LUẬN văn THẠC sĩ HAY 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➯♥ ✕ ✷✵✶✻ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖ ◆●❯❨➍◆ ❚❍➚ ▲■➊◆ ▲➑ ❚❍❯❨➌❚ ◆❊❱❆◆▲■◆◆❆ ❱⑨ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ P✲❆❉■❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ữớ ữợ ●❙✳❚❙❑❍ ❍⑨ ❍❯❨ ❑❍❖⑩■ ❚❤→✐ ◆❣✉②➯♥ ✕ ✷✵✶✻ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝✱ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝ ✈➔ ❝→❝ t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ữủ ró ỗ ố t❤→♥❣ ✹ ♥➠♠ ✷✵✶✻ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ◆❣✉②➵♥ ❚❤à ▲✐➯♥ ✐ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ♥❤í sü ữợ t t ✤➣ ❞➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥✱ ❝ỉ♥❣ sù❝ ❝❤➾ ❜↔♦ tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ◆❤➙♥ ❞à♣ ♥➔② tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛②✳ ❚ỉ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❜❛♥ ❧➣♥❤ ✤↕♦ tr÷í♥❣ ✣❍❙P ❚❤→✐ ◆❣✉②➯♥✱ ❧➣♥❤ ✤↕♦ ❦❤♦❛ ❚♦→♥✱ ❧➣♥❤ ✤↕♦ ❦❤♦❛ ❙❛✉ ✣↕✐ ❍å❝ ❝õ❛ tr÷í♥❣ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ tỉ✐ ❤♦➔♥ t❤➔♥❤ tèt ♥❤✐➺♠ ✈ư ❤å❝ t➟♣ ❝õ❛ ♠➻♥❤✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ sð ●✐→♦ ❉ö❝ ✈➔ ✣➔♦ ❚↕♦ t➾♥❤ ◗✉↔♥❣ ◆✐♥❤✱ ❇❛♥ ❣✐→♠ ❤✐➺✉ tr÷í♥❣ ❚❍P❚ ▲➯ ❈❤➙♥✱ ✤➦❝ t ỗ ✈✐➯♥✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï tỉ✐ ✈➲ ♠å✐ ♠➦t tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚r♦♥❣ q✉→ tr➻♥❤ ✈✐➳t ❧✉➟♥ ✈➠♥ ❝ơ♥❣ ♥❤÷ tr♦♥❣ ✈✐➺❝ ①û ❧➼ ✈➠♥ ❜↔♥ ❝❤➢❝ ❝❤➢♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳ ✈➔ t❤✐➳✉ sât✳ ❘➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sỹ õ ỵ t ổ ỗ ♥❣❤✐➺♣ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✻ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ◆❣✉②➵♥ ❚❤à ▲✐➯♥ ✐✐ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ✐ ▲í✐ ❝↔♠ ì♥ ✐✐ ▼ư❝ ❧ư❝ ✐✐✐ ▼ð ✤➛✉ ✶ ✶ ❈ì sð ❧➼ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ✷ ✶✳✶ ▲➼ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♣✲❛❞✐❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ◗✉❛♥ ❤➺ sè ❦❤✉②➳t ❝❤♦ ♠ö❝ t✐➯✉ ❞✐ ✤ë♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❳→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ìợ ữủ t p✲❛❞✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ p✲❛❞✐❝ ✶✾ ✷✳✶ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè p✲❛❞✐❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ✣à♥❤ ❧➼ ▼❛❧♠q✉✐st ❦✐➸✉ ✭■✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ qst ❦✐➸✉ ✭■■✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✹ ◆❣❤✐➺♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✳ ✷✾ ❑➳t ❧✉➟♥ ❝❤✉♥❣ ✸✻ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✼ ✐✐✐ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ▼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♥❣ ❤đ♣ tê♥❣ q✉→t✳ ❍✉✲❨❛♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü p✲❛❞✐❝ ✈➲ q✉❛♥ ❤➺ sè ❦❤✉②➳t ❝❤♦ ♠ư❝ t✐➯✉ ❞✐ ✤ë♥❣✱ ✤à♥❤ ❧➼ ❝ì ❜↔♥ t❤ù ❤❛✐ ❝❤♦ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ✈➔ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ✈ỵ✐ sè ♣❤➛♥ tû ❤ú✉ ❤↕♥✳ ❈❤❡rr②✲❨❛♥❣ ❬✹❪ ✤➣ ♠ỉ t↔ ♠ët sè t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ✈ỵ✐ sè ♣❤➛♥ tû ❤ú✉ ❤↕♥ ❝õ❛ ❝→❝ ❤➔♠ ♥❣✉②➯♥ p✲❛❞✐❝✳✳✳ ▲✉➟♥ ✈➠♥ ♥➔② ♥❤➡♠ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ♥❣➢♥ ❣å♥ ✈➲ ❧➼ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ✈➔ ù♥❣ ❞ư♥❣ ❝õ❛ ♥â ✤è✐ ✈ỵ✐ ữỡ tr p ỗ ✷ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ■✿ ❚r➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❧➼ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ✈➔ ♠ët sè ❦➳t q✉↔ ✈➲ q✉❛♥ ❤➺ sè ❦❤✉②➳t✱ ❜➔✐ t♦→♥ ①→❝ ✤à♥❤ t t p ữợ ❧÷đ♥❣ ❝➜♣ t➠♥❣ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝✳ ❈❤÷ì♥❣ ■■✿ ●✐ỵ✐ t❤✐➺✉ ✤à♥❤ ♥❣❤➽❛✱ ❝→❝ t➼♥❤ ❝❤➜t ✈➔ ♠ët sè t q ữỡ tr p ỗ ✣à♥❤ ❧➼ ▼❛❧♠q✉✐st ❦✐➸✉ ✭■✮✱ ✣à♥❤ ❧➼ ▼❛❧♠q✉✐st ❦✐➸✉ ✭■■✮ ✈➔ ❝❤➾ r❛ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè p✲❛❞✐❝ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✻ ✶ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❈❤÷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 ✤➣ ✤÷đ❝ ❝❤✉➞♥ ❤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✐ ❝❤✉➞♥ ❦❤ỉ♥❣ ❆❝s✐♠❡t✱ tê♥❣ ✈ỉ ❤↕♥ ❤ë✐ tư ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ sè ❤↕♥❣ tê♥❣ q✉→t ❞➛♥ ✤➳♥ 0✳ ❑❤✐ ✤â tự õ ữợ an z n f (z) = n=0 ①→❝ ✤à♥❤ ✤ó♥❣ ✤➢♥ ❦❤✐ |an z n |p → ✣à♥❤ ♥❣❤➽❛ ❜→♥ ❦➼♥❤ ❤ë✐ tö ρ ❜ð✐ 1 = lim sup |an |pn ρ n→∞ ❑❤✐ ✤â✱ ❝❤✉é✐ ❤ë✐ tö ♥➳✉ |z|p < ρ ✈➔ ♣❤➙♥ ❦➻ ♥➳✉ |z|p > ρ✳ ◆❣♦➔✐ r❛✱ ❤➔♠ f (z) ✤÷đ❝ ❣å✐ ❧➔ ❣✐↔✐ t➼❝❤ p✲❛❞✐❝ tr➯♥ B (ρ) ♥➳✉ ❝❤✉é✐ ❤ë✐ tö tr➯♥ B (ρ) = {z ∈ Cp | |z|p < ρ} ✷ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ◆➳✉ ρ = ∞✱ ❤➔♠ f (z) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ♥❣✉②➯♥ p✲❛❞✐❝ tr➯♥ Cp ✳ ❈❤♦ f ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ p✲❛❞✐❝ ❦❤→❝ ❤➡♥❣ tr➯♥ B (ρ) (0 < ρ ≤ ∞) ❇↔♥ ❝❤➜t ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❲✐♠❛♥✲❱❛❧✐r♦♥ ❧➔ ♣❤➙♥ t➼❝❤ ❞→♥❣ ✤✐➺✉ ❝õ❛ ❤➔♠ ❜➡♥❣ sè ❤↕♥❣ ❝ü❝ ✤↕✐ ✿ µ (r, f ) = max |an |p rn n≥0 (0 < r < ρ) , ❝ị♥❣ ✈ỵ✐ ❝❤➾ sè tr✉♥❣ t➙♠ ✿ ν (r, f ) = max{n| |an |p rn = µ (r, f )} n≥0 ✣à♥❤ ♥❣❤➽❛ ν (0, f ) = lim ν (r, f ) ỡ ỳ ú t ú ỵ r h ❧➔ r→0 ♠ët ❤➔♠ ❣✐↔✐ t➼❝❤ p✲❛❞✐❝ ❦❤→❝ tr➯♥ B (ρ) t❤➻ µ (r, f h) = µ (r, f ) µ (r, h) (1) ❇ê ✤➲ ✶✳✶✳✶✳ ❈❤➾ sè tr✉♥❣ t➙♠ ν (r, f ) t➠♥❣ ❦❤✐ r → ρ ✈➔ t❤ä❛ ♠➣♥ ❝ỉ♥❣ t❤ù❝✿ r log µ (r, f ) = log aν(0,f ) p + ν (t, f ) − ν (0, f ) dt+ν (0, f ) log r t (0 < r < ρ) ỵ rstrss ỗ t t tự P õ (r, f ) ✈➔ ♠ët ❤➔♠ ❣✐↔✐ t➼❝❤ p✲❛❞✐❝ g tr➯♥ B [r] s❛♦ ❝❤♦ f = gP ✱ ð ✤â✿ B [r] = {z ∈ Cp | |z|p ≤ r} ❍ì♥ ♥ú❛✱ g ❦❤ỉ♥❣ ❝â ❜➜t ❦➻ ❦❤ỉ♥❣ ✤✐➸♠ ♥➔♦ tr♦♥❣ B [r]✱ ✈➔ P ❝â ✤ó♥❣ ν (r, f ) ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐ tr➯♥ B [r] ●å✐ n r, f1 ❧➔ sè ❦❤æ♥❣ ✤✐➸♠ ✭❦➸ ❝↔ ❜ë✐✮ ❝õ❛ f ✈ỵ✐ ❣✐→ trà t✉②➺t ✤è✐ ≤ r ✈➔ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ✤➳♠ ❝õ❛ f ✤è✐ ✈ỵ✐ ✵ ❜ð✐✿ r N r, f = n t, f1 − n 0, f1 dt + n 0, t f logr (0 < r < ρ) ✸ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❇ê ✤➲ ✶✳✶✳✷ ❝❤➾ r❛ r➡♥❣ n r, f = ν (r, f ) ❚ø ❜ê ✤➲ ✶✳✶✳✶ s✉② r❛ ❝æ♥❣ t❤ù❝ ❏❡♥s❡♥ ✿ N r, f = log µ (r, f ) − log an(0, ) f p (2) ❈❤ó♥❣ t❛ ❝ơ♥❣ ❦➼ ❤✐➺✉ sè ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛ f tr➯♥ B [r] ❜ð✐ n r, f1 ✈➔ ✤à♥❤ ♥❣❤➽❛✿ r N r, f = n t, f1 − n 0, f1 dt + n 0, t f log r (0 < r < ρ) ❱ỵ✐ ♠é✐ n t❛ ỗ t (t) t ordp (an z n ) ♥❤÷ ❤➔♠ ❝õ❛ t = ordp (z) ❑❤✐ õ (t) ữớ t ợ n ●å✐ γ (t, f ) ❧➔ ❜✐➯♥ ❝õ❛ ♠✐➲♥ ❣✐❛♦ tt ỷ t ữợ ✤÷í♥❣ t❤➥♥❣ γn (t) ✣÷í♥❣ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ✣❛ ❣✐→❝ ◆❡✇t♦♥ ❝õ❛ ❤➔♠ f (z) ❈→❝ ✤✐➸♠ t ♠➔ t↕✐ ✤â γ (t, f ) ❝â ❝→❝ ✤➾♥❤ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ tỵ✐ ❤↕♥ ❝õ❛ f (z).✣♦↕♥ ❤ú✉ ❤↕♥ [α, β] ❝❤➾ ❝❤ù❛ ❤ú✉ ❤↕♥ ❝→❝ ✤✐➸♠ tỵ✐ ❤↕♥✳ ❘ã r➔♥❣ r➡♥❣ ♥➳✉ t ❧➔ ✤✐➸♠ tỵ✐ ❤↕♥ t❤➻ ordp (an )+nt ✤↕t tỵ✐ ❣✐→ trà ♥❤ä ♥❤➜t t↕✐ ❤❛✐ ❣✐→ trà n ❍✐➸♥ ♥❤✐➯♥✱ ❝❤ó♥❣ t❛ ❝â✿ µ(r, f ) = p−γ(t,f ) tr♦♥❣ ✤â r = p−t ❚➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤❛ ❣✐→❝ ◆❡✇t♦♥ ❧➔ ♥➳✉ t = ordp (z) ❦❤ỉ♥❣ ❧➔ ✤✐➸♠ tỵ✐ ❤↕♥ t❤➻ |f (z)|p = p−γ(t,f ) ❦➨♦ t❤❡♦ |f (z)|p = µ(r, f ) ❍➔♠ ♣❤➙♥ ❤➻♥❤ f tr➯♥ B(ρ) ✤÷đ❝ ❤✐➸✉ ❧➔ t❤÷ì♥❣ g h ❝õ❛ ❤❛✐ ❤➔♠ ❣✐↔✐ t➼❝❤ p✲❛❞✐❝ g ✈➔ h s❛♦ ❝❤♦ g ✈➔ h ❦❤æ♥❣ ❝â ♥❤➙♥ tû ❝❤✉♥❣ tr♦♥❣ ✈➔♥❤ ❝→❝ ❤➔♠ ❣✐↔✐ t➼❝❤ p✲❛❞✐❝ tr➯♥ B [ρ] ❱➻ ❤➔♠ µ t❤ä❛ ♠➣♥ ✭✶✮ ✈➔ ✈➻ ❯❈▲◆ ❝õ❛ ❤❛✐ ❤➔♠ ✹ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❣✐↔✐ t➼❝❤ p✲❛❞✐❝ tỗ t t õ t rở t ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f= g h ❜➡♥❣ ❝→❝❤ ✤à♥❤ ♥❣❤➽❛ µ(r, f ) = µ(r, g) µ(r, h) ❚❛ ❝ô♥❣ ✤➦t γ(t, f ) = γ(t, g) − γ(t, h) ❘ã r➔♥❣ r➡♥❣✱ ♥➳✉ t = ordp (z) ❦❤ỉ♥❣ ❧➔ ✤✐➸♠ tỵ✐ ❤↕♥ ❝õ❛ f (z)✱ ♥â✐ ♠ët ❝→❝❤ ❦❤→❝ t ❦❤ỉ♥❣ ❧➔ ✤✐➸♠ tỵ✐ ❤↕♥ ❝õ❛ g(z) ❤♦➦❝ h(z) t❤➻ |f (z)|p = p−γ(t,f ) = µ(r, f ) ✣à♥❤ ♥❣❤➽❛ |Cp | = {|z|p |z ∈ Cp } ú ỵ r {pw |w Q} |C|p } |C|p trò ♠➟t tr♦♥❣ R[0, +∞) ◆➳✉ a : R[0, +∞) −→ R ✈➔ b : |C|p −→ R ❧➔ ❝→❝ ❤➔♠ ❣✐→ trà t❤ü❝ t❤➻ ||a(r)|| ≤ b(z) ợ t số ữỡ ỳ ♥➔♦ < R < ρ ❝â ♠ët t➟♣ ❤ú✉ ❤↕♥ E tr♦♥❣|Cp | ∩ [0, R] s❛♦ ❝❤♦ a(r) ≤ b(r), r = |z|p ∈ |Cp | ∩ [0, R] − E ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❦➼ ❤✐➺✉ ♥➔②✱ t❛ ❝â ||µ(r, f )|| = |f (z)|p ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ f tr➯♥ B(ρ) ✣à♥❤ ♥❣❤➽❛ ❤➔♠ ✤➳♠ n(r, f ) ✈➔ N (r, f ) ❝õ❛ f ✤è✐ ✈ỵ✐ ❝ü❝ ✤✐➸♠ ❜ð✐ 1 n(r, f ) = n(r, ), N (r, f ) = N (r, ) h h ❙❛✉ ✤â →♣ ❞ö♥❣ ✭✷✮ ❝❤♦ g ✈➔ h✱ t❛ t❤✉ ✤÷đ❝ ❝ỉ♥❣ t❤ù❝ ❏❡♥s❡♥ ✿ N (r, ) − N (r, f ) = log µ(r, f ) − Cf , f (3) ✺ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ð ✤â Rj (z) = R(z, aj ), t❤➻ R(z, w) ❧➔ ✤❛ t❤ù❝ ❝õ❛ w ✈ỵ✐ degw (R) ≤ min{Γ(Ω), deg(Ω) + γ(Ω)(1 − ❈❤ù♥❣ ♠✐♥❤✳ ✣à♥❤ ♥❣❤➽❛ ϕ[a1 ] = w (∞))} Ω − R1 , w − a1 ϕ[a1 ] − ϕ[a2 ] a1 − a2 R1 R2 Ω − + , = (w − a1 )(w − a2 ) (a1 − a2 )(w − a1 ) (a1 − a2 )(w − a2 ) ϕ[a1 , a2 ] = ✈➔ ✤à♥❤ ♥❣❤➽❛ q✉② ♥↕♣ ϕ[a1 , , al ] = ϕ[a1 , , al−1 ] − ϕ[a1 , , al−2 , al ] al−1 − al Ω = + (w − a1 ) (w − al ) = Ω − Q1 (z, w) (w − a1 ) (w − al ) l j=0 a ˆlj Rj w − aj (l ≥ 3), ð ✤â aˆlj ❧➔ ❝→❝ ❤➡♥❣ sè ♣❤ö t❤✉ë❝ ✈➔♦ {a1 , , al }, ✈➔ Ql (z, w) ❧➔ ✤❛ t❤ù❝ ❝õ❛ w ❝â ❜➟❝ ≤ l − ✈ỵ✐ ❤➺ sè ❧➔ tê ❤đ♣ t✉②➳♥ t➼♥❤ tr♦♥❣ Rj (1 ≤ j ≤ j) Ð ✤➙② t❛ ✈✐➳t ν = Γ(Ω), ϕl = ϕ[al(ν+1)+1 , , a(l+1)(ν+1) ], l = 0, 1, ❚❛ ❝❤ù♥❣ tä r➡♥❣ ϕl ≡ t↕✐ l ≥ ♥➔♦ ✤â✳ ●✐↔ sỷ ữủ l = ợ l ❑❤✐ ✤â T (r, w) = T (r, w − aν+1 ) + O(1) (27) ≤ T (r, (w − aν+1 )ϕ0 ) + T (r, ϕ0 ) + O(1) ú ỵ r w(z) w(z) aj a max 1, p , |w(z) − aj |p j = 1, , ν + 1, ✷✹ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ð ✤â aˆ = max1≤j≤ν+1 {1 + |aj |p } ❇➡♥❣ ❝→❝❤ ❞ò♥❣ ❜ê ✤➲ ✤↕♦ ❤➔♠ ❧♦❣❛r✐t✱ t❛ ❝â ν+1 a ˆν+1,j Rj Ω + m r, m(r, ϕ0 ) ≤ m r, (w − a1 ) (w − aν+1 ) w − aj j=1 ν+1 ≤2 m r, j=1 + w − aj (28) ν+1 m(r, ci ) + m(r, Rj ) + O(1), j=1 i∈I ν+1 ν + m(r, ci ) + m(r, Rj ) + O(1) m(r, (w − aν+1 )ϕ0 ) ≤ m r, w − a j j=1 j=1 i∈I (29) ❇➙② ❣✐í ❝❤ó♥❣ t❛ ①➨t ❝→❝ ❝ü❝ ✤✐➸♠ ❝õ❛ ϕ0 ❈è ✤à♥❤ z0 ∈ Cp ❉♦ w ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✽✮ t❛ ❝â sup µaw1 ⊂ sup µ0Ω−R1 ❱➻ ✈➟② ♥➳✉ µaw1 (z0 ) > 0, t❤➻ a1 µ∞ ϕ[a1 ] (z0 ) ≤ µw (z0 ) − ❇➡♥❣ q✉② ♥↕♣✱ àaw1 (z0 ) > ợ j õ ≤ j ≤ ν + 1, ♥❤÷♥❣ ci (z0 ) = ∞(i ∈ I), Rl (z0 ) = ∞(1 ≤ l ≤ ν + 1), t❤➻ t❛ ❝â aj µ∞ ϕ0 (z0 ) ≤ µw (z0 ) − ◆➳✉ µ∞ w (z0 ) > t❤➻ ∞ ∞ ∞ µ∞ w (z0 ) ≤ max{0, max{µΩ (z0 ), µQν+1 (z0 ) } − (ν + 1)µw (z0 )} ν+1 µ∞ ci (z0 ) ≤ i∈I µ∞ Ri (z0 ) + j=1 ❱➻ t❤➳ ν+1 N (r, ϕ0 ) ≤ j=1 1 − N r, N r, w − aj w − aj ν+1 + N (r, ci ) + i∈I N (r, Rj ) j=1 (30) ✷✺ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❚÷ì♥❣ tü✱ t❛ ❝â ν+1 N (r, (w − aν+1 )ϕ0 ) ≤ N r, j=1 1 − N r, w − aj w − aj (31) ν+1 + N (r, ci ) + N (r, Rj ) j=1 i∈I ❱➻ ✈➟②✱ ❞♦ ✭✷✼✮✲✭✸✶✮✱ t❛ ❝â ν+1 ν+1 1 T (r, w) ≤ m r, N r, +2 −N r, w − aj w − aj w − aj j=1 j=1 +o(T (r, w)) ❚÷ì♥❣ tü✱ t❛ ❝â✱ ✈ỵ✐ l ≥ 0, (l+1)(ν+1) T (r, w) ≤ {2m(r, j=l(ν+1)+1 1 )+N (r, )−N (r, )}+o(T (r, w)) w − aj w − aj w − aj ❱➻ ✈➟②✱ l(ν+1) lT (r, w) ≤ {2m(r, j=1 1 )+N (r, )−N (r, )}+o(T (r, w)) w − aj w − aj w − aj sỷ ỵ ỡ tự t❛ ✤÷đ❝ (l − 8)T (r, w) ≤ o(T (r, w)) ✣✐➲✉ ♥➔② ❦❤æ♥❣ ①↔② r❛ ♥➳✉ l > ❱➻ t❤➳ ϕl ≡ ✈ỵ✐ l ≥ ♥➔♦ õ w tọ ữỡ tr ữợ (z, w, w , , w(n) ) = Qν+1 (z, w) ✣à♥❤ ♥❣❤➽❛ H(z, w) = R(z, w) − Qν+1 (z, w), Hj (z) = H(z, aj ) ◆➳✉ Hj ≡ 0, t❤➻ N r, 1 ≤ N r, ) ≤ T (r, Hj ) + O(1) w − aj Hj ν+1 ≤ T (r, Rj ) + T (r, Rl ) + O(1) = o(T (r, w)) l=1 ✷✻ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ỵ ❝ì ❜↔♥ t❤ù ❤❛✐✱ ❝â ➼t ♥❤➜t ❤❛✐ ❣✐→ trà aj s❛♦ ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤ó♥❣✳ ❉♦ ✤â Hj ≡ ❤♦➦❝ R(Z, aj ) = Qν+1 (z, aj ), z ∈ (C)p , ♥❣♦↕✐ trø ❤❛✐ ❣✐→ trà aj P❤➛♥ ❝á♥ ❧↕✐ ❝õ❛ ✤à♥❤ ❧➼ ✤÷đ❝ s r tứ ỵ qst ❦✐➸✉ ✭■■✮ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ t❛ ①❡♠ ①➨t ♣❤÷ì♥❣ tr ữợ (z, w, w , , w(n) ) = R(z, w)Φ(z, w, w , , w(n) ), (32) ð ✤â di wi0 (w )i1 (w(n) )in Φ(z, w, w , , w(n) ) = (#J < ∞, di ∈ M(Cp )) i∈J (33) ❇ê ✤➲ ✷✳✸✳✶✳ ❈❤♦ R(z, w) ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✶✹✮ ✈➔ w ❧➔ ♥❣❤✐➺♠ ❝õ❛✭✸✷✮✳ ◆➳✉ q ≥ k t❤➻ Ω T r, ≤ T (r, Φ) + Φ q k T (r, ci ) + T (r, aj ) + O j=0 i∈I T (r, bj ) j=0 ◆➳✉ q ≥ k + deg(Φ) t❤➻ q k m(r, Ω) ≤ m(r, ci )+ i∈I m(r, di )+ m(w, aj )+O j=0 i∈J m(r, bj )+m(r, j=0 ) bq ❈❤ù♥❣ ♠✐♥❤✳ ❉ü❛ t❤❡♦ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❜ê ✤➲ ✷✳✶✳✶✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ Ω m r, ≤m r, + Φ Φ k m(r, ci ) + i∈I q m(r, aj ) j=0 (34) +O m(r, bj ) + m(r, ) bq j=0 ✷✼ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com Ω N r, ≤ N r, + Φ Φ q k N (r, ci ) + N (r, aj ) + O j=0 i∈I N (r, j=0 ) , (35) bj ♥➳✉ q ≥ k ◆❤÷ ✈➟② ❜➜t ✤➥♥❣ t❤ù❝ ✤➛✉ t✐➯♥ ✤÷đ❝ s✉② r❛ tø ✭✸✹✮ ✈➔ ✭✸✺✮✳ ❚÷ì♥❣ tü✱ ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❜➜t ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐✳ ✣à♥❤ ỵ tỗ t w ợ R(z, w) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✶✹✮ s❛♦ ❝❤♦ q k T (r, ci ) + T (r, di ) + i∈I T (r, aj ) + j=0 i∈J T (r, bj ) = o(T (r, w)), j=0 t❤➻ q ≤ min{Γ(Φ), deg(Φ) + γ(Φ)(1 − w (∞))}, k ≤ min{Γ(Ω), deg(Ω) + γ(Ω)(1 − w (∞))} ❈❤ù♥❣ ♠✐♥❤✳ P❤÷ì♥❣ tr õ t t ữợ (z, w, w , , w(n) ) = A1 (z, w) + A2 (z, w) Φ(z, w, w , , w(n) ), B(z, w) ð ✤â deg(A1 ) = k − q ♥➳✉ j ≥ q, ✈➔ deg(A2 ) = k2 < q ❚ø ❜ê ✤➲ ✷✳✸✳✶ t❛ ❝â T r, Ω − A1 Φ ≤ T (r, Φ) + o(T (r, w)) Φ ✣à♥❤ ❧➼ ✶✳✹✳✹ ❦➨♦ t❤❡♦ Ω − A1 Φ A2 T r, = T r, = qT (r, w) + o(T (r, w)) Φ B ❉♦ ✤â t❛ t❤✉ ✤÷đ❝ qT (r, w) ≤ T (r, Φ) + o(T (r, w)) ❇➡♥❣ ❝→❝❤ ❦➳t ❤đ♣ ✤✐➲✉ ♥➔② ✈ỵ✐ ✭✷✸✮ ✈➔ ✭✷✹✮✱ t❛ t❤✉ ✤÷đ❝ ❝➟♥ tr➯♥ ❝❤♦ q ❱✐➳t ữợ B = = , Ω R A ❜➡♥❣ ❝→❝ ❦➳t ❧✉➟♥ tr➯♥✱ ♥❣÷í✐ t❛ ❝â t❤➸ t❤✉ ✤÷đ❝ ❝➟♥ tr➯♥ ❝❤♦ k ✷✽ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ỵ ▲➜② R ∈ M(Cp ) ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ữợ (z, w, w , , w(n) ) = R(w)Φ(z, w, w , , w(n) ) ❝â ♥❣❤✐➺♠ ❦❤→❝ ❤➡♥❣ w ∈ M(Cp ) t❤ä❛ ♠➣♥ T (r, ci ) + i∈I t❤➻ R = ✷✳✹ A B T (r, di ) = o(T (r, w)), i∈J ❧➔ ❤➔♠ ❤ú✉ t✛ ✈ỵ✐ deg(B) ≤ min{Γ(Φ), deg(Φ) + γ(Φ)(1 − w (∞))}, deg(A) ≤ min{Γ(Ω), deg(Ω) + γ(Ω)(1 − w (∞))} ◆❣❤✐➺♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ ❜➔♥ ❧✉➟♥ ữỡ tr ữợ k (z, w, w , , w (n) aj (z)wj )= (36) j=0 ✈ỵ✐ ♠ët sè ❞↕♥❣ ✤➦❝ ❜✐➺t ❝õ❛ Ω ❈❤♦ w ∈ M(Cp ), ð ✤➙② ✈➔ tr♦♥❣ ♣❤➛♥ t✐➳♣ t❤❡♦ Ω(z, w, w , , w(n) ) ✤÷đ❝ ❣å✐ ❧➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ w ♥➳✉ T (r, ci ) = o(T (r, w)) (i ∈ I) ❇ê ✤➲ ✷✳✹✳✶✳ ◆➳✉ w0 , w1 ∈ M(Cp ) ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✱ t❤➻ T (r, w0 ) ≤m(r, w0 + w1 ) + N (r, w0 ) + N (r, w0 ) 1 +N (r, ) + N (r, w1 ) + N (r, ) + O(1) w0 w1 ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t w = w0 + w1 , t❛ ❝â w = ❦➨♦ t❤❡♦ w0 = w w0 w w + w1 , w0 w1 w1 w − w1 w ✷✾ w1 w0 − w1 w0 −1 LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com õ sỷ ỵ ỡ t❤ù ♥❤➜t ✈➔ ❜ê ✤➲ ❝õ❛ ✤↕♦ ❤➔♠ ❧♦❣❛r✐t✱ t❛ ❝â w w1 w w −1 + m r, − − w1 w w1 w0 w w ≤m(r, w) + m r, + m r, w1 w w w w w + m r, − + N r, − + O(1) w1 w0 w1 w0 1 ≤m(r, w) + N (r, w0 ) + N (r, w1 ) + N (r, ) + N (r, ) + O(1) w0 w1 m(r, w0 ) ≤m(r, w) + m r, ❉♦ ✤â ❜ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✷✳✹✳✷✳ ●✐↔ sû Ω(z, w, w , , w(n) ) = (P (z, w, w , , w(n) ) + Q(z, w, w , , w(n) ))l , (37) ð ✤â P ❧➔ ✤ì♥ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ w ✈➔ ◗ ❧➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ w ✈ỵ✐ deg(P ) ≥ deg(Q), γ(P ) > γ(Q) ◆➳✉ k < l ✈➔ ♥➳✉ ✭✸✻✮ ❝â ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ t❤➻ ✭✸✻✮ õ ữợ (z, w, w , , w(n) ) = ak (z)(w + b(z))k , b(z) = ak−1 (z) kak (z) (38) ❈❤ù♥❣ ♠✐♥❤✳ ❚r÷í♥❣ ❤đ♣ k = ❧➔ rã r➔♥❣✳ ●✐↔ sû < k < l ✈➔ ♥❣÷đ❝ ❧↕✐ k m j k Aj wj ≡ 0, aj w − ak (w + b) = j=0 b= j=0 ak−1 , kak ð ✤â ≤ m ≤ k − 2, Am ≡ ✈➔ Aj ❧➔ ❤➔♠ ❤ú✉ t✛ ❝õ❛ {aj } t❤➻ w0 = −ak (w + b)k ✈➔ w1 = Ω = (P + Q)l ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ●✐↔ sû r➡♥❣✱ αw0 + βw1 ≡ 0, {α, β} ⊂ Cp − {0} ✸✵ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❑❤✐ ✤â k βaj wj = αak (w + b)k = αak wk + αak−1 wk−1 + + αak bk j=0 Mw (Cp ) ❧➔ t➟♣ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ f t❤ä❛ ♠➣♥ T (r, f ) = o(T (r, w)) ❑❤✐ ✤â 1, w, w2 , , wk ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ tr➯♥ Mw (Cp ) ữ = ú ỵ r m Aj wj ≡ w0 + w = j=0 ❱➻ t❤➳ α = β = ❚❤❡♦ ❝→❝ ỵ t õ N (r, ) = lN (r, P + Q) = l{deg(P )N (r, w) + γ(P )N (r, w) + o(T (r, w))} k aj wj ) = kN (r, w) + o(T (r, w)) = N (r, j=0 ❚ø deg(P ) > 0, l > k t❛ t❤✉ ✤÷đ❝ N (r, w) = o(T (r, w)) ❚÷ì♥❣ tü✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ k T (r, P + Q) = T (r, w) + o(T (r, w)) l ú ỵ r T (r, w0 ) = kT (r, w) + o(T (r, w)), T (r, w1 ) = lT (r, P + Q) = kT (r, w) + o(T (r, w)), N (r, w0 ) = o(T (r, w)), N r, w0 N (r, w1 ) = o(T (r, w)), = N r, + o(T (r, w)) ≤ T (r, w) + o(T (r, w)), w+b ✸✶ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com k 1 ) ≤ T (r, P + Q) = T (r, w) + o(T (r, w)), = N r, w1 P +Q l m(r, w0 + w1 ) = mm(r, w) + o(T (r, w)) ≤ mT (r, w) + o(T (r, w)) N r, ❚ø ❜ê ✤➲ ✷✳✹✳✶ t❛ t❤✉ ✤÷đ❝ k kT (r, w) ≤ mT (r, w) + T (r, w) + T (r, w) + o(T (r, w)), l ✤✐➲✉ ♥➔② ❧➔ ❦❤æ♥❣ t❤➸ ✈➻ k > m + + kl ❍➺ q✉↔ ✷✳✹✳✸✳ ◆➳✉ k aj (z)wj (w ) = (k < n) (39) j=0 ❝â ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ t❤➻ ✭✸✾✮ õ t t ữợ (w ) = ak (z)(w + b(z))k , b(z) = ak−1 (z) kak (z) (40) ❍➺ q✉↔ ✷✳✹✳✹✳ ◆➳✉ n > k ✈➔ ♥➳✉ n − k ❦❤æ♥❣ ♣❤↔✐ ❧➔ ♠ët ♥❤➙♥ tû ❝õ❛ n, t❤➻ ✭✸✾✮ ✈ỵ✐ ❤➺ sè ❤➡♥❣ aj ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✳ ❇ê ✤➲ ✷✳✹✳✺✳ ●✐↔ sû Ω(z, w, w , , w(n) ) = B(z, w)P (z, w, w , , w(n) ) + Q(z, w, w , , w(n) ) ≡ 0, (41) ð ✤â P ≡ ✈➔ Q ≡ ❧➔ ❝→❝ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ w, ✈➔ B(z, w) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✶✸✳✮ ◆➳✉ q = deg(B) > min{Γ(Q), deg(Q) + γ(Q)(1 − w (∞))}, t❤➻ (q − deg(Q))T (r, w) ≤(Γ(Q) − deg(Q) + 1)N (r, w) 1 + N r, + N r, + o(T (r, w)) Ω B ✸✷ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❈❤ù♥❣ ♠✐♥❤✳ ✣à♥❤ ❧➼ ✷✳✸✳✷ ❦➨♦ t❤❡♦ BP Q ❧➔ ❦❤→❝ ❤➡♥❣✱ ✈➔ ❞♦ ✤â Ω Q ❧➔ ❦❤→❝ ❤➡♥❣ ❤❛② Ω, Q ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ◆❤÷ ✈➟② Q∗ = Ω Q − Q Q ú ỵ r BP + Q = Ω = B P + BP + Q Ω Ω BP ∗ = Q∗ , tr♦♥❣ ✤â Ω B P P∗ = − − P Ω B P ❇➡♥❣ ❝→❝❤ ❞ò♥❣ ❜ê ✤➲ ✷✳✸✳✶✱ t❛ t❤➜② m(r, P ∗ ) = o(T (r, w)) ❚÷ì♥❣ tü✱ ❝❤ó♥❣ t❛ ❝ơ♥❣ ❝â ✤→♥❤ ❣✐→ m(r, Q∗ ) ≤ deg(Q)m(r, w) + o(T (r, w)) ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ ✤à♥❤ ❧➼ ❝ì ❜↔♥ t❤ù ♥❤➜t✱ t❛ t❤✉ ✤÷đ❝ m r, 1 ∗ ∗ = m(r, P ) + N (r, P ) − N r, + O(1) P∗ P∗ = N (r, P ∗ ) − N r, ∗ + o(T (r, w)) P ❱➻ t❤➳ qm(r, w) = m(r, B) + o(T (r, w)) ≤ m(r, Q∗ ) + m r, ∗ + o(T (r, w)) P ≤ deg(Q)m(r, w) + N (r, P ∗ ) − N r, + o(T (r, w)) P∗ ❈è ✤à♥❤ z0 ∈ Cp ◆➳✉ µ0B (z0 ) > ♥❤÷♥❣ z0 ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❝ü❝ ✤✐➸♠ ❤♦➦❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ ❤➺ sè ❝õ❛ B, P ✈➔ Q, t❤➻ w(z0 ) = ∞, ✈➔ µ∞ P ∗ (z0 ) ≤ ✸✸ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com w (z0 ) > ữ z0 ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❝ü❝ ✤✐➸♠ ❤♦➦❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ ❤➺ sè ❝õ❛ B, P ✈➔ Q, t❤➻ ∞ µ∞ Q∗ (z0 ) ≤ deg(Q)µw (z0 ) + Γ(Q) − deg(Q) + ✈➔ ❤ì♥ ♥ú❛✱ ♥➳✉ µ∞ P ∗ (z0 ) > 0, t❤➻ ∞ ∞ ∞ µ∞ P ∗ (z0 ) = µQ∗ /B (z0 ) ≤ (Q)µw (z0 ) + Γ(Q) − deg(Q) + − qµw (z0 ), ◆➳✉ µ∞ P ∗ (z0 ) = 0, t❤➻ ∞ ∞ ∞ µ∞ 1/P ∗ (z0 ) = µB/Q∗ (z0 ) ≥ qµw (z0 ) − {deg(Q)µw (z0 ) + Γ(Q) − deg(Q) + 1} ❉♦ ✤â✱ ❜ê ✤➲ ✤÷đ❝ s✉② r❛ tø 1 + N r, − (q − deg(Q))N (r, w) N (r, P ∗ ) − N r, ∗ ≤ N r, P Ω B + (Γ(Q) − deg(Q) + 1)N (r, w) + o(T (r, w)) ỵ sỷ (z, w, w , , w(n) ) = (wq P (z, w, w , , w(n) ) + Q(z, w, w , , w(n) ))N , (42) ð ✤â P ≡ ✈➔ Q ≡ ❧➔ ❝→❝ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ w ✈ỵ✐ q > max{deg(Q) + 2, Γ(Q)} ◆➳✉ k < N, t❤➻ ✭✸✻✮ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû✱ ♥❣÷đ❝ ❧↕✐ r➡♥❣ ✭✸✻✮ ❝â ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ s✐➯✉ ✈✐➺t ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ w ❑❤✐ ✤â k aj w j N (r, Ω) = N r, = kN (r, w) + o(T (r, w)), j=0 N (r, Ω) = N (r, wq P + Q) ≥ N qN (r, w) + o(T (r, w)), ✈➔ ❞♦ ✤â N (r, w) = o(T (r, w)) ✸✹ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ✤à♥❤ ❧➼ ✷✳✹✳✷✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ✭✸✻✮ ❝â ❞↕♥❣ ✭✸✽✮✳ ◆❤÷ ✈➟②✱ ❜ê ✤➲ ✷✳✹✳✺ ❦➨♦ t❤❡♦ 1 + + o(T (r, w)) N r, wq P + Q w 1 = N r, + N r, + o(T (r, w)) w+b w ≤ 2T (r, w) + o(T (r, w)) (q − deg(Q))T (r, w) ≤ N r, ✤✐➲✉ ♥➔② ❧➔ ❦❤æ♥❣ t❤➸ ①↔② r❛ ✈➻ q deg(Q) > ỵ sỷ r ữủ ợ q > (Q) + ❑❤✐ ✤â ✭✸✽✮ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ s t ữủ ợ số ữỡ k ✈➔ N ✸✺ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❑➳t ❧✉➟♥ ❝❤✉♥❣ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ởt số s ỡ s ỵ tt ỵ ỡ tự t tự q số t ố ợ trữớ ❤đ♣ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤✱ ❝ơ♥❣ ♥❤÷ tr÷í♥❣ ❤đ♣ ♠ư❝ t✐➯✉ ❞✐ ✤ë♥❣ ✈➔ ❤➔♠ ♥❤ä✳ ✷✴ ởt số ự ỵ tt tr ✈➜♥ ✤➲ t➻♠ ❝→❝ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ỵ tt tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♣✲❛❞✐❝✳ ❚r➻♥❤ ỵ qst ✈➲ ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♣✲❛❞✐❝✳ t q ợ sỹ tỗ t ❤➻♥❤ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♣✲❛❞✐❝ ✤÷đ❝ tr➻♥❤ ❜➔② t❤❡♦ ❜➔✐ ❜→♦ ❝õ❛ ❈✲❈✳ ❨❛♥❣ ✈➔ P✲❈✳ ❍✉ ✧❆ s✉r✈❡② ♦♥ ✲❛❞✐❝ ◆❡✈❛♥❧✐♥♥❛ t❤❡♦r② ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥s t♦ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✧ ✭❚❛✐✇❛♥❡s❡ ❏✳ ▼❛t❤✱ ❱♦❧✳ ✸✱ ◆♦✳ ✶✱ ♣♣✳ ✶✲✸✹✱ ✶✾✾✾✮✳ ✸✻ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❆♥❤ ❬✶❪ ❲✳ ❲✳ ❆❞❛♠s ❛♥❞ ❊✳ ●✳ ❙tr❛✉s ✭✶✾✼✶✮✱ ✧◆♦♥✲❆r❝❤✐♠❡❞✐❛♥ ❛♥❛❧②t✐❝ ❢✉♥❝✲ t✐♦♥s t❛❦✐♥❣ t❤❡ s❛♠❡ ✈❛❧✉❡s ❛t t❤❡ s❛♠❡ ♣♦✐♥ts✧✱ ■❧❧✐♥♦✐s ❏✳ ▼❛t❤✳✱ ✶✺✱ ♣♣✳ ✹✶✽ ✲ ✹✷✹✳ ❬✷❪ ✳❆✳ ❇♦✉t❛❜❛❛ ✭✶✾✾✶✮✱ ✧❆♣♣❧✐❝❛t✐♦♥s ❞❡ ❧❛ t❤❡♦r✐❡ ❞❡ ◆❡✈❛♥❧✐♥♥❛ ♣✲❛❞✐❝✧✱ ❈♦❧❧❡❝t✳ ▼❛t❤✳✱ ♣♣✳ ✼✺ ✲ ✾✸✳ ❬✸❪ ✳❆✳ ❇♦✉t❛❜❛❛✱ ❆✳ ❊s❝❛ss✉t ❛♥❞ ▲✳ ❍❛❞❞❛❞ ✭✶✾✾✼✮✱ ✧❖♥ ✉♥✐q✉❡♥❡ss ♦❢ ♣✲❛❞✐❝ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✧✱ ■♥❞❛❣✳ ▼❛t❤✳✱ ✽✱ ♣♣✳ ✶✹✺ ✲ ✶✺✺✳ ❬✹❪ ❲✳ ❈❤❡rr② ❛♥❞ ❈✳ ❈✳ ❨❛♥❣ ✭✶✾✾✾✮✱ ✧❯♥✐q✉❡♥❡ss ♦❢ ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ❡♥✲ t✐r❡ ❢✉♥❝t✐♦♥s s❤❛r✐♥❣ s❡ts ♦❢ ✈❛❧✉❡s ❝♦✉♥t✐♥❣ ♠✉❧t✐♣❧✐❝✐t②✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✷✼ ✭✹✮✱ ♣♣✳ ✾✻✼ ✲ ✾✼✶✳ ❬✺❪ ✳●✳ ❋r❛♥❦ ❛♥❞ ▼✳ ❘❡✐♥❞❡rs ✭✶✾✾✽✮✱ ✧❆ ✉♥✐q✉❡ r❛♥❣❡ s❡t ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ✇✐t❤ ✶✶ ❡❧❡♠❡♥ts✧✱ ❈♦♠♣❧❡① ❱❛r✐❛❜❧❡s ❚❤❡♦r② ❆♣♣❧✳✱ ✸✼ ✭✶✲✹✮✱ ♣♣✳ ✶✽✺ ✲ ✶✾✸✳ ❬✻❪ ❍❛ ❍✉② ❑❤♦❛✐ ✭✶✾✽✸✮✱ ✧❖♥ ♣✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❉✉❦❡ ▼❛t❤✳ ❏✳✱ ✺✵✱ ♣♣✳ ✻✾✺ ✲ ✼✶✶✳ ❬✼❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ▼② ❱✐♥❤ ◗✉❛♥❣ ✭✶✾✽✽✮✱ ✧❖♥ ♣✲❛❞✐❝ ◆❡✈❛♥❧✐♥♥❛ t❤❡✲ ♦r②✧✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳ ✱ ✶✸✺✶✱ ♣♣✳ ✶✹✻ ✲ ✶✺✽✳ ✸✼ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❬✽❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ▼❛✐ ❱❛♥ ❚✉ ✭✶✾✾✺✮✱ ✧P✲❛❞✐❝ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ t❤❡✲ ♦r❡♠✧✱ ■♥t❡r♥❛t✳ ❏✳ ▼❛t❤✳✱ ✻✱ ♣♣✳ ✼✶✾ ✲ ✼✸✶✳ ❬✾❪ P✳ ▲✐ ❛♥❞ ❈✳ ❈✳ ❨❛♥❣ ✭✶✾✾✻✮✱ ✧❖♥ t❤❡ ✉♥✐q✉❡ r❛♥❣❡ s❡ts ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✷✹✱ ♣♣✳ ✶✼✼ ✲ ✶✽✺✳ ❬✶✵❪ P✳ ▲✐ ❛♥❞ ❈✳ ❈✳ ❨❛♥❣ ✭✶✾✾✺✮✱ ✧❙♦♠❡ ❢✉rt❤❡r r❡s✉❧ts ♦♥ t❤❡ ✉♥✐q✉❡ r❛♥❣❡ s❡ts ♦❢ ♠❡r♦✲ ♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❑♦❞❛✐ ▼❛t❤ ❏✳✱ ✶✽✱ ♣♣✳ ✹✸✼ ✲ ✹✺✵✳ ❬✶✶❪ ❊✳ ▼✉❡s ❛♥❞ ▼✳ ❘❡✐♥❞❡rs ✭✶✾✾✺✮✱ ✧▼❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ s❤❛r✐♥❣ ♦♥❡ ✈❛❧✉❡ ❛♥❞ ✉♥✐q✉❡ r❛♥❣❡ s❡ts✧✱ ❑♦❞❛✐ ▼❛t❤✳ ❏✳✱ ✶✽✱ ♣♣✳ ✺✶✺ ✲ ✺✷✷✳ ❬✶✷❪ ❈✳❈✳ ❨❛♥❣ ❛♥❞ P✳❈✳ ❍✉ ✭✶✾✾✾✮✱ ✧❆ s✉r✈❡② ♦♥ ✲❛❞✐❝ ◆❡✈❛♥❧✐♥♥❛ t❤❡♦r② ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥s t♦ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✧✱ ❚❛✐✇❛♥❡s❡ ❏✳ ▼❛t❤✳✱ ✸✭✶✮✱ ♣♣✳ ✶ ✲ ✸✹✳ ❬✶✸❪ ❍✳ ❳✳ ❨✐ ✭✶✾✾✺✮✱ ✧❖♥ ❛ q✉❡st✐♦♥ ♦❢ ●r♦ss✧✱ ❙❝✐✳ ❈❤✐♥❛✱ ❙❡r✳ ❆✱ ✸✽✱ ♣♣✳ ✽ ✲ ✶✻✳ ❬✶✹❪ ❍✳ ❳✳ ❨✐ ✭✶✾✾✺✮✱ ✧❚❤❡ ✉♥✐q✉❡ r❛♥❣❡ s❡ts ♦❢ ❡♥t✐r❡ ♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝✲ t✐♦♥s✧✱ ❈♦♠♣❧❡① ❱❛r✐❛❜❧❡s ❚❤❡♦r② ❆♣♣❧✳✱ ✷✽✱ ♣♣✳ ✶✸ ✲ ✷✶✳ ✸✽ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ... |bq (z) |p B(z) = max 0≤j B(z), t❛ t❤➜② |bj (z) |p |w(z)|jp ≤ |bq (z) |p B(z)q−j |w(z)|jp < |bq (z) |p |w(z)|qp ❉♦ ✤â |B(z, w(z)) |p = |bq (z) |p |w(z)|qp ❑❤✐ ✤â |Ω(z) |p = |A(z,... ♣✲❛❞✐❝✳ ❈❤♦ 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 ✤➣ ✤÷đ❝ ❝❤✉➞♥ ❤â❛ s❛♦ ❝❤♦ |p| p = p? ??1 ✳ ❚❛ t✐➳♣... ✤ë♥❣✳ ●å✐ Pn (Cp ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤ n ❝❤✐➲✉ tr➯♥ Cp ▼ët ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ f : Cp −→ Pn (Cp ), t❛ ❤✐➸✉ ❧➔ ♠ët ợ tữỡ ữỡ (n + 1) ♥❣✉②➯♥ p? ??❛❞✐❝ f˜ = (f0 , , fn ) : Cp −→ Cn+1 p s❛♦ ❝❤♦