ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM NGUYỄN TRƯỜNG GIANG VỀ DẠNG ĐỊNH LÝ CƠ BẢN THỨ HAI KIỂU CARTAN CHO CÁC ĐƯỜNG CONG CHỈNH HÌNH LUẬN VĂN THẠC SĨ KHOA HỌC TOÁN HỌC THÁI NGUYÊN – 2008 ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM NGUYỄN TRƯỜNG GIANG VỀ DẠNG ĐỊNH LÝ CƠ BẢN THỨ HAI KIỂU CARTAN CHO CÁC ĐƯỜNG CONG CHỈNH HÌNH Chuyên ngành: GIẢI TÍCH Mã số: 60.46.01 LUẬN VĂN THẠC SĨ KHOA HỌC TOÁN HỌC Người hướng dẫn khoa học: TS TẠ THỊ HỒI AN THÁI NGUN – 2008 ▼ư❝ ❧ư❝ ỵ tt ❤➻♥❤ ✻ ✶✳✶ ❍➔♠ ♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳ ✽ ✶✳✷✳✶ ✶✳✷✳✷ ▼ët sè ✈➼ ❞ö ✈➲ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷✳✸ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ✳ ✳ ỵ ỡ tự t ỵ ỡ tự ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ỡ t❤ù ❤❛✐ ❦✐➸✉ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ❝❤♦ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ✷✸ ✷✳✶ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ỵ ỡ tự ữớ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ❝➢t ❝→❝ s✐➯✉ ♠➦t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷✳✶ ▼ët sè ❜ê ✤➲ q✉❛♥ trå♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ỡ tự ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tt ♣❤➙♥ ❜è ❣✐→ trà ❝õ❛ ◆❡✈❛♥❧✐♥♥❛ ✤÷đ❝ ✤→♥❤ ❣✐→ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ t❤➔♥❤ tü✉ ✤➭♣ ✤➩ ✈➔ s➙✉ s➢❝ ❝õ❛ t♦→♥ ❤å❝ tr♦♥❣ t❤➳ ❦✛ ❤❛✐ ♠÷ì✐✳ ✣÷đ❝ ❤➻♥❤ t tứ ỳ ừ t ỵ tt õ ỗ ố tứ ỳ ổ tr ❍❛❞❛♠❛r❞✱ ❇♦r❡❧ ✈➔ ♥❣➔② ❝➔♥❣ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ỹ t ỵ tt ♣❤➙♥ ❜è ❣✐→ trà ❝ê ✤✐➸♥ ❧➔ sü tê♥❣ q✉→t õ ỵ ỡ số ỡ ỵ tt ự sỹ ố tr ❝õ❛ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tø C ✈➔♦ C∪{∞}✳ ❚r✉♥❣ t ỵ tt ỗ ỵ ỡ ỵ ỡ tự t ♠ët ❝→❝❤ ✈✐➳t ❦❤→❝ ❝õ❛ ❝æ♥❣ t❤ù❝ P♦✐ss♦♥ ✲ ❏❡♥s❡♥✱ ỵ õ r trữ T (r, a, f ) ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ a ♥➳✉ t➼♥❤ s❛✐ ❦❤→❝ ♠ët ✤↕✐ ❧÷đ♥❣ ❜à ❝❤➦♥✱ tr♦♥❣ ✤â a ởt số ự tũ ỵ ỵ ỡ t❤ù ❤❛✐ t❤➸ ❤✐➺♥ ♥❤ú♥❣ ❦➳t q✉↔ ✤➭♣ ♥❤➜t✱ s➙✉ s t ỵ tt ố tr ỵ ữ r ố q ỳ tr÷♥❣ ✈➔ ❤➔♠ ①➜♣ ①➾✳ ◆➠♠ ✶✾✸✸✱ ❍✳ ❈❛rt❛♥ ❬✸❪ ự ỵ s f : C −→ Pn(C) ❧➔ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ❦❤ỉ♥❣ s✉② ❜✐➳♥ t✉②➳♥ t➼♥❤✱ Hi✱ i = 1, , q✱ ❧➔ ❝→❝ s✐➯✉ ♣❤➥♥❣ ð ✈à tr➼ tê♥❣ q✉→t✳ ❱ỵ✐ ✷ ♠é✐ ε > t❛ ❝â q m(r, Hj , f ) ≤ (n + + ε)T (r, f ), j=1 tr♦♥❣ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ✤ó♥❣ ✈ỵ✐ ♠å✐ r > ♥➡♠ ♥❣♦➔✐ ♠ët t➟♣ ❝â ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ ❤↕♥✳ ❑➳t q✉↔ tr➯♥ ❝õ❛ ❍✳ ❈❛rt❛♥ ❧➔ ổ tr t rở ỵ tt ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤✳ ❙û ❞ư♥❣ ❦➳t q✉↔ ✤â ổ ữ r ữợ ữủ số t ữớ ợ s ð ✈à tr➼ tê♥❣ q✉→t✳ ❈æ♥❣ tr➻♥❤ ♥➔② ❝õ❛ æ♥❣ ✤➣ ✤÷đ❝ ✤→♥❤ ❣✐→ ❧➔ ❤➳t sù❝ q✉❛♥ trå♥❣ ✈➔ r ởt ữợ ự ợ t tr ỵ tt ỵ tt ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ s❛✉ ♥➔② ✤÷đ❝ ♠❛♥❣ t➯♥ ❤❛✐ ♥❤➔ t♦→♥ ❤å❝ ♥ê✐ t✐➳♥❣ ❝õ❛ t❤➳ ❦✛ ✷✵✱ õ ỵ tt rt ỳ ✤➙②✱ ✈✐➺❝ ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ❈❛rt❛♥ ❝❤♦ tr÷í♥❣ ủ s t t út ữủ sỹ ú ỵ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝✳ ◆➠♠ ✷✵✵✹✱ ▼✳ ❘✉ ❬✶✷❪ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❣✐↔ t❤✉②➳t ❝õ❛ ❇✳ ❙❤✐❢❢♠❛♥ ❬✶✹❪ ✤➦t r❛ ❈❤♦ f : C → Pn(C) ❧➔ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ❦❤æ♥❣ s✉② ❜✐➳♥ ✤↕✐ sè✱ Dj , j = 1, , q, ❧➔ ❝→❝ s✐➯✉ ♠➦t ❜➟❝ dj ð ✈à tr➼ tê♥❣ q✉→t✳ ❑❤✐ ✤â ✈➔♦ ♥➠♠ ✶✾✼✾✳ ❈ư t❤➸✱ ỉ♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣✿ q d−1 j N (r, Dj , f ) + o(T (r, f )), (q − (n + 1) − ε)T (r, f ) ≤ j=1 tr♦♥❣ ✤â ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤ó♥❣ ✈ỵ✐ ♠å✐ r ✤õ ❧ỵ♥ ♥➡♠ ♥❣♦➔✐ ♠ët t➟♣ ❝â ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ ❤↕♥✳ ❑➳t q✉↔ tr➯♥ ✤➣ ✤÷đ❝ ◗✳ ❨❛♥ ✈➔ ✸ ❩✳ ❈❤❡♥ ❬✹❪ ♠ð rë♥❣ ❝❤♦ tr÷í♥❣ ❤đ♣ ❤➔♠ ✤➳♠ t➼♥❤ ✤➳♥ ❜ë✐ ❝❤➦♥ ✭❤❛② ❝á♥ ❣å✐ ❧➔ ❤➔♠ ✤➳♠ ❝ưt✮✳ ❑➳t q✉↔ ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿ ●✐↔ sû f : C → Pn(C) ❧➔ ♠ët →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❦❤æ♥❣ s✉② ❜✐➳♥ ✤↕✐ sè ✈➔ Dj ✱ ≤ j ≤ q ❧➔ q s✐➯✉ ♠➦t tr♦♥❣ Pn(C) ❝â ❜➟❝ dj t÷ì♥❣ ù♥❣✱ ð ✈à tr➼ tê♥❣ q✉→t✳ ❑❤✐ ✤â ✈ỵ✐ ♠é✐ ε > tỗ t ởt số ữỡ M s q M d−1 j N (r, Dj , f ) + o (T (r, f )) , q − (n + 1) − ε)T (r, f ) ≤ j=1 tr♦♥❣ ✤â ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤ó♥❣ ✈ỵ✐ ♠å✐ r ✤õ ❧ỵ♥ ♥➡♠ ♥❣♦➔✐ ♠ët t➟♣ ❝â ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ ự sỹ tỗ t ❝õ❛ ❝→❝ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ t❤ỉ♥❣ q✉❛ ↔♥❤ ♥❣÷đ❝ ❝õ❛ ❝→❝ s✐➯✉ ♠➦t✱ ♥❣÷í✐ t❛ t❤÷í♥❣ sû ❞ư♥❣ ✤à♥❤ ỵ ỡ tự rt tổ q✉❛ ❤➔♠ ✤➳♠ t➼♥❤ ✤➳♥ ❜ë✐ ❝❤➦♥✳ ◆❣♦➔✐ r❛ ✤à♥❤ ỵ rt ỏ t t t➼♥❤ s✉② ❜✐➳♥ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤✳ ▼ư❝ t✐➯✉ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ❝→❝ ❦➳t q✉↔ ✤➣ ✤÷đ❝ ✤÷❛ r❛ ❝õ❛ ◗✳ ❨❛♥ ✈➔ ❩✳ ợ ổ ự ỵ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛ ✲ ❈❛rt❛♥ ❝❤♦ ❝→❝ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ tø C ✈➔♦ Pn (C) ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ữỡ ũ ợ t ❞❛♥❤ ♠ư❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ❝→❝ ✤à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛✳ r ự ỵ ỡ tự ❝õ❛ ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ự ởt ỵ ỡ tự ✹ ❝❤♦ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❝➢t ❝→❝ s✐➯✉ ♠➦t ð ✈à tr➼ tê♥❣ q✉→t✳ ❈❤÷ì♥❣ ♥➔② ✤÷đ❝ ✈✐➳t ❞ü❛ tr➯♥ ❝æ♥❣ tr➻♥❤ ❝õ❛ ◗✳ ❨❛♥✱ ❩✳ ❈❤❡♥ ❬✹❪✳ ▲✉➟♥ ữủ t ữợ sỹ ữợ ❝õ❛ ❚❙✳ ❚↕ ❚❤à ❍♦➔✐ ❆♥ ✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❚❙ ✈➲ sü ❣✐ó♣ ✤ï ❦❤♦❛ ❤å❝ ♠➔ ❚❙ ✤➣ ❞➔♥❤ ❝❤♦ t→❝ ❣✐↔ ✈➔ ✤➣ t↕♦ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ♥❤➜t ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ t❤✉ë❝ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ✤➦❝ ❜✐➺t ❧➔ ❚❤➔② P❤÷ì♥❣ ❍➔ ❚r➛♥ ✈➔ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✈➔ ❝→❝ t❤➛② ❝æ ❣✐→♦ ❱✐➺♥ ❚♦→♥ ❤å❝ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ tr÷í♥❣ ❈❛♦ ✤➥♥❣ ❈ỉ♥❣ ♥❣❤➺ ✈➔ ❑✐♥❤ t➳ ❈æ♥❣ ♥❣❤✐➺♣✱ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ♥❤➜t ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ t ữỡ ỵ tt ♣❤➙♥ ❤➻♥❤ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ s➩ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❝→❝ ♣❤➛♥ s❛✉✳ ❈→❝ ❦✐➳♥ t❤ù❝ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ tr➼❝❤ ❞➝♥ tø ❬✶❪✱ ❬✺❪✱ ❬✼❪✱ ❬✾❪✱ ✳✳✳ ✶✳✶ ❍➔♠ ♣❤➙♥ ❤➻♥❤ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ D ❧➔ ♠ët ♠✐➲♥ tr♦♥❣ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✱ f (z) = u(x, y) + iv(x, y) ✤÷đ❝ ❣å✐ ❧➔ C✲❦❤↔ ✈✐ t↕✐ z0 C tỗ f (z0 + h) f (z0 ) t↕✐ ❣✐ỵ✐ ❤↕♥ ❤ú✉ ❤↕♥ lim ✳ h→0 h ●✐→ trà ✤â ✤÷đ❝ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ ♣❤ù❝ ❝õ❛ ❤➔♠ f (z) t↕✐ z0 ✳ ❤➔♠ ❍➔♠ f (z) ✤÷đ❝ ❣å✐ ❧➔ C✲❦❤↔ ✈✐ tr♦♥❣ D ♥➳✉ ♥â C ✲ ❦❤↔ ✈✐ t↕✐ ♠å✐ z0 ∈ D ✶✳✶✳✷ ✣à♥❤ ♥❣❤➽❛✳ ♥â C ❍➔♠ ❍➔♠ f (z) ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾♥❤ ❤➻♥❤ t↕✐ z0 ∈ C ♥➳✉ ✲ ❦❤↔ ✈✐ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ♥➔♦ ✤â ❝õ❛ f (z) ✤÷đ❝ ❣å✐ ❧➔ z0 ✳ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ D ♥➳✉ ♥â ❝❤➾♥❤ ❤➻♥❤ t↕✐ ♠å✐ ✻ ✤✐➸♠ z t❤✉ë❝ D✳ ❚➟♣ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ ♠✐➲♥ ✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ♣❤ù❝ C ✤÷đ❝ ❣å✐ ❧➔ f (z) ❍➔♠ D✱ ❦➼ ❤✐➺✉ ❧➔ H(D)✳ ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣ t♦➔♥ ♠➦t ♣❤➥♥❣ ❤➔♠ ỵ f (z) = u(x, y) + iv(x, y) ❝❤➾♥❤ ❤➻♥❤ tr➯♥ D ♥➳✉ ❝→❝ ❤➔♠ u(x, y) ✈➔ v(x, y) ❧➔ R2 ✲ ❦❤↔ ✈✐ tr➯♥ D ✈➔ tr➯♥ ✤â ❝→❝ ❤➔♠ u(x, y)✱ v(x, y) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❈❛✉❝❤② ✲ ❘✐❡♠❛♥♥✱ tù❝ ❧➔ ∂u ∂v ∂u ∂v = , = − , ∀ (x, y) ∈ D ∂x ∂y ∂y ∂x ✶✳✶✳✺ ✣à♥❤ ỵ sỷ f (z) ởt tr♦♥❣ ♠✐➲♥ ❤ú✉ ❤↕♥ D ⊂ C✳ ❑❤✐ ✤â tr♦♥❣ ♠é✐ ❧➙♥ ❝➟♥ ❝õ❛ ♠é✐ ✤✐➸♠ z ∈ D✱ ❤➔♠ f (z) ✤÷đ❝ ❦❤❛✐ tr✐➸♥ t❤➔♥❤ ❝❤✉é✐ (z − z0 ) (z − z0 )2 f (z) = f (z0 ) + f (z0 ) + f (z0 ) + 1! 2! ❍ì♥ ♥ú❛✱ ❝❤✉é✐ tr➯♥ ❤ë✐ tö ✤➲✉ ✤➳♥ ❤➔♠ |z − z0 | ≤ ρ tũ ỵ tr D ộ ữủ ❝õ❛ ✤✐➸♠ ✣✐➸♠ ✭❤❛② ❦❤æ♥❣✲✤✐➸♠ ❝➜♣ n = 1, , m − ❝❤✉é✐ ❚❛②❧♦ ❝õ❛ ❤➔♠ f (z) tr♦♥❣ ❧➙♥ ❝➟♥ ♥➳✉ f= g h z0 ∈ C m > 0✮ ✈➔ ✶✳✶✳✼ ✣à♥❤ ♥❣❤➽❛✳ D⊂C tr♦♥❣ ❤➻♥❤ trá♥ z0 ✶✳✶✳✻ ✣à♥❤ ♥❣❤➽❛✳ ♠å✐ f (z) ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ❝õ❛ ❤➔♠ f (z) ❦❤æ♥❣ ✤✐➸♠ ❜➟❝ m > ♥➳✉ f (n) (z0 ) = 0, ❝❤♦ f (m) (z0 ) = ❍➔♠ tr♦♥❣ ✤â f (z) g, h ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❧➔ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣ ✼ tr♦♥❣ D ◆➳✉ ❧➔ D = C t❤➻ t❛ ♥â✐ f (z) ♣❤➙♥ ❤➻♥❤ tr➯♥ C ❤❛② ✤ì♥ ❣✐↔♥ ❧➔ f (z) ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ✶✳✶✳✽ ✣à♥❤ ♥❣❤➽❛✳ ✣✐➸♠ z0 ✤÷đ❝ ❣å✐ ❝ü❝ ✤✐➸♠ ❝➜♣ ❧➔ h(z)✱ (z − z0 )m z0 ✈➔ h(z0 ) = 0✳ m > ❝õ❛ ❤➔♠ f (z) ♥➳✉ tr♦♥❣ ❧➙♥ ❝➟♥ ❝õ❛ z0 ❤➔♠ f (z) = tr♦♥❣ ✤â h(z) ❧➔ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣ ỵ ổ tự Ps ❏❡♥s❡♥✮✳ ●✐↔ sû f (z) ≡ ❧➔ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr♦♥❣ ❤➻♥❤ trá♥ {|z| ≤ R} ✈ỵ✐ < R < ∞✳ ●✐↔ sû aµ✱ µ = 1, , M, ❧➔ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❦➸ ❝↔ ❜ë✐✱ bν , ν = 1, 2, , N, ❧➔ ❝→❝ ❝ü❝ ✤✐➸♠ ❝õ❛ f tr♦♥❣ ❤➻♥❤ trá♥ ✤â✱ ❝ô♥❣ ❦➸ ❝↔ ❜ë✐✳ ❑❤✐ ✤â✱ ♥➳✉ z = reiθ (0 < r < R), f (z) = 0, f (z) = ∞ t❤➻ 2π log |f (z)| = 2π log f (Reiφ ) M + R2 − r dφ R2 − 2Rr cos(θ − φ) + r2 ✭✶✳✷✮ N log µ=1 R(z − aµ ) R(z − bν ) − log 2−b z R − aµ z R =1 ỵ tt ❤➻♥❤ ✶✳✷✳✶ ❈→❝ ❤➔♠ ◆❡✈❛♥❧✐♥♥❛ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ●✐↔ sỷ f ỵ tr ❜→♥ ❦➼♥❤ r ✈➔ r < R✳ n(r, ∞, f ) ✭t÷ì♥❣ ù♥❣✱ n(r, ∞, f ), ❧➔ sè ❝→❝ ❝ü❝ ✤✐➸♠ t➼♥❤ ❝↔ ❜ë✐✱ ✭t÷ì♥❣ ù♥❣✱ ❦❤ỉ♥❣ t➼♥❤ ❜ë✐✮✮✱ ❝õ❛ ❤➔♠ ❦➼♥❤ R ●✐↔ sû a ∈ C✱ f t❛ ✤à♥❤ ♥❣❤➽❛ n(r, a, f ) = n r, ∞, ✽ , f −a tr♦♥❣ ✤➽❛ ✤â♥❣ ❜→♥ ợ ( ) ữủ ữ tr♦♥❣ ❇ê ✤➲ ✷✳✷✳✹✳ ❚❛ ❝â ✐ θ := (✐) N n+1 δ(✐) ij ≥ , d(n + 1)! tr♦♥❣ ✤â tê♥❣ ✤÷đ❝ ❧➜② tr➯♥ t➜t ❝↔ (n + 1) ❜ë sè ♥❣✉②➯♥ ❦❤ỉ♥❣ ➙♠ (i) ✈ỵ✐ tê♥❣ ✤ó♥❣ ❜➡♥❣ N/d✳ ❈❤ù♥❣ ♠✐♥❤✳ ❑➳t ❤đ♣ ✈ỵ✐ ❇ê ✤➲ ✷✳✷✳✹ t❛ ❝â✱ (i) dn δ(i) ij = n+1 dn = n+1 = n+1 (i) (i) dn N ij = n+1 d j=1 (i) n N d N/d + n N = d n+1 d n N (N + d) + + (N + nd) N n+1 ≥ (n + 1)!d d(n + 1)! ❚❛ ❝â ✤✐➲✉ ự ỵ ỡ tự ❝❤♦ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ▼ët sè ♣❤→t ❜✐➸✉ ỵ ỡ tự rữợ t t ỵ rt ữ s ỵ sỷ f ởt ❦❤æ♥❣ s✉② ❜✐➳♥ t✉②➳♥ t➼♥❤ ✈➔ Hj ✱ ≤ j ≤ q✱ ❧➔ q s✐➯✉ ♣❤➥♥❣ tr♦♥❣ Pn (C) ð ✈à tr➼ tê♥❣ q✉→t✳ ❑❤✐ ✤â ✈ỵ✐ ♠é✐ ε > t❛ ❝â : C → Pn (C) q m(r, Hj , f ) ≤ (n + + ε)T (r, f ), j=1 ✷✾ tr♦♥❣ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ✤ó♥❣ ✈ỵ✐ ♠å✐ r > ♥➡♠ ♥❣♦➔✐ ♠ët t õ s ỳ ỵ tr ữủ t ữợ tr ỵ sỷ f ♠ët →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ♠➔ ↔♥❤ ❝õ❛ ♥â ❦❤æ♥❣ ❜à ❝❤ù❛ tr♦♥❣ ❜➜t ❦➸ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ♥➔♦✳ ●å✐ H1, , Hq ❧➔ ❝→❝ s✐➯✉ ♣❤➥♥❣ ♣❤➙♥ ❜✐➺t tr♦♥❣ Pn(C)✱ Lj ✱ ≤ j ≤ q✱ ❧➔ ❝→❝ ❞↕♥❣ t✉②➳♥ t➼♥❤ ✤à♥❤ ♥❣❤➽❛ H1 , , Hq ✳ ❑➼ ❤✐➺✉ W (f0 , , fn ) ❧➔ ❲r♦♥s❦✐❛♥ ❝õ❛ f0 , , fn ✳ ❑❤✐ ✤â ✈ỵ✐ ♠é✐ ε > 2π max log K j∈K = (f0 : : fn ) : C → Pn (C) f (reiθ ) Lj dθ +NW (r, 0) ≤ (n+1+ )T (r, f )+o(T (r, f )), |Lj (f )(reiθ )| 2π tr♦♥❣ ✤â tê♥❣ tr➯♥ ❧➜② tr➯♥ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❑ ❝õ❛ 1, , q s❛♦ ❝❤♦ ❝→❝ ❞↕♥❣ t✉②➳♥ t➼♥❤ Lj ✱ j ∈ K ✱ ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✱ f (z) ❧➔ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ |fj (z)|✱ ≤ j ≤ n ✈➔ Lj ❧➔ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❣✐→ trà t✉②➺t ✤è✐ ❝õ❛ ❝→❝ ❤➺ sè tr♦♥❣ Lj ✳ ◆➠♠ ✷✵✵✹✱ ▼✳ ❘✉ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ữủ ỵ ọ tự t s t tr tờ qt ỵ ❣✐↔✐ q✉②➳t trå♥ ✈➭♥ ❣✐↔ t❤✉②➳t ❝õ❛ ❙❤✐❢❢♠❛♥ ❬✶✹❪✳ ✷✳✷✳✽ ỵ sỷ f ởt ❤➻♥❤ ❦❤æ♥❣ s✉② ❜✐➳♥ ✤↕✐ sè ✈➔ Dj ✱ ≤ j ≤ q ❧➔ q s✐➯✉ ♠➦t tr♦♥❣ Pn(C) ❝â ❜➟❝ dj t÷ì♥❣ ù♥❣✱ ð ✈à tr➼ tê♥❣ q✉→t✳ ❑❤✐ ✤â ✈ỵ✐ ♠é✐ ε > 0✱ t❛ ❝â q j=1 : C → Pn (C) d−1 j m(r, Dj , f ) ≤ (n + + ε)T (r, f ), ✸✵ tr♦♥❣ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ✤ó♥❣ ✈ỵ✐ ♠å✐ r > ♥➡♠ ♥❣♦➔✐ ♠ët t➟♣ ❊ ❝â s ỳ ỵ ❝❤÷❛ t➼♥❤ ✤➳♥ ②➳✉ tè ❜ë✐ ❝õ❛ ❦❤ỉ♥❣ ✤✐➸♠✳ ❚✐➳♣ t❤❡♦ t❛ s➩ tr➻♥❤ ❜➔② ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ♠ët ❞↕♥❣ ỵ ỡ tự õ q ửt ỵ ỵ sỷ f : C Pn(C) ❧➔ ♠ët →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❦❤æ♥❣ s✉② ❜✐➳♥ ✤↕✐ sè ✈➔ Dj ✱ ≤ j ≤ q ❧➔ q s✐➯✉ ♠➦t tr♦♥❣ Pn(C) ❝â ❜➟❝ dj t÷ì♥❣ ù♥❣✱ ð ✈à tr➼ tê♥❣ q✉→t✳ ❑❤✐ ✤â ✈ỵ✐ ♠é✐ > tỗ t ởt số ữỡ M s❛♦ ❝❤♦ q M d−1 j N (r, Dj , f ) + o(T (r, f )), (q − (n + 1) − ε)T (r, f ) ≤ j=1 tr♦♥❣ ✤â ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤ó♥❣ ✈ỵ✐ ♠å✐ r ✤õ ❧ỵ♥ ♥➡♠ ♥❣♦➔✐ ♠ët t➟♣ ❝â ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❤ú✉ rữợ ự ỵ ú t ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❜ê ✤➲ s❛✉✳ ✷✳✷✳✶✵ ❇ê ✤➲✳ ●✐↔ sû f ❧➔ ♠ët →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❦❤æ♥❣ s✉② ❜✐➳♥ ✤↕✐ sè ✈➔ Dj ✱ ≤ j ≤ q ❧➔ q s✐➯✉ ♠➦t tr♦♥❣ Pn(C) ❝â ❜➟❝ ♥❤÷ ♥❤❛✉ ❧➔ d ✱ ð ✈à tr➼ tê♥❣ q✉→t✳ ❑❤✐ ✤â : C → Pn (C) q n m(r, Qj , f ) ≤ max i1 , ,in j=1 k=1 f (z) d + O(1) log |Qik ◦ f (z)| ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû Q1, , Qq ❧➔ ❝→❝ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t (n+1) ❜✐➳♥ ✈ỵ✐ ❝→❝ ❤➺ sè tr♦♥❣ C ❝â ❜➟❝ ♥❤÷ ♥❤❛✉ ❧➔ ✸✶ d, ✤à♥❤ ♥❣❤➽❛ ❝→❝ s✐➯✉ ♠➦t D1 , , Dq ✳ sè i1 , , iq zC t õ tỗ t↕✐ ♠ët ❝→❝❤ s➢♣ ①➳♣ ❝→❝ ❝❤➾ ❝õ❛ ❝→❝ ❝❤➾ sè 1, , q s❛♦ ❝❤♦ |Qi1 ◦ f (z)| ≤ |Qi2 ◦ f (z)| ≤ ≤ Qiq ◦ f (z) ❉♦ Qj ✱ ≤ j ≤ n tr tờ qt t ỵ ❍✐❧❜❡rt✬s ◆✉❧❧st❡❧❡♥s❛t③✱ t❛ ❝â✿ ✈ỵ✐ ♠é✐ sè ♥❣✉②➯♥ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✭✷✳✶✮ mk ≥ d k ✱ ≤ k n tỗ t ởt s n+1 k xm k = bjk (x0 , , xn )Qij (x0 , , xn ), j=1 tr♦♥❣ ✤â bjk ✱ ≤ j ≤ n + 1✱ ≤ k ≤ n✱ ❤➺ sè tr♦♥❣ C ❝â ❜➟❝ mk − d |fk (z)|mk ≤ c1 f (z) mk −d ❧➔ ❝→❝ ❞↕♥❣ t t ợ tữỡ ự ữ max |Qi1 f (z)| , , Qin+1 ◦ f (z) , ✭✷✳✷✮ tr♦♥❣ ✤â f (z) := max {|f0 (z)| , , |fn (z)|}✱ c1 ❧➔ ❤➡♥❣ sè ❞÷ì♥❣ ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ ❝→❝ ❤➺ sè ❝õ❛ bjk ✱1 ≤ j ≤ n + 1✱ ≤ k ≤ n✱ ♣❤ö t❤✉ë❝ ✈➔♦ ❝→❝ ❤➺ sè ❝õ❛ Qj ✱ ≤ j n + ú ỵ r ú ợ ♠å✐ f (z) mk k = 0, , n✳ tù❝ ❧➔ ❝❤➾ ◆❤÷ ✈➟② = max |f (z)|mk k=0, ,n ≤ c1 f (z) mk −d max |Qi1 ◦ f (z)| , , Qin+1 ◦ f (z) , ✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ f (z) d ≤ c1 max |Qi1 ◦ f (z)| , , Qin+1 ◦ f (z) ✸✷ ✭✷✳✸✮ ❚❤❡♦ ✭✷✳✶✮ ✈➔ ✭✷✳✸✮✱ t❛ ❝â q n f (z) d = Qij ◦ f (z) j=1 k=1 f (z) d |Qik ◦ f (z)| n ≤ cq−n k=1 q k=n+1 f (z) d |Qik ◦ f (z)| f (z) d |Qik ◦ f (z)| ◆❤÷ ✈➟②✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ①➜♣ ①➾ t❛ s✉② r❛ q q f (z) dr m (r, Qj , f ) = log Qij ◦ f (z) j=1 j=1 n ≤ max {i1 , ,in } tr♦♥❣ ✤â c1 k=1 r f (z) d log |Qik ◦ f (z)| + (q − n) log c1 , ❧➔ ❤➡♥❣ sè ❞÷ì♥❣ ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ ❝→❝ ❤➺ sè ❝õ❛ j ≤ n + 1✱ ≤ k ≤ n✱ bjk ✱1 ≤ tù❝ ❧➔ ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ ❝→❝ ❤➺ sè ❝õ❛ Qj ✱ ≤ j ≤ n + ❚ø ✤â✱ t❛ ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ❤ñ♣ N sè ♥❣✉②➯♥ ❞÷ì♥❣ ♥➔♦ ✤â✳ ❚❛ s➩ ①➙② ❞ü♥❣ ♠ët ❝ì sð t❤➼❝❤ ψ1 , , ψM ❝õ❛ VN ữ s tr õ ợ ởt ổ ✤➛✉ t✐➯♥ M := dim VN ✳ W(✐ ) ❚❛ ❜➢t ✤➛✉ ✈➔ ❧➜② ♠ët ❝ì sð ❜➜t ❦ý ❝õ❛ ♥â✳ ❚❛ t✐➳♣ tư❝ ①➙② ❞ü♥❣ ❜➡♥❣ q✉② ♥↕♣ ♥❤÷ s❛✉✿ ❣✐↔ sû ❧➔ ❤❛✐ n ❜ë ❧✐➯♥ t✐➳♣ s❛♦ ❝❤♦ ❝❤å♥ ✤÷đ❝ ♠ët ❝ì sð ❝õ❛ ❞✐➵♥ tr♦♥❣ ❝â ❞↕♥❣ W(✐) ✐ ✐ dσ( )✱ dσ( ) ≤ N W(✐ ) ✳ ❚ø ✤à♥❤ ♥❣❤➽❛ t❛ ❝â t❤➸ ❧➜② ữủ tr õ ỹ trữợ tr ❣✐↔ sû r➡♥❣ t❛ ✤➣ ❝õ❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ t❤÷ì♥❣ r1i1 rnin η ✱ ✐ ( )>() W(✐ ) η ∈ VN −dσ(i) ✳ W(i) /W(i ) ❚❛ ♠ð rë♥❣ ❝ì sð ✤➣ ✤÷đ❝ ❜➡♥❣ ❝→❝❤ t❤➯♠ ✈➔♦ ❝→❝ ❜✐➸✉ ❞✐➵♥ ✤â ✈➔ t❛ s➩ t❤✉ ✤÷đ❝ ❝ì sð ❝❤♦ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✸✸ W(✐) ✈➔ q✉→ tr➻♥❤ q✉② ♥↕♣ ✤÷đ❝ t✐➳♣ tư❝ ❝❤♦ ✤➳♥ ❦❤✐ W(✐) = VN ✱ ♥❤÷ ✈➟②✱ t❛ s➩ t❤✉ ✤÷đ❝ ♠ët ❝ì sð ❦❤✐ ✤â t❛ ❞ø♥❣ ❧↕✐✳ ❇➡♥❣ ❝→❝❤ ψ1 , , ψM ❝õ❛ VN ✳ ✷✳✷✳✶✶ ❇ê ✤➲✳ ❱ỵ✐ tt tr ỵ õ MN qd − T (r, f ) ≤ θ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ ✤â ψ1 , , ψM L1 , , LM sû q N (r, Qj , f ) − NW (r, 0) + o (T (r, f )) θ j=1 φ1 , , φM ❧➔ ♠ët ❝ì sð ❝è ✤à♥❤ ❝õ❛ φ1 , , φM ✱ ♥❤÷ ✈➟② (φ1 (f ) : : φM (f )) : C → PM −1 (C)✳ ψt (f ) = Lt (F )✱ L1 , , LM ❦❤æ♥❣ s✉② ❜✐➳♥ M z C t ữợ ữủ log |Lt (F )(z)| = log t=1 ψ f F = ❦❤æ♥❣ s✉② ❜✐➳♥ t✉②➳♥ t➼♥❤✳ M ❱ỵ✐ tr♦♥❣ ✤â ❈→❝ ❞↕♥❣ t✉②➳♥ t➼♥❤ ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ ❜✐➳t r➡♥❣ t❤❡♦ ❣✐↔ t❤✐➳t✱ F õ t ữủ ữợ tờ ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ✤↕✐ sè ♥➯♥ VN ✳ |ψt (f )(z)|✳ ●å✐ t=1 ❧➔ ♠ët ♣❤➛♥ tû tr♦♥❣ ❝ì sð✱ ✤÷đ❝ ①➙② ❞ü♥❣ tø ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❝ì sð ❝õ❛ W(i) /W(i ) ✱ ❦❤✐ ✤â ψ = r1i1 rnin η, tr♦♥❣ ✤â η ∈ VN −dσ(i) ✳ ◆❤÷ ✈➟② t❛ ❝â ♠ët ❝❤➦♥ |ψ(f )(z)| ≤ |r1 (f )(z)|i1 |rn (f )(z)|in |η(f )(z)| ≤ c2 |r1 (f )(z)|i1 |rn (f )(z)|in f (z) tr♦♥❣ ✤â c2 N −dσ(i) ❧➔ ♠ët ❤➡♥❣ sè ❞÷ì♥❣ ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ ✸✹ ψ✱ , ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ f ✈➔ z ✈➔ ❝â ✤ó♥❣ δ(i) ❤➔♠ ψ ♥❤÷ t❤➳✳ ◆❤÷ ✈➟② log |ψt (f )(z)| ≤ i1 log |r1 (f )(z)| + + in log |rn (f )(z)| + (N − dσ(i)) log f (z) + c3 d ≤ i1 log |r1 (f )(z)| − log f (z) + in log |rn (f )(z)| − log f (z) + d + N log f (z) + c3 f (z) d f (z) d ≤ −i1 log − − in log |r1 (f )(z)| |rn (f )(z)| + N log f (z) + c3 , tr♦♥❣ ✤â f t❤✉ë❝ c3 ✈➔ ❧➔ ♠ët ❤➡♥❣ sè ❞÷ì♥❣ ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ z✳ M |ψt (f )(z)| |Lt (F )(z)| = log log t=1 t=1 ≤− δ(i) (i) f (z) d f (z) d i1 log + + in log |r1 (f )(z)| |rn (f )(z)| + M N log f (z) + M c3 n d f (z) =− log |rj (f )(z)| j=1 ð tr➯♥ t❤➻ M θ ✭✷✳✸✮ δ(i) ij + M N log f (z) + M c3 (i) tr♦♥❣ ✤â tê♥❣ tr➯♥ ✤÷đ❝ ❧➜② tr➯♥ t➜t ❝↔ j ❦❤ỉ♥❣ ♣❤ư ❑➨♦ t❤❡♦✱ M sè ψ✱ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ n ✐ n ❜ë ✈ỵ✐ σ( ) ≤ N/d✳ ❱ỵ✐ ♠é✐ j✳ ◆➯♥ ✭✷✳✸✮ trð t❤➔♥❤ f (z) d log + M N log f (z) + M c3 |Lt (F )(z)| ≤ −θ log |r (f )(z)| j t=1 j=1 ✸✺ ✣✐➲✉ ✤â ❦➨♦ t❤❡♦ n M f (z) d F (z) M log ≤ log − log F (z) |r (f )(z)| θ |L (F )(z)| θ j t t=1 j=1 + MN M c3 log f (z) + θ θ ❉♦ ❝â ♠ët sè ❤ú✉ ❤↕♥ ❝→❝❤ ❝❤å♥ r1 , , rn ∈ {Q1 , , Qq } ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ ❞↕♥❣ t✉②➳♥ t➼♥❤ n max log {i1 , ,in } k=1 ❝→❝ ❞↕♥❣ max K L1 , , Lu ✳ f (z) dr ≤ max log |gik (f )(z)|r θ K − tr♦♥❣ ✤â j∈K F : C → PM −1 (C) ❚ø ✭✷✳✹✮ t❛ ❝â M MN T (r, F ) + T (r, f ) + c4 , θ θ ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✱ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ♥➯♥ t❛ ❝â F (z) r − |Lj (F )(z)|r ✤÷đ❝ ❧➜② tr➯♥ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ Lj ✱ j ∈ K ✭✷✳✹✮ c4 K ❝õ❛ 1, , u s❛♦ ❝❤♦ ❧➔ ♠ët ❤➡♥❣ sè ữỡ r ỵ ❝❤➾♥❤ ❤➻♥❤ ✈➔ ❝→❝ ❞↕♥❣ t✉②➳♥ t➼♥❤ L1 , , Lu ✈➔ ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✸✮ t❛ ❝â q n m(r, Qi , f ) ≤ max log {i1 , ,in } i=1 k=1 f (z) dr + (q − c) log c1 |rik (f )(z)|r MN ≤ − NW (r, 0) + T (r, f ) + o(T (r, F )) θ θ tr♦♥❣ ✤â W ❧➔ ❲r♦♥s❦✐❛♥ ❝õ❛ F1 , , FM ✳ ✷✳✷✳✶✷ ❇ê ✤➲✳ ❱ỵ✐ tt tr ỵ õ q N (r, Qj , f ) − NW (r, 0) ≤ θ j=1 q N M (r, Qj , f ) j=1 ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ z ∈ C✱ ❦❤ỉ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ Qj ◦ f tr✐➺t t✐➯✉ t↕✐ z ✸✻ ✈ỵ✐ ≤ j ≤ q1 ✈➔ Qj ◦ f ❦❤æ♥❣ tr✐➺t t✐➯✉ t↕✐ z tê♥❣ q✉→t ♥➯♥ ✈ỵ✐ j > q1 ✳ q1 ≤ n ✳ Qj ð ✈à tr➼ ❑❤✐ õ tỗ t ởt số ữỡ ❝❤➾♥❤ ❤➻♥❤ z ❚❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣✿ ❝→❝ γj kj ≥ ❦❤æ♥❣ tr✐➺t t✐➯✉ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❯ ❝õ❛ s❛♦ ❝❤♦ Qj ◦ f = (ζ − z)kj γj , j = 1, , q ✱ ❱ỵ✐ {Q1 , , Qn } ∈ {Q1 , , Qq } t❛ ❝â t❤➸ t❤✉ ✤÷đ❝ ♠ët ❝ì sð ψ1 , , ψM ❝õ❛ VN Lt (F )✳ tr♦♥❣ ✤â kj = q1 < j ≤ q ✳ ✈ỵ✐ ♥➳✉ ✈➔ ❝→❝ ❞↕♥❣ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ L1 , , LM s❛♦ ❝❤♦ ψt (f ) = ❚❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ✤à♥❤ t❤ù❝ ❲r♦♥s❦✐❛♥✱ t❛ ❝â W = W (F1 , , FM ) = CW (L1 (F ), , LM (F )) =C ψ1 (f ) ··· ψM (f ) (ψ1 (f )) ··· (ψM (f )) ✳✳ ✳ ✳✳ ✳✳ ✳ ✳ (ψ1 (f ))(M −1) · · · (ψM (f ))(M −1) ●å✐ ψ ❧➔ ♠ët ♣❤➛♥ tû tr♦♥❣ ❝ì sð✱ ✤÷đ❝ ①➙② ❞ü♥❣ tø ❝ì sð ❝õ❛ W(✐) /W(✐ ) ✱ t❛ ❝â t❤➸ ✈✐➳t ψ = Qi11 Qinn η,η ∈ VN −dσ(i) ✳ ❚❛ ❝â ψ(f ) = (Q1 (f ))i1 (Qn (f ))in η(f ), tr♦♥❣ ✤â i (Qj (f ))ij = (ζ − z)ij kj γjj , j = 1, , n ❍ì♥ ♥ú❛ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ kj ≥ M ❚❛ t❤➜② r➡♥❣ ❝â ♥➳✉ δ(i) ≤ j ≤ q0 ♣❤➛♥ tû ❞↕♥❣ ✸✼ ✈➔ ψ ≤ kj < M ♥➳✉ q0 < j ≤ q1 ✳ tr♦♥❣ ❝ì sð✳ ◆❤÷ ✈➟② W tr✐➺✉ t✐➯✉ t↕✐ z ✈ỵ✐ ❜ë✐ ➼t ♥❤➜t ❧➔ q0 q0 ij (kj − M ) δ(i) = (i) j=1 (kj − M ) ij δ(i) j=1 (i) q0 (kj − M ) =θ j=1 ◆❤÷ ✈➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ t t s ự ỵ ự ỵ sỷ f = (f0 : : fn) : C → Pn(C) ❧➔ ♠ët →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❦❤æ♥❣ s✉② ❜✐➳♥ ✤↕✐ sè✳ ●å✐ Pn (C) s✐➯✉ ♠➦t ð ✈à tr➼ tê♥❣ q✉→t tr♦♥❣ ❧➔ ❝→❝ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t tr♦♥❣ ♥❣❤➽❛ ❝→❝ Dj ✳ ❚❛ t❤❛② t❤➳ ♥❤ä ♥❤➜t ❝õ❛ ❝→❝ ♥❣✉②➯♥ ❞÷ì♥❣ Qj ❝â ❜➟❝ C[x0 , , xn ] ❜ð✐ d/dj Qj dj D1 , , Dq ✈➔ dj ❜➟❝ ✱ tr♦♥❣ ✤â ❧➔ ❝→❝ Qj ✱ ≤ j ≤ q ✱ t÷ì♥❣ ù♥❣✱ ✤à♥❤ d ❧➔ ❜ë✐ sè ❝❤✉♥❣ dj ✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ✈ỵ✐ ♠é✐ ε > tỗ t ởt số M s q d/dj M d−1 j N (r, Qj (q − (n + 1) − ε)T (r, f ) ≤ , f ) + o(T (r, f ) j=1 ú ỵ r z∈C t❤➻ ③ ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ d/dj N M (r, Qj , f) ≤ ❧➔ ♠ët ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ Qj ◦ f ✈ỵ✐ ❜ë✐ αdj /d✳ d/dj Qj ◦f ✈ỵ✐ ❜ë✐ ✣✐➲✉ ✤â ❦➨♦ t❤❡♦ d d [M d/dj ] N (r, Qj , f ) ≤ N M (r, Qj , f ) dj dj ◆❤÷ ✈➟② q d/dj M d−1 j N (r, Qj (q − (n + 1) − ε)T (r, f ) ≤ j=1 ✸✽ , f ) + o(T (r, f )) α ❇ð✐ ✈➟②✱ ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ ❝→❝ ✤❛ t❤ù❝ Q1 , , Qq ❝â ❜➟❝ ♥❤÷ ♥❤❛✉ ✈➔ ❜➡♥❣ d✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✶✵ t❛ s✉② r❛ q n m (r, Qj , f ) ≤ max {i1 , ,in } j=1 ▲➜② n ✤❛ t❤ù❝ ♣❤➙♥ ❜✐➺t k=1 f (z) d log |Qik ◦ f (z)| r1 , , rn ∈ {Q1 , , Qq }✳ + (q − n) log c1 ❚❤❡♦ ❣✐↔ t❤✐➳t ð ✈à tr➼ tê♥❣ q✉→t✱ ❝❤ó♥❣ ✤à♥❤ ♥❣❤➽❛ ♠ët ✤❛ t↕♣ ❝♦♥ ❝â sè ❝❤✐➲✉ ❜➡♥❣ ✵ tr♦♥❣ Pn (C)✳ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ s➩ ❝â ♠ët ❝➜✉ tró❝ ❧å❝ W(i) ❝õ❛ VN δ(i) := dim ✈ỵ✐ ♠é✐ ❝➦♣ n ❜ë ❧✐➯♥ t✐➳♣ M := dim VN ✳ ✣➦t N ❚❛ ❝è ✤à♥❤ ♠ët sè ♥❣✉②➯♥ ❧ỵ♥ MN qd − θ ✐ N ✭s➩ ❝❤å♥ s❛✉✮✱ ❣å✐ VN C[x0 , , xn ]✳ ❚❛ tr♦♥❣ ✈➔ W(i) = dn ✱ W(i ) ✐ ( ) > ( )✳ ◆❤÷ ✈➟② t❤❡♦ ❇ê ✤➲ ✷✳✷✳✶✶✱ t❛ ❝â q T (r, f ) ≤ N (r, Qj , f ) − NW (r, 0) + o (T (r, f )) θ j=1 ✭✷✳✺✮ ▼➦t ❦❤→❝ ❇ê ✤➲ ✷✳✷✳✶✷ ❝❤♦ t❛ q N (r, Qj , f ) − NW (r, 0) ≤ θ j=1 q N M (r, Qj , f ) j=1 tử t s ữợ ữủ tr→✐ ❝õ❛ ✭✷✳✺✮✳ ●✐↔ sû N ❝❤✐❛ ❤➳t ❝❤♦ d✳ ❑❤✐ ✤â M = N +n n n = (N + n)! = N + O N n−1 N !n! n! ✸✾ ✭✷✳✼✮ ❇➡♥❣ ❝→❝❤ ❦❤→❝✱ ✈➻ ❜➡♥❣ ✈ỵ✐ ✈➔ ❜➡♥❣ m ❜ë sè ♥❣✉②➯♥ ❦❤ỉ♥❣ ➙♠ ✈ỵ✐ tê♥❣ (m + 1) ❜ë sè ♥❣✉②➯♥ ❦❤ỉ♥❣ ➙♠ ✈ỵ✐ tê♥❣ T +m ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✺✱ t❛ ❝â m ≤T ✤ó♥❣ ❜➡♥❣ ✤ó♥❣ T ∈Z N n+1 + O(N n ) MN ≤ n! n+1 = d(n + 1) + O(N −1 ) θ N d(n + 1)! tù❝ ❧➔✱ (qd − MN )T (r, f ) ≥ d(q − (n + 1) − O(N −1 ))T (r, f ) θ ✣✐➲✉ ✤â s✉② r❛✱ ✈ỵ✐ ♠é✐ (qd − ♥➳✉ t❛ ❧➜② N >0 tr ỵ MN )T (r, f ) ≥ d(q − (n + 1) − ε)T (r, f ), θ ✤õ ❧ỵ♥ s❛♦ ❝❤♦ O(N −1 ) ≤ ε ✈➔ N ❝❤✐❛ ❤➳t ❝❤♦ ❑➳t ❤đ♣ ❝→❝ ❝ỉ♥❣ t❤ù❝ ✭✷✳✺✮✱ ✭✷✳✻✮ ✈➔ ✭✷✳✽✮ t❛ ❝â✱ ✈ỵ✐ ♠é✐ t↕✐ sè M ✭✷✳✽✮ d ε>0 s❛♦ ❝❤♦ q (q − (n + 1) − ε)T (r, f ) ≤ j=1 ỵ ữủ ự M N (r, Qj , f ) + o(T (r, f )) d tỗ t tr ởt số t t ỵ tt ❤➻♥❤ ✈➔ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤✳ ▼ư❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ❝õ❛ ◗✳ ỵ ỡ tự ❤❛✐ ❝❤♦ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ❝➢t ❝→❝ s✐➯✉ ♠➦t ð ✈à tr➼ tê♥❣ q✉→t✱ ❝â t➼♥❤ ✤➳♥ ②➳✉ tè ❤➔♠ ✤➳♠ ❜ë✐ ❝❤➦♥✳ ✹✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❍➔ ❍✉② ❑❤♦→✐✭✷✵✵✵✮✱ ●✐→♦ tr➻♥❤ ❣✐↔✐ t➼❝❤ ♣❤ù❝✱ ❱✐➺♥ ❚♦→♥ ❤å❝✳ ❬✷❪ ◆❣✉②➵♥ ❱➠♥ ❑❤✉➯✱ ▲➯ ▼➟✉ ❍↔✐ ✭✶✾✾✼✮✱ ❍➔♠ ❜✐➳♥ ♣❤ù❝✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❙✉r ❧❡s ③❡r♦s ❞❡s ❝♦♠❜✐♥❛✐s✐♦♥s ❧✐♥❡❛r✐r❡s ❞❡ p ❢♦♥❝t✐♦♥s ❤♦❧♦♠♦r♣❡s ❞♦♥♥❡❡s✱ ▼❛t❤❡♠❛t✐❝❛ ✭❈❧✉❥✮✳ ✼✱ ✽✵✲✶✵✸✳ ❬✸❪ ❈❛rt❛♥✳❍✭✶✾✸✸✮✱ ❬✹❪ ❈❤❡♥ ❩✳ ❛♥❞ ❨❛♥ ◗✳ ✱ ❲❡❛❦ ❈❛rt❛♥✲t②♣❡ ▼❛✐♥ ❚❤❡♦r❡♠ ❢♦r ❍♦❧♦✲ ♠♦r♣❤✐❝ ❈✉r✈❡s✱ ◆❛t✐♦♥❛❧ ◆❛t✉r❛❧ ❙❝✐❡♥❝❡ ❋♦✉♥❞❛t✐♦♥ ♦❢ ❈❤✐♥❛✱ ◆♦✳ ✶✵✺✼✶✶✸✺✳ ◆❡✈❛♥❧✐♥♥❛✬s ❚❤❡♦r② ♦❢ ❱❛❧✉❡ ❉✐s✲ tr✐❜✉t✐♦♥✳ ❚❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❚❤❡♦r❡♠ ❛♥❞ ✐ts ❊rr♦r ❚❡r♠s✱ ❬✺❪ ❈❤❡rr② ❲✳ ❛♥❞ ❨❡ ③✭✷✵✵✺✮✳✱ ❙♣r✐♥❣❡r ▼♦♥♦❣r❛♣❤s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✷✵✵✶✳ ❬✻❪ ❈♦r✈❛❥❛ P✳ ❛♥❞ ❩❛♥♥✐❡r ✭✷✵✵✹✮ ❯✳▼✳✱ ❖♥ ❛ ❣❡♥❡r❛❧ ❚❤✉❡✬s ❡q✉❛✲ t✐♦♥✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳ ✶✷✻✱ ♥♦✳ ✺✱ ✶✵✸✸✕✶✵✺✺✳ ❬✼❪ ❍❛rts❤♦r♥❡ ❘✳✱ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡tr② ✶✾✾✼ ✈♦❧✳ ✺✷✱❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✳ ✹✷ ●r❛❞✳ ❚❡①ts ✐♥ ▼❛t❤✳ ❬✽❪ ❍❛②♠❛♥ ❲✳ ❑✳ ✭✶✾✻✹✮✱ ▼❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✱ ❈❧❛r❡♥❞♦ Pr❡ss✱ ❖①❢♦r❞✳ ❬✾❪ ◆❡✈❛♥❧✐♥♥❛ ❘✳ ✭✶✾✷✻✮✱ ❊✐♥✐❣❡ ❊✐♥❞❡✉t✐❣❦❡✐tss❛t③❡ ✐♥ ❞❡r ❚❤❡♦r✐❡ ❞❡r ♠❡r♦♠♦r♣❤❡♥ ❋✉♥❝t✐♦♥✱ ❆❝t❛✳ ▼❛t❤✳ ❬✶✵❪ ◆♦❝❤❦❛ ❊✳ ■✳ ✭✶✾✽✸✮ ✱ ✹✽ ✱ ✸✻✼✲✸✾✶✳ ❖♥ t❤❡ t❤❡♦r② ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❝✉r✈❡s✱ ❙♦✲ ✈✐❡t ▼❛t❤✳ ❉♦❦❧✳ ✷✼ ✱ ♥♦✳ ✷✱ ✸✼✼✕✸✽✶✳ ❬✶✶❪ ❑♦❜❛②❛s❤✐ ❙✳ ✭✶✾✼✵✮✱ ♣✐♥❣s✱ ▼❛r❝❡❧ ❉❡❦❦❡r✳ ❍②♣❡r❜♦❧✐❝ ♠❛♥✐❢♦❧❞s ❛♥❞ ❤♦❧♦♠♦♣❤✐❝ ♠❛♣✲ ❆ ❞❡❢❡❝t r❡❧❛t✐♦♥ ❢♦r ❤♦❧♠♦r♣❤✐❝ ❝✉r✈❡s ✐♥t❡rs❡❝t✲ ✐♥❣ ❤②♣❡rs✉r❢❛❝❡s✱ ❆♠❡r✳ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤✳✶✷✻ ✱ ✷✶✺✲✷✷✻✳ ❬✶✷❪ ❘✉ ▼✳ ✭✷✵✵✹✮✱ ❬✶✸❪ ❘✉ ▼✳✭✶✾✾✼✮✱ ❖♥ ❛ ❣❡♥❡r❛❧ ❢♦r♠ ♦❢ t❤❡ s❡❝♦♥❞ ♠❛✐♥ t❤❡♦r❡♠✱ ✸✹✾ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✱ ✺✵✾✸✲✺✶✵✺✳ ❖♥ ❤♦❧♦♠♦r♣❤✐❝ ❝✉r✈❡s ❛♥❞ ♠❡r♦♠♦r♣❤✐❝ ♠❛♣s ✐♥ ♣r♦❥❡❝t✐✈❡ s♣❛❝❡✱ ■♥❞✐❛♥❛ ❯♥✐✈✳ ▼❛t❤✳ ❏✳ ✷✽✱ ♥♦✳ ✹✱ ✻✷✼✕ ❬✶✹❪ ❙❤✐❢❢♠❛♥ ❇✳ ✭✶✾✼✾✮✱ ✻✹✶✳ ✹✸ ...ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM NGUYỄN TRƯỜNG GIANG VỀ DẠNG ĐỊNH LÝ CƠ BẢN THỨ HAI KIỂU CARTAN CHO CÁC ĐƯỜNG CONG CHỈNH HÌNH Chuyên ngành: GIẢI TÍCH Mã số: 60.46.01 LUẬN VĂN THẠC SĨ