f2 s❛✐ ❦❤→❝ ♥❤❛✉ ♠ët ♥❤➙♥ tû ❤➡♥❣ sè✳ ❚❤❡♦ t➼♥❤ ❝❤➜t t❤✉➛♥ ♥❤➜t ❝õ❛ tå❛ ✤ë ①↕ ↔♥❤ t❛ s✉② r❛ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ f ❧➔ →♥❤ ①↕ ❤➡♥❣✳ ❚✐➳♣ t❤❡♦✱ ú tổ tr tữỡ tỹ ỵ s rữợ t t t rã✱ ❣✐ú❛ t➟♣ ❤ñ♣ ❝→❝ sè ♥❣✉②➯♥ ✈➔ t➟♣ ❝→❝ ✤❛ t❤ù❝ ❝â ♥❤ú♥❣ t➼♥❤ ❝❤➜t r➜t ❣✐è♥❣ ♥❤❛✉ s❛✉ ✤➙②✿ ✶✳ ❈→❝ q✉② t➢❝ ❝ë♥❣✱ trø✱ ♥❤➙♥✱ ❝❤✐❛ ❤♦➔♥ t♦➔♥ ♥❤÷ ♥❤❛✉ ❝❤♦ ❝↔ ❤❛✐ t➟♣✳ ✷✳ ◆➳✉ ✤è✐ ✈ỵ✐ sè ♥❣✉②➯♥✱ t❛ ❝â ❝→❝ sè ♥❣✉②➯♥ tè✱ t❤➻ ✤è✐ ✈ỵ✐ ❝→❝ ✤❛ t❤ù❝✱ t❛ ❝â ✤❛ t❤ù❝ ❜➜t q ố ợ số ụ ữ ố ợ tự õ t ữợ ợ t ỡ ỳ tr trữớ ủ ÷ỵ❝ ❝❤✉♥❣ ❧ỵ♥ ♥❤➜t ♥➔② t➻♠ ✤÷đ❝ ❜➡♥❣ t❤✉➟t t♦→♥ ❊✉❝❧✐❞❡✳ ✹✳ ▼é✐ sè ♥❣✉②➯♥ ❝â t❤➸ ♣❤➙♥ t➼❝❤ t❤➔♥❤ ❝→❝ t❤ø❛ sè ♥❣✉②➯♥ tè✱ ♠é✐ ✤❛ t❤ù❝ ❝â t❤➸ ♣❤➙♥ t➼❝❤ t❤➔♥❤ t➼❝❤ ❝→❝ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉②✳ ✺✳ ▼é✐ sè ♥❣✉②➯♥ ❝â ❣✐→ trà t✉②➺t ✤è✐ ❝õ❛ ♥â✱ ❝ơ♥❣ ♥❤÷ ♠é✐ ✤❛ t❤ù❝ ❦❤→❝ ❦❤ỉ♥❣ ✤➲✉ ❝â ❜➟❝✳ ❈❤ó♥❣ t❛ ❝â t❤➸ ❦➨♦ ❞➔✐ ❜↔♥ ❞❛♥❤ s→❝❤ ♥➔②✳ Ð ✤➙② ❝❤ó♥❣ tỉ✐ ✤✐ ✈➔♦ ♠ët ✈➔✐ sü tữỡ tỹ õ t ỡ ỵ sỹ t÷ì♥❣ tü ❣✐ú❛ ♣❤➙♥ t➼❝❤ r❛ t❤ø❛ sè ♥❣✉②➯♥ tè ✈➔ ♣❤➙♥ t➼❝❤ ❜➜t ❦❤↔ q✉②✳ ◆➳✉ ❣✐↔ t❤✐➳t k ❧➔ ✸✺ tr÷í♥❣ ✤â♥❣ ✤↕✐ sè t❤➻ ♠é✐ ✤❛ t❤ù❝ f (x) k[x] õ t t ữủ ữợ ❞↕♥❣ f (x) = pα1 pα2 pαnn tr♦♥❣ ✤â pi (x) = x − , ∈ k ✳ ◆❤÷ ✈➟②✱ ❝â t❤➸ t❤➜② r➡♥❣✱ tr♦♥❣ sü ♣❤➙♥ t➼❝❤ ❜➜t ❦❤↔ q✉② ✈➔ ♣❤➙♥ t➼❝❤ r❛ t❤ø❛ sè ♥❣✉②➯♥ tè✱ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ t÷ì♥❣ ù♥❣ ợ ữợ tố số õ sè ❝→❝ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t ❝õ❛ ✤❛ t❤ù❝ ❝â ✈❛✐ trỏ tữỡ tỹ ữ số ữợ tố ❜✐➺t ❝õ❛ sè ♥❣✉②➯♥✳ ❚ø ♥❤➙♥ ①➨t ✤â t❛ ✤✐ s ỵ a ❧➔ ♠ët sè ♥❣✉②➯♥✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ ❝➠♥ ❝õ❛ ❛✱ No(a) t ữợ tố ❜✐➺t ❝õ❛ a No (a) = p p|a ❚❛ s➩ t sỹ tữỡ tỹ tr ũ ợ t t tự ủ ỵ ởt ữớ ự ỵ rt s ự ởt ỵ rt s tự ỵ ✳ ●✐↔ sû a(t), b(t), c(t) ❧➔ ❝→❝ ✤❛ t❤ù❝ ợ ỵ s số ự tố ũ ♥❤❛✉ tø♥❣ ❝➦♣ ✈➔ t❤ä❛ ♠➣♥ ❤➺ t❤ù❝ a(t) + b(t) = c(t) ❑❤✐ ✤â✱ ♥➳✉ ❦➼ ❤✐➺✉✱ no(f ) ❧➔ sè ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t ❝õ❛ ♠ët ✤❛ t❤ù❝ f ✱ t❤➻ t❛ ❝â max {deg a, deg b, deg c} no (abc) ỵ s t ởt ự ỡ ỵ rt tr tự ỵ ỵ rt tr tự ổ tỗ t t❤ù❝ a, b, c ❦❤→❝ ❤➡♥❣ sè✱ ❤➺ sè ♣❤ù❝✱ ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉✱ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ an + bn = cn ✈ỵ✐ n ≥ 3✳ ✸✻ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ❝→❝ ✤❛ t❤ù❝ a, b, c t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ♥â✐ tr➯♥✳ ❘ã r➔♥❣ sè ❝→❝ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t ❝õ❛ ✤❛ t❤ù❝ q✉→ deg a + deg b + deg c✳ an bn cn ❦❤ỉ♥❣ ✈÷đt ⑩♣ ❞ư♥❣ ỵ s t õ n deg a deg a + deg b + deg c − ❱✐➳t t tự tr ợ b, c rỗ tứ ✈➳ ❜❛ ❜➜t ✤➥♥❣ t❤ù❝✱ t❛ ✤÷đ❝ n(deg a + deg b + deg c) ≤ 3(deg a + deg b + deg c) − ❚❛ ❝â ♠➙✉ t❤✉➝♥ n ỵ s sỹ tữỡ tỹ ỳ số tự ủ ỵ ❝❤♦ ●✐↔ t❤✐➳t ✑❛❜❝✑✭❖❝st❡r❧❡✱ ✶✾✽✻✮ a, b, c ❧➔ ❝→❝ sè ♥❣✉②➯♥✱ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ✈➔ t❤ä❛ ♠➣♥ ❤➺ t❤ù❝ a + b = c✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ > tỗ t số C s sû max(|a|, |b|, |c|) < CN 1+ε , tr♦♥❣ ✤â N= p ❧➔ ❝➠♥ ❝õ❛ abc✳ p|abc ❍♦➔♥ t♦➔♥ t÷ì♥❣ tü ♥❤÷ tr➯♥✱ tø ❣✐↔ t❤✐➳t ✧❛❜❝✧ ❝â t❤➸ s✉② r ỵ rt t ợ ợ ♣❤÷ì♥❣ tr➻♥❤ ❋❡r♠❛t ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥ ✳ p ❧➔ ÷ỵ❝ ♥❣✉②➯♥ tè ♥➔♦ ✤â ❝õ❛ ♠ët tr♦♥❣ ❝→❝ sè a, b, c ❝❤➥♥❣ ❤↕♥ p | a✳ ❑❤✐ ✤â ♥➳✉ p ❧ỵ♥ t❤➻ tr♦♥❣ sü ♣❤➙♥ t➼❝❤ ❝õ❛ a r❛ t❤ø❛ sè ♥❣✉②➯♥ tè✱ p ♣❤↔✐ ❝â sè ♠ô t÷ì♥❣ ✤è✐ ♥❤ä ✭✤➸ |a| ❦❤ỉ♥❣ ✈÷đt q✉→ ①❛ ❝➠♥ ❝õ❛ abc t❤❡♦ ❣✐↔ t❤✐➳t ð tr➯♥✳ ✣✐➲✉ ♥➔② ❝ô♥❣ ❣✐↔✐ t❤➼❝❤ ❧➼ ❞♦ t↕✐ s❛♦ ❚❤➟t ✈➟②✱ ❣✐↔ sû ữỡ tr rt ổ õ ợ ợ õ ữợ tố an , bn , cn s➩ t❤❛♠ ❣✐❛ ✈ỵ✐ ❜➟❝ q✉→ ❧ỵ♥✳ ❈â ỵ s ✤➲ ❧✐➯♥ q✉❛♥ ❧➔ ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ sæ✐ ✤ë♥❣ tớ sỹ rữợ t ú tổ ữ r ởt t q tốt ỡ t q ỵ tr ỵ f1, , fn+1 ❧➔ n + ❤➔♠ ♥❣✉②➯♥ ♣✲❛❞✐❝ s❛♦ ❝❤♦ f1 , f2 , , fn ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣✱ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ n fi = fn+1 i=1 ❑❤✐ ✤â n+1 max {Tfi (r)} ≤ 1≤i≤n+1 ❈❤ù♥❣ ♠✐♥❤✳ t➼♥❤ ✈➔ Nn−1,fi (r) − i=1 n(n − 1) log r + O(1) f1 , , fn ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣✱ ✤ë❝ ❧➟♣ t✉②➳♥ f1 + + fn = fn+1 ♥➯♥ f2 , , fn+1 ❝ô♥❣ ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❱➻ ❝❤✉♥❣ ✈➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ❉♦ ✤â f = (f1 , f2 , , fn ) ✈➔ g = (f2 , f3 , , −fn+1 ) ❧➔ ❝→❝ ❜✐➸✉ ❞✐➵♥ t÷ì♥❣ ù♥❣ ❝õ❛ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ❦❤ỉ♥❣ s✉② ❜✐➳♥ t✉②➳♥ t➼♥❤ f, g tø Cp ✤➳♥ Pn−1 (Cp ) = Pn−1 ✳ ✣➦t Xi = (z1 , , zn ) ∈ Pn−1 : zi = ✈ỵ✐ i = 1, , n✱ Xn+1 = (z1 , , zn ) ∈ Pn−1 : z1 + + zn = X1 , X2 , , Xn+1 ❧➔ ❝→❝ s✐➯✉ ♣❤➥♥❣ ð ✈à tr➼ tờ qt tr Pn1 (Cp ) ỵ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ♣ ✲❛❞✐❝ ❝❤♦ f ✈➔ n + s✐➯✉ ♣❤➥♥❣ X1 , X2 , , Xn , Xn+1 t❛ ❝â ❑❤✐ ✤â n+1 ((n + 1) − (n − 1) − 1)Tf (r) ≤ Nn−1,f (Xi , r) − i=1 n(n − 1) log r + O(1), ❍❛② Tf (r) ≤ Nn−1,f1 (r) + Nn−1,f2 (r) + + Nn−1,fn (r) + Nn−1,f1 + +fn (r) − n(n − 1) log r + O(1) ❉♦ ✤â Tf (r) ≤ Nn−1,f1 (r) + Nn−1,f2 (r) + + Nn−1,fn (r) + Nn−1,fn+1 (r) ✸✽ − ✭❱➻ n(n − 1) log r + O(1) f1 + f2 + + fn = fn+1 ✮✳ ❱➟② n+1 Tf (r) ≤ Nn−1,fi (r) − i=1 n(n − 1) log r + O(1) ữỡ tỹ ỵ rt ♣❤➥♥❣ X1 , X2 , , Xn , Xn+1 ♣ ✲❛❞✐❝ ❝❤♦ g ✈➔ n + s✐➯✉ t❛ ❝â n+1 ((n + 1) − (n − 1) − 1)Tg (r) ≤ Nn−1,g (Xi , r) − i=1 ✭❱➻ n(n − 1) log r + O(1), Tg (r) ≤ Nn−1,f2 (r) + + Nn−1,fn+1 (r) + Nn−1,f2 + +fn −fn+1 (r) n(n − 1) − log r + O(1), Tg (r) ≤ Nn−1,f2 (r) + + Nn−1,fn+1 (r) + Nn−1,f1 (r) n(n − 1) − log r + O(1) f2 + + fn − fn+1 = −f1 ✮✳ ❉♦ ✤â n+1 Tg (r) ≤ Nn−1,fi (r) − i=1 n(n − 1) log r + O(1) ▼➦t ❦❤→❝ Tf (r) = max {Tf1 (r), Tf2 (r), , Tfn (r)} , Tg (r) = max Tf2 (r), , Tfn+1 (r) ❉♦ ✤â n+1 max {Tfi (r)} ≤ 1≤i≤n+1 Nn−1,fi (r) − i=1 n(n − 1) log r + O(1) ỵ ữủ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✷✳✶✻✳ ●✐↔ sû X1, X2, , Xq ❧➔ ❝→❝ s✐➯✉ ♣❤➥♥❣ ð ✈à tr➼ tê♥❣ q✉→t tr♦♥❣ Pn(Cp)✱ k1, k2, , kq ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ f : Cp −→ Pn (Cp ) ✸✾ ❧➔ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ❦❤ỉ♥❣ s✉② ❜✐➳♥ t✉②➳♥ t➼♥❤✳ ❑❤✐ ✤â q q−n−1− i=1 − ❈❤ù♥❣ ♠✐♥❤✳ n ki + q Tf (r) ≤ i=1 ki ≤ki Nn,f (Xi , r) ki + n(n + 1) log r + O(1) X ∈ {X1 , X2 , , Xq }✱ k ∈ {k1 , k2 , , kq } F = t õ ợ õ ữỡ tr➻♥❤ ❣✐↔ sû X ≤k >k Nn,f (X, r) = Nn,f (X, r) + Nn,f (X, r) k ≤k ≤k >k ≤ Nn,f (X, r) + Nn,f (X, r) + nN1,f (X, r) k+1 k+1 n n k ≤k ≤k Nn,f (X, r) + N1,f (X, r) + Nf>k (X, r) ≤ k+1 k+1 k+1 k n n ≤k ≤ Nn,f (X, r) + Nf≤k (X, r) + Nf>k (X, r) k+1 k+1 k+1 n k ≤k (X, r) + Nn,f Nf (X, r) = k+1 k+1 ⑩♣ ❞ö♥❣ ❈æ♥❣ t❤ù❝ P♦✐s♦♥❣✲❥❡♥s❡♥ ♣ ✲❛❞✐❝ t❛ ❝â Nf (X, r) = NF ◦f (r) = TF ◦f (r) + O(1) = Tf (r) + O(1) ❉♦ ✤â Nn,f (X, r) ≤ k n ≤k Nn,f (X, r) + Tf (r) + O(1) k+1 k+1 ỵ rt X1 , , Xq ♣ ✲❛❞✐❝ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ f ✈➔ q s✐➯✉ t❛ ❝â q (q − n − 1)Tf (r) ≤ Nn,f (Xi , r) − i=1 q ≤ i=1 − n(n − 1) log r + O(1) ki ≤ki Nn,f (Xi , r) + ki + n(n − 1) log r + O(1) q i=1 n Tf (r) ki + ✹✵ ❱➟② q q−n−1− i=1 n ki + q Tf (r) ≤ i=1 − ki ≤ki Nn,f (Xi , r) ki + n(n − 1) log r + O(1) ❇ê ữủ ự ỵ fi i = 1, 2, , n + ❧➔ n + ❤➔♠ ♥❣✉②➯♥ ♣✲❛❞✐❝ s❛♦ ❝❤♦ f1, , fn ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣✱ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ f1 + f2 + + fn = fn+1 ✳ ❑❤✐ õ ợ ộ số k ữỡ t õ n2 − 1− k+1 k max {Tfi (r)} ≤ 1≤i≤n+1 k+1 ❈❤ù♥❣ ♠✐♥❤✳ n+1 ≤k Nn−1,f (r)− i i=1 n(n − 1) log r+O(1) f1 , , fn ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣✱ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ f1 + f2 + + fn = fn+1 ♥➯♥ f2 , , −fn+1 ❝ô♥❣ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❉♦ ✈➔ ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣✳ ❉♦ ✤â f = (f1 , f2 , , fn ), g = (f2 , , fn , −fn+1 ) ❧➔ ❝→❝ ❜✐➸✉ ❞✐➵♥ t÷ì♥❣ ù♥❣ ❝õ❛ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ❦❤ỉ♥❣ s✉② f, g tø Cp −→ Pn−1 (Cp ) = Pn−1 ✳ Xi = (z1 , , zn ) ∈ Pn−1 : zi = ✈ỵ✐ i = 1, , n✱ ❜✐➳♥ t✉②➳♥ t➼♥❤ ✣➦t Xn+1 = (z1 , , zn ) ∈ Pn−1 : z1 + + zn = X1 , X2 , , Xn+1 ❧➔ ❝→❝ s✐➯✉ ♣❤➥♥❣ ð ✈à tr➼ tê♥❣ q✉→t tr♦♥❣ Pn−1 (Cp )✳ ❱ỵ✐ ♠é✐ k ♥❣✉②➯♥ ❞÷ì♥❣✱ →♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✶✻ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ f ✈➔ n + s✐➯✉ ♣❤➥♥❣ X1 , X2 , , Xn+1 t❛ ❝â ❑❤✐ ✤â n + − (n − 1) − − n+1 ≤ i=1 (n + 1)(n − 1) Tf (r) k+1 k n(n − 1) ≤k Nn−1,f (Xi , r) − log r + O(1) k+1 ✹✶ ❉♦ ✤â n2 − k 1− Tf (r) ≤ k+1 k+1 n+1 ≤k Nn−1,f (r) − i i=1 n(n − 1) log r + O(1) ❚÷ì♥❣ tü✱ →♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✶✻ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ♣❤➥♥❣ X1 , X2 , , Xn+1 ≤ i=1 ✈➔ n + s✐➯✉ t❛ ❝â n + − (n − 1) − − n+1 g (n + 1)(n − 1) Tg (r) k+1 k n(n − 1) ≤k (Xi , r) − Nn−1,g log r + O(1) k+1 ❱➟② n2 − k 1− Tg (r) ≤ k+1 k+1 n+1 ≤k Nn−1,f (r) − i i=1 n(n − 1) log r + O(1) ▼➦t ❦❤→❝ Tf (r) = max {Tf1 (r), Tf2 (r), , Tfn (r)} , Tg (r) = max Tf2 (r), Tf3 (r) , Tfn+1 (r) ❚ø ✤â t❛ s✉② r❛ n2 − 1− k+1 k max Tfi (r) ≤ 1≤i≤n+1 k+1 − n+1 ≤k Nn−1,f (r) i i=1 n(n − 1) log r + O(1) ỵ ữủ ự t ❑❤✐ k −→ +∞ t❤➻ n2 − −→ k+1 ✈➔ ≤k Nn−1,f (r) −→ Nn−1,fi (r) i ❑❤✐ õ ỵ tr t n+1 max {Tfi (r)} ≤ 1≤i≤n+1 Nn−1,fi (r) − i=1 n(n − 1) log r + O(1) ✹✷ ✣â ❝❤➼♥❤ ❧➔ ❦➳t q✉↔ ❝õ❛ ❍✉✲❨❛♥❣✳ ▼➦t ❦❤→❝ ♥➳✉ fi ❧➔ ✤❛ t❤ù❝ ❦❤→❝ t❛ ❝â Tfi (r) = degfi r→+∞ log r lim ✈➔ N fi (r) = n(fi ) r→+∞ log r lim õ ỵ õ t ữ tữỡ tỹ ỵ s ❤➔♠ ♥❣✉②➯♥ ♣✲❛❞✐❝✳ ❇ê ✤➲ ✷✳✶✾✳ ◆➳✉ f ❧➔ ❤➔♠ ổ ỗ t tr Cp õ ỹ a ❜ë✐ m t❤➻ f ❝â ❝ü❝ ✤✐➸♠ a ❜ë✐ m + 1✳ ❈❤ù♥❣ ♠✐♥❤✳ ❧➙♥ ❝➟♥ ❝õ❛ a ●✐↔ sû ✈➔ f (z) = ϕ(a) = 0✳ ϕ(z) ✱ ϕ(z) (z − a)m ❧➔ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣ ❚❛ ❝â✿ ϕ (z − a)m − mϕ(z − a)m−1 f = (z − a)2m ϕ (z − a) − mϕ = (z − a)m+1 ❱➟② a ❧➔ ❝ü❝ ✤✐➸♠ ❜ë✐ m+1 ❝õ❛ f ✳ ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚ø ❜ê ✤➲ tr➯♥ ✈➔ ❞ị♥❣ q✉② ♥↕♣ ♥❤➟♥ ✤÷đ❝ ❤➺ q✉↔ s❛✉✿ ❍➺ q✉↔ ✷✳✷✵✳ ◆➳✉ f ❧➔ ❤➔♠ ổ ỗ t ổ tr Cp õ ỹ ✤✐➸♠ a ❜ë✐ m t❤➻ f (k) ❝â ❝ü❝ ✤✐➸♠ a m + k ỵ f1, f2, , fn+1 ❧➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣✲❛❞✐❝ s❛♦ ❝❤♦ f1 , , fn ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣✱ ❝ü❝ ✤✐➸♠ ❝❤✉♥❣✱ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✱ f1 + f2 + + fn = fn+1 ✱ |fi |r ≥ Ai ✱ Ai ❧➔ ❝→❝ ❤➡♥❣ sè ❞÷ì♥❣✳ ❑❤✐ ✤â n+1 max {Tfi (r)} ≤ 1≤i≤n+1 Nn−1,fi (r) + (n − 1) max i=1 n+1 1≤j≤n+1 i=1 i=j N1, f1 (r) i ✹✸ − n(n − 1) log r + O(1), ❝❤➦♥ ❦❤✐ r −→ +∞✳ ð ✤â O(1) ❧➔ ✤↕✐ ❧÷đ♥❣ ❜à ❈❤ù♥❣ ♠✐♥❤✳ ❉♦ tr♦♥❣ ❦➳t ❧✉➟♥ ỵ õ O(1) Ai Ai = ✈ỵ✐ i = 1, 2, , n + 1✳ ❚❤❡♦ ❣✐↔ t❤✐➳t f1 , , fn ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ f1 + + fn − fn+1 = fj ổ ỗ t ổ Wj (f1 , , fj−1 , fj+1 , , fn+1 ) ≡ 0✱ j = 1, 2, , n + 1✳ sè ❞÷ì♥❣ ♥➯♥ ❦❤ỉ♥❣ ❣✐↔♠ tê♥❣ q✉→t ❣✐↔ sû rữợ t t ự n+1 Tf1 (r) n+1 Nn−1, f1 (r) − Nn−1,fi (r) + i=1 i i=2 ❳➨t g= n(n − 1) log r + O(1) W1 (f2 , , fn+1 ) f1 fn+1 ❚❛ ❝â ··· fn+1 f2 ··· f2 W1 (f2 , , fn+1 ) f n+1 ϕ= = det ✳✳ ✳✳ ✳✳✳ ✳ ✳ f2 fn+1 (n−1) (n−1) f f2 · · · n+1 f2 fn+1 ❙✉② r❛ (k ) (k ) n fn+1 f2 + + log + O(1) log |ϕ|r ≤ log f2 fn+1 ❱ỵ✐ (k1 , k2 , , kn ) ❧➔ ❤♦→♥ ✈à ❜➜t ❦➻ ❝õ❛ {0, 1, , n − 1}✳ ❚❤❡♦ ❇ê ✤➲ ✤↕♦ ❤➔♠ ❧♦❣❛ t❛ ❝â (k ) fi i log fi ≤ −ki log r ❍❛② log |ϕ|r ≤ −(1 + + + n − 1) log r ✹✹ ❱➟② Tϕ (r) ≤ − ❚❛ ❝â ❉♦ g= f1 ϕ f1 f1 = ✈➔ ϕ g ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♥➯♥ ✤➦t n(n − 1) log r f1 = a1 b1 ✈ỵ✐ a1 , b1 ❧➔ ❝→❝ ❤➔♠ ♥❣✉②➯♥ ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣✳ ❙✉② r❛ a1 = f b = ϕ b1 ✳ g ❚❛ ❝â Tf1 (r) = max {log |a1 |r , log |b1 |r } + O(1) = max T ϕg b1 (r), Tb1 (r) + O(1) ❉♦ |f1 |r ≥ ♥➯♥ ϕ b1 g = r ϕ |b1 |r ≥ |b1 |r ✳ g r ❱➟② Tf1 (r) = log ϕ ϕ b1 + O(1) = log + log |b1 |r + O(1) g r g r = log |ϕ|r − log |g|r + log |b1 |r + O(1) = log |ϕ|r − (log |W1 |r − log |f1 f2 fn+1 |r ) + Hb1 (r) + O(1) n+1 = log |ϕ|r − NW1 (r) + N W1 (r) + log |fi |r + Hb1 (r) + O(1) i=1 n+1 Nfi (r) − = log |ϕ|r − NW1 (r) + N W1 (r) + n+1 i=1 N f1 (r) i=1 i + Nb1 (r) + O(1) n+1 n+1 Nfi (r) − NW1 (r) + N W1 (r) − = i=1 N f1 (r) i=2 i + log |ϕ|r + O(1) ❉♦ W1 (f2 , , fn+1 ) = W (f2 , , fn , −f2 − − fn + fn+1 ) = W (f2 , , fn , f1 ) ✹✺ ✈➔ f1 , , fn ❦❤æ♥❣ ❝â ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ♥➯♥ n+1 n+1 Nfi (r) − NW1 (r) ≤ i=1 ❚❤❡♦ ❤➺ q✉↔ ✷✳✷✵ ✈➔ Nn−1,fi (r) i=1 f1 , , fn ❦❤æ♥❣ ❝â ❝ü❝ ✤✐➸♠ ❝❤✉♥❣ t❛ ❝â n+1 N W1 (r) − ✈➔ log |ϕ|r ≤ n+1 N f1 (r) ≤ (n − 1) i i=2 N1, f1 (r), i i=2 −n(n − 1) log r✳ ❱➟② n+1 Tf1 (r) ≤ n+1 Nn−1,fi (r) + (n − 1) N1, f1 (r) − i=1 i i=2 n(n − 1) log r + O(1) ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝â n+1 Tfi (r) ≤ n+1 i=1 ✈ỵ✐ N1, f1 (r)− Nn−1,fi (r)+(n−1) i i=1 i=j n(n − 1) log r+O(1) (∗∗) j = 1, 2, , n + 1✳ ❚ø ✭✯✯✮ s✉② r❛ n+1 max {Tfi (r)} ≤ 1≤i≤n+1 Nn−1,fi (r) + (n − 1) max 1≤j≤n+1 i=1 − n+1 i=1 i=j N1, f1 (r) i n(n − 1) log r + O(1) ỵ ữủ ự ♠✐♥❤✳ ◆❤➟♥ ①➨t ✷✳✷✷✳ ◆➳✉ fi✱ i = 1, 2, , n + ❧➔ ❝→❝ ❤➔♠ ♥❣✉②➯♥ ♣✲❛❞✐❝ ❦❤æ♥❣ ỗ t ổ t õ N1, f1 (r) = 0, ✈ỵ✐ i max ♠å✐ ✐❂✶✱✷✱✳✳✳✱♥✰✶✳ n+1 1≤j≤n+1 i=1 i=j N1, f1 (r) = i ✹✻ ❱➟② n+1 max {Tfi (r)} ≤ 1≤i≤n+1 Nn−1,fi (r) − i=1 n(n − 1) log r + O(1) ✣â ❝❤➼♥❤ ❧➔ ❦➳t q õ õ t ỵ rở ỵ s ♣❤➙♥ ❤➻♥❤ ♣✲❛❞✐❝✳ ✹✼ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ữủ ởt số t q s ã r ữỡ ❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ỵ ũ q ỵ tt ố tr • ♣ ✲❛❞✐❝✳ ❚r♦♥❣ ❈❤÷ì♥❣ ✷✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ỵ rt ự ỵ ✈➔♦ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ sü s✉② ❜✐➳♥ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ♣ ✲❛❞✐❝ ✈➔ t÷ì♥❣ tü ❝õ❛ ✣à♥❤ ỵ s ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❍➔ ❚r➛♥ P❤÷ì♥❣✲❱ơ ❍♦➔✐ ❆♥ ✭✷✵✶✹✮✱ ●✐↔✐ t➼❝❤ ♣✲❛❞✐❝✱ ●✐→♦ tr➻♥❤ ❝❛♦ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ❍✉✱ P✳❈✳ ❛♥❞ ❨❛♥❣✱ ❈✳❈✳ ✭✷✵✵✵✮✱ ▼❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ♦✈❡r ♥♦♥✲ ❆r❝❤✐♠❡❞❡❛♥ ❢✐❡❧❞s✱ ❑❧✉✇❡r✳ ❬✸❪ ❍✉✱ P✳❈✳ ❛♥❞ ❨❛♥❣✱ ❈✳❈✳✱ ❆ ●❡♥❡r❛❧✐③❡❞ ❛❜❝✲❈♦♥❥❡t✉r❡ ♦✈❡r ❋✉♥❝✲ t✐♦♥ ❋✐❡❧❞s✱ ❏♦✉r♥❛❧ ♦❢ ◆✉♠❜❡r ❚❤❡♦r② ✾✹✱ ✷✽✻✲✷✾✽ ✭✷✵✵✷✮✳ ❬✹❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ❱✉ ❍♦❛✐ ❆♥ ✭✷✵✵✸✮✱ ❱❛❧✉❡ ❞✐str✐❜✉t✐♦♥ ❢♦r ♣✲❛❞✐❝ ❤②♣❡rs✉r❢❛❝❡s✱ ❚❛✐✇❛♥❡s❡ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳✼✱ ◆♦✳✶✱ ♣♣✳ ✺✶✲✻✼✳ ❬✺❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ▼❛✐ ❱❛♥ ❚✉ ✭✶✾✾✺✮✱ ♣✲❛❞✐❝ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ❚❤❡♦r❡♠✱ ■♥t❡r♥❛t✳ ❏✳ ▼❛t❤✱ ♣♣✳ ✼✶✾✲✼✸✶✳ ... ∈ A(ρ (Cp ), h ≡ h ❚❛ ❝ô♥❣ ✈✐➳t M(ρ (Cp ) = M(Cp (0; ρ)) ✣➦❝ ❜✐➺t✱ ♠ët ♣❤➛♥ tû ❝õ❛ t➟♣ ❤ñ♣ M(∞ (Cp ) = M(Cp (0; ∞)) = M(Cp ) ✤÷đ❝ ❣å✐ ❧➔ ♠ët t➾ tr➯♥ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ Cp✳ ❑➼ ❤✐➺✉ Cp(z) ❧➔ t➟♣... tø Cp ✤➳♥ Pn−1 (Cp ) = Pn−1 ✳ ✣➦t Xi = (z1 , , zn ) ∈ Pn−1 : zi = ✈ỵ✐ i = 1, , n✱ Xn+1 = (z1 , , zn ) ∈ Pn−1 : z1 + + zn = X1 , X2 , , Xn+1 ❧➔ ❝→❝ s✐➯✉ ♣❤➥♥❣ ð ✈à tr➼ tê♥❣ q✉→t tr♦♥❣ Pn−1 (Cp... tr÷í♥❣ sè Qp✳ ❑➼ ❤✐➺✉ Cp = Qp ❧➔ ❜ê s✉♥❣ ❝õ❛ ❜❛♦ ✤â♥❣ ✤↕✐ sè ❝õ❛ R✱ Qp ✳ Q✳ ♠ð rë♥❣ t❤❡♦ ❚❛ ❣å✐ Cp ❧➔ ✹ tr÷í♥❣ sè ♣❤ù❝ ♣ ✲❛❞✐❝✳ ❈❤✉➞♥ tr➯♥ ✤÷đ❝ ♠ð rë♥❣ tü ♥❤✐➯♥ ❝õ❛ ❝❤✉➞♥ Cp ♣ ✲❛❞✐❝ tr➯♥ Qp✳ ❑➼ ❤✐➺✉✿