Luận án tiến sĩ toán học một số dạng của định lý ritt và ứng dụng vào vấn đề duy nhất

ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM PHẠM NGỌC HOA MỘT SỐ DẠNG CỦA ĐỊNH LÝ RITT VÀ ỨNG DỤNG VÀO VẤN ĐỀ DUY NHẤT LUẬN ÁN TIẾN SĨ TOÁN HỌC THÁI NGUYÊN - NĂM 2018 ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM PHẠM NGỌC HOA MỘT SỐ DẠNG CỦA ĐỊNH LÝ RITT VÀ ỨNG DỤNG VÀO VẤN ĐỀ DUY NHẤT Ngành: Tốn Giải tích Mã số: 46 01 02 LUẬN ÁN TIẾN SĨ TOÁN HỌC Người hướng dẫn khoa học: TS Vũ Hoài An GS.TSKH Hà Huy Khoái THÁI NGUYÊN - NĂM 2018 ✐ ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ổ tr ự tổ ữợ sỹ ữợ ❞➝♥ ❝õ❛ ●❙✳❚❙❑❍ ❍➔ ❍✉② ❑❤♦→✐ ✈➔ ❚❙ ❱ô ❍♦➔✐ ❆♥✳ ❈→❝ ❦➳t q✉↔ ✈✐➳t ❝❤✉♥❣ ✈ỵ✐ t→❝ ❣✐↔ ❦❤→❝ ữủ sỹ t tr ỗ t ✤÷❛ ✈➔♦ ❧✉➟♥ →♥✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❧✉➟♥ →♥ ợ ữ tứ ữủ ổ ố tr t ❦ý ❝æ♥❣ tr➻♥❤ ❦❤♦❛ ❤å❝ ❝õ❛ ❛✐ ❦❤→❝✳ ❚→❝ ❣✐↔ P❤↕♠ ◆❣å❝ ❍♦❛ ✐✐ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ →♥ ✤÷đ❝ t❤ü❝ ❤✐➺♥ ✈➔ ❤♦➔♥ t❤➔♥❤ t↕✐ ❦❤♦❛ ❚♦→♥ t❤✉ë❝ tr÷í♥❣ ữ ữợ sỹ ữợ t t ❚❙❑❍✳ ❍➔ ❍✉② ❑❤♦→✐ ✈➔ ❚❙✳ ❱ô ❍♦➔✐ ❆♥✳ ❈→❝ t❤➛② ✤➣ tr✉②➲♥ ❝❤♦ t→❝ ❣✐↔ ❦✐➳♥ t❤ù❝✱ ❦✐♥❤ ♥❣❤✐➺♠ ❤å❝ t➟♣ ✈➔ sü s❛② ♠➯ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✳ ❱ỵ✐ t➜♠ ❧á♥❣ tr✐ ➙♥ s➙✉ s➢❝✱ t→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ♥❤➜t ✤è✐ ✈ỵ✐ ❤❛✐ t❤➛②✳ ❚→❝ ❣✐↔ ①✐♥ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ✤è❝ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❇❛♥ ✣➔♦ t↕♦ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❇❛♥ ●✐→♠ ❤✐➺✉ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❝→❝ P❤á♥❣ ❇❛♥ ❝❤ù❝ ♥➠♥❣✱ P❤á♥❣ ✣➔♦ t↕♦✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥ ❝ò♥❣ t♦➔♥ t❤➸ ❣✐→♦ ✈✐➯♥ tr♦♥❣ ❦❤♦❛✱ ✤➦❝ ❜✐➺t ❧➔ tê ●✐↔✐ t➼❝❤ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ →♥✳ ❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ tr÷í♥❣ ❈❛♦ ✤➥♥❣ ❍↔✐ ❉÷ì♥❣✱ P❤á♥❣ ❇❛♥ ❝❤ù❝ ♥➠♥❣✱ P❤á♥❣ ✣➔♦ t↕♦✱ ❝→❝ ❣✐↔♥❣ ✈✐➯♥ tr♦♥❣ ❑❤♦❛ ❚ü ◆❤✐➯♥ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ →♥✳ ❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ✱ ❜↕♥ ❜➧ tr♦♥❣ ❝→❝ ❙❡♠✐♥❛r t↕✐ ❇ë ♠æ♥ ❚♦→♥ ●✐↔✐ t➼❝❤ ✈➔ ❚♦→♥ ù♥❣ ❞ư♥❣ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❚❤➠♥❣ ▲♦♥❣ ✈➔ ❚r÷í♥❣ ❈❛♦ ✤➥♥❣ ❍↔✐ ❉÷ì♥❣ ✤➣ ❧✉ỉ♥ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ỏ t ỡ tợ ỳ ữớ t tr t ỗ ũ tr ỳ ữớ ✤➣ ❝❤à✉ ♥❤✐➲✉ ❦❤â ❦❤➠♥✱ ✈➜t ✈↔ ✈➔ ❞➔♥❤ ❤➳t t➻♥❤ ❝↔♠ ②➯✉ t❤÷ì♥❣✱ ✤ë♥❣ ✈✐➯♥✱ ❝❤✐❛ s➫✱ ❦❤➼❝❤ ❧➺ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ →♥✳ ❚→❝ ❣✐↔ P❤↕♠ ◆❣å❝ ❍♦❛ ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ▲í✐ ❝↔♠ ì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ▼ö❝ ❧ö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐✐ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ỵ ❝õ❛ ❘✐tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tt ố ợ ✤❛ t❤ù❝ ❦✐➸✉ ❋❡r♠❛t✲❲❛r✐♥❣ ❝õ❛ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tự ❝õ❛ ❘✐tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ❈❤÷ì♥❣ ✷✳ ✣à♥❤ ỵ tự tt t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✷✳ ỵ tự tt ♥❤➜t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tự ❘✐tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ♥❤✐➲✉ ❜✐➳♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ữỡ ỵ tự tt ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ t➼❝❤ q✲s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✸✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tự tt ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ t➼❝❤ q✲s❛✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tự tt ✤➲ ❞✉② ♥❤➜t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ✈➔ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ❑➳t ❧✉➟♥ ✈➔ ❦✐➳♥ ♥❣❤à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ❉❛♥❤ ♠ư❝ ❝ỉ♥❣ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✶ ▼ð ✤➛✉ ỵ t ỵ ỡ ỵ tt số t r số n t ữợ t➼❝❤ ❝→❝ sè ♥❣✉②➯♥ tè ❝â ❞↕♥❣ mk n = pm pk , ✈ỵ✐ k ≥ 1, ð ✤â ❝→❝ t❤ø❛ sè ♥❣✉②➯♥ tè p1 , , pk ✤ỉ✐ ♠ët ♣❤➙♥ ❜✐➺t ✈➔ ❝→❝ sè ♠ơ t÷ì♥❣ ù♥❣ m1 ≥ 1, , mk ≥ ✤÷đ❝ ①→❝ ✤à♥❤ ♠ët ❝→❝❤ ❞✉② ♥❤➜t t❤❡♦ n ❘✐tt ❧➔ ♥❣÷í✐ ✤➛✉ t tữỡ tỹ ỵ ố ợ t❤ù❝✳ ✣➸ ♠æ t↔ ❦➳t q✉↔ ❝õ❛ ❘✐tt✱ t❛ ❦➼ ❤✐➺✉ M(C) ✭t÷ì♥❣ ù♥❣✱ A(C)✮ ❧➔ t➟♣ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭t÷ì♥❣ ù♥❣✱ ♥❣✉②➯♥✮ tr➯♥ C ✈➔ ❦➼ ❤✐➺✉ L(C) ❧➔ t➟♣ ❝→❝ ✤❛ t❤ù❝ ❜➟❝ 1✳ ✣➦t E, F ❧➔ ❝→❝ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ M(C)✱ ❦❤✐ ✤â ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ F (z) ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ t ữủ tr Eì F t ý ✈✐➳t t❤➔♥❤ ♥❤➙♥ tû F (z) = f ◦ g(z) ✈ỵ✐ f (z) ∈ E ✈➔ g(z) ∈ F ✤➲✉ ❦➨♦ t❤❡♦ ❤♦➦❝ f ❧➔ t✉②➳♥ t➼♥❤ ❤♦➦❝ g ❧➔ t✉②➳♥ t➼♥❤✳ ◆➠♠ ✶✾✷✷✱ ❘✐tt ❬✹✻❪ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ỵ s ỵ ỵ tự t ❘✐tt✮✳ ❈❤♦ F ❧➔ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ C[z] \ L(C) ◆➳✉ ♠ët ✤❛ t❤ù❝ F (z) ❝â ❤❛✐ ❝→❝❤ ♣❤➙♥ t➼❝❤ ❦❤→❝ ♥❤❛✉ t❤➔♥❤ ❝→❝ ✤❛ t❤ù❝ ❦❤æ♥❣ t ữủ tr Fì F F = ◦ ϕ2 ◦ · · · ϕr = ψ1 ◦ ψ2 ◦ · · · ψs , t❤➻ r = s, ✈➔ ❜➟❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ψ ❧➔ ❜➡♥❣ ✈ỵ✐ ❜➟❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ϕ ♥➳✉ ❦❤ỉ♥❣ t➼♥❤ ✤➳♥ t❤ù tü ①✉➜t ❤✐➺♥ ❝õ❛ ❝❤ó♥❣✳ ❈ơ♥❣ tr♦♥❣ ❬✹✻❪✱ tt ự ỵ s ỵ ỵ tự tt sỷ r a, b, c, d ∈ C[x]\ C t❤ä❛ ♠➣♥ a◦b = c◦d ✈➔ gcd(deg(a); deg(c)) = gcd(deg(b); deg(d)) = ❑❤✐ õ tỗ t t t lj C[x] s❛♦ ❝❤♦ (l1 ◦ a ◦ l2 , l2−1 ◦ b ◦ l3 , l1 ◦ c ◦ l2 , l4−1 ◦ d ◦ l3 ) ❝â ♠ët tr♦♥❣ ❝→❝ ❞↕♥❣ (Fn , Fm , Fm , Fn ) ❤♦➦❝ ✷ (xn , xs h(xn ), xs h(x)n , xn ), ð ✤â m, n > ❧➔ ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉✱ s > ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉ ✈ỵ✐ n, ✈➔ h ∈ C[x]\xC[x], lj−1 ❧➔ ❤➔♠ ♥❣÷đ❝ ❝õ❛ lj ✱ Fn , Fm ❧➔ ❝→❝ ✤❛ t❤ù❝ ❈❤❡❜②❝❤❡✈✳ Ð ✤➙②✱ ♣❤➨♣ ♣❤➙♥ t➼❝❤ F (z) = f ◦ g(z) ❝❤➼♥❤ ❧➔ ♣❤➨♣ ❤ñ♣ t❤➔♥❤ F (z) = f (g(z)) õ t t r ỵ tự ❝õ❛ ❘✐tt ♠ỉ t↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ a(b) = c(d)✱ ð ✤â a, b, c, d ❧➔ ❝→❝ ✤❛ t❤ù❝ ✈➔ ❜➟❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ❧➔ ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉✳ ❘ã r➔♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤❛ t❤ù❝ ✤÷đ❝ ❘✐tt ♥❣❤✐➯♥ ❝ù✉ ❧➔ tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ P (f ) = Q(g), ð ✤â P, Q ❧➔ ❝→❝ ✤❛ t❤ù❝ ✈➔ f, g ❧➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ P (f ) = Q(g) ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ ♥❤✐➲✉ t→❝ ❣✐↔ ♥❤÷ ❚↕ ❚❤à ❍♦➔✐ ❆♥✲◆❣✉②➵♥ ❚❤à ◆❣å❝ ❉✐➺♣ ❬✸❪✱ ❍✳❋✉❥✐♠♦t♦ ❬✶✾❪✱ ❍➔ P ỵ r➡♥❣✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧✐➯♥ q✉❛♥ ♠➟t t❤✐➳t ✤➳♥ ✈➜♥ ✤➲ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤✲♠ët ự ỵ tt ố tr ✤➲ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❧➛♥ ✤➛✉ t✐➯♥ ❜ð✐ ❘✳◆❡✈❛♥❧✐♥♥❛✳ ◆➠♠ ✶✾✷✻✱ ❘✳◆❡✈❛♥❧✐♥♥❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ữủ r ợ f g tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✱ ♥➳✉ ❝❤ó♥❣ ❝â ❝❤✉♥❣ ♥❤❛✉ ↔♥❤ ♥❣÷đ❝ ✭❦❤ỉ♥❣ t➼♥❤ ❜ë✐✮ ❝õ❛ ✺ ✤✐➸♠ ♣❤➙♥ ❜✐➺t t f = g ỵ ❝❤ó♥❣ ❝â ❝❤✉♥❣ ♥❤❛✉ ↔♥❤ ♥❣÷đ❝ ✭❝â t➼♥❤ ❜ë✐✮ ❝õ❛ ✹ af + b ✭a, b, c, d ❧➔ ❝→❝ sè ♣❤ù❝ ♥➔♦ ✤â s❛♦ ❝❤♦ cf + d ad bc = ỵ ỗ tứ ỵ ỵ t ữủ ự tử ợ ữợ ự õ rt ♥❤✐➲✉ ❦➳t q✉↔ s➙✉ s➢❝ ❝õ❛ ●✳❉❡t❤❧♦❢❢✱ ✣é ✣ù❝ ❚❤→✐✱ ▼✳ ❙❤✐r♦s❛❦✐✱ ❍✳❳✳❨✐✱ P✳❈✳❍✉✲❈✳❈✳❨❛♥❣✱ ❍➔ ❍✉② ❑❤♦→✐✱ ❍➔ ❍✉② ❑❤♦→✐✲❱ô ❍♦➔✐ ❆♥✱ ❍➔ ❍✉② ❑❤♦→✐✲❱ô ❍♦➔✐ ❆♥✲▲➯ ◗✉❛♥❣ ◆✐♥❤✱ ❚↕ ❚❤à ❍♦➔✐ ❆♥✱ ❚↕ ❚❤à ❍♦➔✐ ❆♥✲❍➔ ❚r➛♥ P❤÷ì♥❣✱ ▲✳▲❛❤✐r✐✱ ❚r➛♥ ❱➠♥ ❚➜♥✱ ❙➽ ✣ù❝ ◗✉❛♥❣✱ ❆✳❊s❝❛ss✉t✱ ❍✳❋✉❥✐♠♦t♦✱✳✳✳ ✤✐➸♠ ♣❤➙♥ ❜✐➺t t❤➻ g = ❚✐➳♣ t❤❡♦✱ sü ♥❣❤✐➯♥ ❝ù✉ ✤÷đ❝ rở s ởt ỵ tt ❞✉② ♥❤➜t ✤â ❧➔ ①❡♠ ①➨t t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✳ ❱➔ ♥❣÷í✐ ✤➛✉ t ữợ ữợ ự ◆➠♠ ✶✾✻✼✱ ❍❛②♠❛♥ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❦➳t q✉↔ ♥ê✐ t✐➳♥❣ r➡♥❣ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ f tr➯♥ tr÷í♥❣ sè ♣❤ù❝ C ❦❤æ♥❣ ♥❤➟♥ ❣✐→ trà ✵ ✈➔ ✤↕♦ ❤➔♠ ❜➟❝ k ❝õ❛ f ✱ ✈ỵ✐ k ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ ❦❤ỉ♥❣ ♥❤➟♥ ❣✐→ trà ✶ t❤➻ f ❧➔ ❤➔♠ ❤➡♥❣✳ ❍❛②♠❛♥ ❝ơ♥❣ ✤÷❛ r❛ ❣✐↔ t❤✉②➳t s❛✉✳ ●✐↔n t❤✉②➳t ❍❛②♠❛♥✳ ❬✷✶❪ ◆➳✉ ♠ët ❤➔♠ ♥❣✉②➯♥ f t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (z)f (z) = ✈ỵ✐ n ❧➔ sè ữỡ ợ z C t f ❧➔ ✸ ❤➔♠ ❤➡♥❣✳ ●✐↔ t❤✉②➳t ♥➔② ✤➣ ✤÷đ❝ ❝❤➼♥❤ tr ợ n > ữủ ❦✐➸♠ tr❛ ✈ỵ✐ n ≥ 1✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➣ ❤➻♥❤ t❤➔♥❤ ♠ët ữợ ự ữủ sỹ ỹ ổ tr q trồ tú ữợ ự ♥➔② t❤✉ë❝ ✈➲ ❨❛♥❣✲❍✉❛ ❬✺✶❪✱ ❤❛✐ æ♥❣ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤ì♥ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ♥â ❝â ❞↕♥❣ f n f ổ ự ữủ r ợ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣✱ n ❧➔ sè ♥❣✉②➯♥✱ n ≥ 11 ♥➳✉ f n f ✈➔ g n g ❝ò♥❣ ♥❤➟♥ ❣✐→ trà ♣❤ù❝ a t➼♥❤ ❝↔ ❜ë✐ t❤➻ ❤♦➦❝ f, g s❛✐ ❦❤→❝ ♥❤❛✉ ♠ët ❝➠♥ ❜➟❝ n + ❝õ❛ ✤ì♥ ✈à✱ ❤♦➦❝ f, g ✤÷đ❝ t➼♥❤ t❤❡♦ ❝→❝ ❝ỉ♥❣ t❤ù❝ ❝õ❛ ❤➔♠ ♠ơ ✈ỵ✐ ❝→❝ ❤➺ sè t❤ä❛ ♠➣♥ ♠ët ✤✐➲✉ ❦✐➺♥ ♥➔♦ ✤â✳ ❚ø ✤â✱ ❝→❝ ❦➳t q✉↔ t✐➳♣ t❤❡♦ ✤➣ ♥❤➟♥ ✤÷đ❝ ❞ü❛ tr➯♥ ①❡♠ ①➨t ❝→❝ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❞↕♥❣ (f n )(k) , [f n (f − 1)](k) ✭❇❤♦♦s♥✉r♠❛t❤ ✲ ❉②❛✈❛♥❛❧ ❬✶✵❪✱ ❋❛♥❣ ❬✶✽❪✮ ✈➔ ❝â ❞↕♥❣ [f n (af m + b)](k) , [f n (f − 1)m ](k) ✭①❡♠ ❩❤❛♥❣ ✈➔ ▲✐♥✱ ❬✺✹❪✮✱ ✈➔ ❝â ❞↕♥❣ (f )( ) P (f ),✭ ①❡♠ ❑✳ ❇♦✉ss❛❢✲ ❆✳ ❊s❝❛ss✉t✲ ❏✳ ❖❥❡❞❛❬✶✶❪✮✳ ◆➠♠ ✶✾✾✼✱ t❤❛② ✈➻ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✤↕♦ ❤➔♠ ❜➟❝ n✱ ■✳ ▲❛❤✐r✐ ❬✸✻❪ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t ❤ì♥ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❦❤æ♥❣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♥❤➟♥ ❣✐→ trà ✶ t➼♥❤ ❝↔ ❜ë✐✳ ❚❤❡♦ ữợ ự ▼✳ ▲✳ ❋❛♥❣ ❬✶✼❪ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣✱ ♥➳✉ n ≥ 13, ✈➔ ✤è✐ ✈ỵ✐ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ f ✈➔ g, ♠➔ f (n) (f − 1)2 f ✈➔ g (n) (g − 1)2 g ♥❤➟♥ ❣✐→ trà ✶ t➼♥❤ ❝↔ ❜ë✐✱ t❤➻ f = g ❱➔♦ ❝✉è✐ ♥❤ú♥❣ ♥➠♠ ❝õ❛ t❤➟♣ ❦✛ ♥➔②✱ ✈➜♥ ✤➲ ♥❤➟♥ tr ụ ữủ t ố ợ tự s❛✐ ♣❤➙♥ ❝õ❛ ❝→❝ ❤➔♠ ♥❣✉②➯♥ ✈➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ▲❛✐♥❡ ✈➔ ❨❛♥❣ ❬✸✼❪ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ ♣❤➙♥ ❜è ❣✐→ trà ❝õ❛ t➼❝❤ s❛✐ ♣❤➙♥ ✤è✐ ✈ỵ✐ ❝→❝ ❤➔♠ ♥❣✉②➯♥✳ ❳✳ ❈✳✲◗✐✱ ▲✳✲❩✳ ❨❛♥❣ ✈➔ ❑✳ ▲✐✉ ❬✹✺❪ ①❡♠ ①➨t ❝→❝ t➼❝❤ s❛✐ ♣❤➙♥ ✈➔ ✈✐ ♣❤➙♥ ❝â ❞↕♥❣ f (z)(n) f (z + c), ✈➔ ✤➣ ❝❤➾ r❛ ✤✐➲✉ ❦✐➺♥ ✤➸ f = tg ✱ ✈ỵ✐ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♥❣✉②➯♥ s✐➯✉ ✈✐➺t ❝â ỳ t t tứ ỵ tự tt P õ ỵ tữ t ữủ t t ố ợ t❤ù❝✳ ➷♥❣ ✤➣ t➻♠ ✤÷đ❝ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❤❛✐ ✤❛ t❤ù❝ f1 , f2 ✈➔ ❤❛✐ t➟♣ ❝♦♠♣❛❝t K1 , K2 t❤ä❛ ♠➣♥ f1−1 (K1 ) = f2−1 (K2 ) ❑➳t q✉↔ ❝õ❛ ❋✳P❛❝❦♦✈✐❝❤ ✤÷đ❝ ✣✐♥❤ ❚✐➳♥ ❈÷í♥❣ ♠ð rë♥❣ tr ứ ỵ tt tự ❦➳t q✉↔ ❝õ❛ ❋✳P❛❦♦✈✐❝❤ ♥â✐ tr➯♥ ❝❤ó♥❣ tỉ✐ ❝â ♥❤➟♥ t t ỵ tt tự õ t ✤÷đ❝ ①❡♠ ❧➔ ❦➳t q✉↔ ✤➛✉ t✐➯♥ ✈➲ ✈➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠ tø ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ P (f ) = Q(g)✱ tø ✤â s✐♥❤ r❛ ❝→❝ ❦➳t q✉↔ ❝❤♦ ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ✤❛ t❤ù❝ t❤æ♥❣ q✉❛ ✤✐➲✉ ❦✐➺♥ ↔♥❤ ♥❣÷đ❝ ❝õ❛ t➟♣ ❤đ♣ ✤✐➸♠✳ ❚ø ♥❤➟♥ ①➨t ♥➔② ✈➔ ❝→❝ ❦➳t q✉↔ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✭①❡♠ ❬✸❪✱ ❬✸✺❪✱ ✹ ❬✹✹❪✮ ♥➯✉ tr➯♥✱ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ ✤÷đ❝ ✤➦t r❛ tü ♥❤✐➯♥ ♥❤÷ s❛✉✳ ❳❡♠ ①➨t sü t÷ì♥❣ tỹ ỵ tt ố ợ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥✳ ❳❡♠ ①➨t ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠✱ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ tự q s ữợ õ ỵ tt ứ õ ú tổ t ởt số ỵ tt ự ✈➜♥ ✤➲ ❞✉② ♥❤➜t✧ ✤➸ ❣✐↔✐ q✉②➳t ❝→❝ ✈➜♥ ✤➲ ự tr ỗ tớ õ ú t t q ự ỵ t❤✉②➳t ◆❡✈❛♥❧✐♥♥❛✳ ❱➜♥ ✤➲ ✶✳ ❱➜♥ ✤➲ ✷✳ ✷✳ ▼ö❝ t✐➯✉ ❝õ❛ ❧✉➟♥ →♥ ✷✳✶✳ ❚❤✐➳t ❧➟♣ ♠ët sè ✤à♥❤ ỵ tữỡ tỹ ỵ tt ố ợ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝ ✈➔ p✲❛❞✐❝✳ ✷✳✷✳ ❚✐➳♣ ❝➟♥ ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠✱ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q s tr trữớ ủ ự p ữợ õ ỵ tt ố tữủ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝ p ữợ õ ỵ tt ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q s tr trữớ ủ ự p ữợ õ ỵ tt Pữỡ ổ ự ỷ ỵ tữỡ tỹ ú ũ ợ r ỵ tt ố tr ✤➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tữỡ tỹ ữ ữỡ tr tr ỵ tt tự ỷ ỵ ❝❤✉②➸♥ ❜➔✐ t♦→♥ ①→❝ ✤à♥❤ ❤➔♠✱ ❜➔✐ t♦→♥ ❞✉② ♥❤➜t ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳ ◆❤í ✤â ✈➔ ❝→❝ ❦➳t q✉↔ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ♥â✐ tr➯♥ ✤➸ ✤÷❛ r❛ ❝→❝ ❦➳t q✉↔ ✈➲ ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠ ✈➔ ❱➜♥ ✤➲ ❞✉② ♥❤➜t✳ ✺✳ Þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ❝õ❛ ❧✉➟♥ →♥ ▲✉➟♥ →♥ ✤➣ ✤÷❛ r❛ ♠ët ❝→❝❤ t✐➳♣ ❝➟♥ ♠ỵ✐ ✤è✐ ✈ỵ✐ ❱➜♥ ✤➲ ①→❝ ✤à♥❤✱ ❱➜♥ ✺ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❤➔♠✱ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ✈➔ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✳ ✣â ❧➔✱ ①❡♠ ①➨t ❝→❝ ✈➜♥ ✤➲ ữợ õ ỵ tt õ tt ữủ t q ợ õ rở t ự ỵ tt ✻✳ ❈➜✉ tró❝ ✈➔ ❦➳t q✉↔ ❝õ❛ ❧✉➟♥ →♥ ▲✉➟♥ ỗ õ ữỡ ũ ợ ♣❤➛♥ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❈❤÷ì♥❣ ✶ ợ tỹ ỵ tt ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤✧✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❈❤÷ì♥❣ ✶ ✤÷đ❝ ✈✐➳t ❞ü❛ tr➯♥ ❝→❝ ❜➔✐ ❜→♦ ❬✺❪✱ ❬✼❪✱ ❬✷✾❪✳ ❱✐➺❝ ự t ỗ ữợ s ữợ ✶✳ ❚❤✐➳t ❧➟♣ ❝→❝ ❦➳t q✉↔ t÷ì♥❣ tü ❤❛✐ ✤à♥❤ ỵ tt ố ợ ữợ ❜➔✐ t♦→♥ ❞✉② ♥❤➜t ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✈➔ ❞ị♥❣ t q ữợ ữ t t tr ỵ tự t tt ự tọ r➡♥❣✿ ❜➜t ❦ý ❤❛✐ sü ♣❤➙♥ t➼❝❤ ❝õ❛ ♠ët ✤❛ tự trữợ t tự ổ t ✤÷đ❝ s➩ ❝❤ù❛ ❝ị♥❣ ♠ët sè ✤❛ t❤ù❝ ♥❤÷ ♥❤❛✉ ✈➔ ❜➟❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ tr♦♥❣ ♠é✐ ❝→❝❤ ♣❤➙♥ t➼❝❤ ❧➔ ♥❤÷ ♥❤❛✉ ♥➳✉ ❦❤ỉ♥❣ t➼♥❤ ✤➳♥ t❤ù tü ❝õ❛ ❝❤ó♥❣ tr♦♥❣ ❝→❝❤ ♣❤➙♥ t➼❝❤✳ ❚ø ✤â✱ ♠ư❝ t✐➯✉ t❤ù ♥❤➜t ❝õ❛ ❈❤÷ì♥❣ ✶ ❧➔✿ ❚❤✐➳t ❧➟♣ ❦➳t q✉↔ tữỡ tỹ ỵ tự t tt ♣❤➙♥ ❤➻♥❤✳ ❚✉② ♥❤✐➯♥✱ t❛ t❤➜② r➡♥❣✱ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ỵ tt tr ữớ ữ ổ tữỡ tỹ ữủ ỵ ð ❝❤é✱ ❘✐tt ✤➣ ❞ò♥❣ ✤➳♥ ✤✐➲✉ ❦✐➺♥ ✧❤ú✉ ❤↕♥✧ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ ✤❛ t❤ù❝ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ æ♥❣✳ õ trữợ t ú tổ tt ỵ ỵ ởt ỵ tt tự ố ợ ữỡ tr ❤➔♠ P (f1 , f2 ) = Q(g1 , g2 ), ð ✤â P, Q ❧➔ ❝→❝ ✤❛ t❤ù❝ ❤❛✐ ❜✐➳♥ ❦✐➸✉ ❨✐ ✈➔ f1 , f2 , g1 , g2 ú ỵ r t q ♥➔② ✤➣ ✤÷đ❝ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ❬✷❪ ✈➔ ❬✸✷❪✱ t✉② ♥❤✐➯♥ ð ✤➙② ❝❤ó♥❣ tỉ✐ ♥❤➻♥ ❦➳t q ữợ õ ỵ tt tự ❤❛✐ ✈➔ ✤÷❛ r❛ ♠ët ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ❦❤→❝✳ ◆❤í ỵ q ú tổ ự ữủ ỵ ởt t q tữỡ tỹ ỵ tt tự t ố ợ r ữỡ ỏ tr ự ỵ õ ỵ ỵ ỵ ❝❤♦ t❛ ❝→❝ ❦➳t q✉↔ ♠ỵ✐ ✈➲ Bi − U RSM ỵ r ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❞↕♥❣ (P (f ))(k) ✱ ð ✤â P ❧➔ ✤❛ t❤ù❝ ✈➔ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ❧➔ ♠ët ❜➔✐ t♦→♥ ❦❤â✳ ❑❤â ❦❤➠♥ ð ✤➙② ❧➔ tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t ❤✐➺♥ ❝❤÷❛ ❝â ♠ët ♠è✐ ❧✐➯♥ ❤➺ tèt ❣✐ú❛ ❤➔♠ ✽✺ ❚÷ì♥❣ tü✱ t❛ ❝â (n − m)T (r, g) ≤ 16T (r, g) + 12T (r, f ) − log r + O(1) ❇ð✐ ✈➟②✱ ❝â (n − m) T (r, f ) + T (r, g) ≤ 28 T (r, f ) + T (r, g) − log r + O(1), (n − m − 28) T (r, f ) + T (r, g) + log r ≤ O(1) ❉♦ n ≥ m + 28 t❛ ❣➦♣ ♠➙✉ t❤✉➝♥✳ A.B = tù❝ ❧➔ f n f m (qz + c).g n g m (qz + c) = 1✳ ❚❤❡♦ ❇ê ❚r÷í♥❣ ❤đ♣ ✷✳ l ✈ỵ✐ ln+m = g A = B tù❝ ❧➔ f n f m (qz + c) = g n g m (qz + c) ❚❤❡♦ ❇ê ✤➲ ✸✳✷✳✶✐✐✮ s✉② r❛ f = hg ✈ỵ✐ hn+m = ✤➲ s r f = rữớ ủ ỵ ✸✳✷✳✼✳ ❈❤♦ f, g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ K, n, m ❧➔ l ✈ỵ✐ ln+m = 1, ❤♦➦❝ g f = hg ✈ỵ✐ hn+m = ♥➳✉ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ①↔② r❛✿ m = 1, n ≥ 13 ✈➔ f n f (qz + c) ✈➔ g n g(qz + c) ♥❤➟♥ ❝â t➼♥❤ ❜ë✐❀ m = 1, n ≥ 25 ✈➔ f n f (qz + c) ✈➔ g n g(qz + c) ♥❤➟♥ ❦❤æ♥❣ t➼♥❤ ❜ë✐❀ m ≥ 2, n ≥ m + 16 ✈➔ f n f m (qz + c) ✈➔ g n g m (qz + c) ♥❤➟♥ ❝â t➼♥❤ ❜ë✐❀ m ≥ 2, n ≥ m + 28 ✈➔ f n f m (qz + c) ✈➔ g n g m (qz + c) ♥❤➟♥ ❦❤ỉ♥❣ t➼♥❤ ❜ë✐✳ ❤❛✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ q, c ∈ K, |q| = 1✳ ❑❤✐ ✤â f = ỵ tự ❝õ❛ ❘✐tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ✈➔ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ❇ê ✤➲ ✸✳✸✳✶✳ ❈❤♦ q, c ∈ K ✈ỵ✐ |q| = 1, n, m, d, k ❧➔ ❝→❝ sè ữỡ ợ n > 2k + 1, m > d ❑❤✐ ✤â ✶✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ f nm f nd (qz + c) (k) g nm g nd (qz + c) (k) =1 ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ (f, g) ✷✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ f nm f nd (qz + c) (k) = g nm g nd (qz + c) (k) ✽✻ ❝â ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ (f, g) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f = hg ✈ỵ✐ h ∈ K ✈➔ hn(m+d) = ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t A = (f nm (z)f nd (qz + c))(k) , B = (g nm (z)g nd (qz + c))(k) ✱ A B C = f m (z)f d (qz + c), D = g m (z)(g d (qz + c), P = n−k , Q = n−k ❑❤✐ C D n (k) n−k n (k) n−k ✤â A = (C ) = C P, B = (D ) = D Q ✶✳ (f nm (z)f nd (qz + c))(k) (g nm (z)g nd (qz + c))(k) = (C n )(k) (Dn )(k) = ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ C = 0✱ C = ∞✱ D = 0✱ D = ∞✳ ●✐↔ sû r➡♥❣ C ❝â ❦❤æ♥❣ ✤✐➸♠✳ ●å✐ a ❧➔ ❦❤ỉ♥❣ ✤✐➸♠ ✈ỵ✐ ω(C, 0, a) = α✱ α ≥ 1✳ ❑❤✐ ✤â a ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ D ✈ỵ✐ ω(D, ∞, a) = β ✱ β ≥ s❛♦ ❝❤♦ nα − k = nβ + k ✈➔ k(m + d) + 16 > 2k + t❛ ❣➦♣ m−d ♠➙✉ t❤✉➝♥✳ ❇➡♥❣ ❝→❝❤ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü✱ t❛ ❝â D = 0✱ C = ∞✱ D = ∞✳ ❉♦ C, D ❦❤→❝ ❤➡♥❣✱ t❛ ❝ô♥❣ ❣➦♣ ♠➙✉ t❤✉➝♥✳ ✷✳ (f nm (z)f nd (qz + c))(k) = (g nm (z)g nd (qz + c))(k) , (C n )(k) = (Dn )(k) ❇ð✐ ✈➻ f, g ❦❤→❝ ❤➡♥❣✱ ✈➔ ❞♦ ❇ê ✤➲ ✸✳✶✳✺ t❛ t❤➜② C, D ❦❤→❝ ❤➡♥❣✳ ❉♦ ✤â C n = Dn + s, Dn = C n − s✱ ✈ỵ✐ s ❧➔ ♠ët ✤❛ t❤ù❝ ❜➟❝ < k ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ s ≡ 0✳ ●✐↔ sû s ≡ ❚❤➳ t❤➻ n(α − β) = 2k ✳ ❚ø ✤➙② ✈➔ ❞♦ n ≥ 2k + nT (r, D) = T (r, Dn ) + O(1) ≤ T (r, C n ) + T (r, s) + O(1) ≤ nT (r, C) + (k − 1) log r + O(1) ❚ø ✤➙② ✈➔ ❞♦ n ≥ 2k + k(m + d) + 16 > 2k + t❛ ♥❤➟♥ ✤÷đ❝ m−d k−1 1 < , T (r, D) ≤ T (r, C) + log r + O(1) n 2 Dn Cn ,G = ❉♦ C, D ❦❤→❝ ❤➡♥❣✱ t❛ ♥❤➟♥ ✤÷đ❝ ✣➦t F = s s F s = C n , nT (r, C) = T (r, C n ) ≤ T (r, F ) + T (r, s) + O(1) ≤ T (r, F ) + (k − 1) log r + O(1), nT (r, C) − (k − 1) log r ≤ T (r, F ) + O(1), 1 N1 (r, ) ≤ N1 (r, ) ≤ T (r, C) + O(1), F C 1 N1 (r, ) ≤ T (r, D) + O(1) ≤ T (r, C) + log r + O(1), F ✭✸✳✶✼✮ ✽✼ N1 (r, F ) ≤ N1 (r, C n ) + N1 (r, ) ≤ N1 (r, C) + (k − 1) log r + O(1) s ≤ T (r, C) + (k − 1) log r + O(1) ❚ø ✤➙② ✈➔ ❞♦ ❇ê ✤➲ ✷✳✶✳✾✱ ❞♦ ❝â F − = G ♥➯♥ t❛ s✉② r❛ nT (r, C) − (k − 1) log r + O(1) ≤ T (r, F ) 1 ) + N1 (r, F ) + N1 (r, ) − log r + O(1) F F −1 ≤ T (r, C) + T (r, C) + (k − 1) log r + N1 (r, ) − log r + O(1) G ≤ 2T (r, C) + (k − 1) log r + N1 (r, ) − log r + O(1) D ≤ 2T (r, C) + T (r, C) + log r + (k − 1) log r − log r + O(1) ❱➟②✱ t❛ ❝â ≤ N1 (r, (n − 3)T (r, C) − 2(k − 1) log r + log r ≤ O(1) ▼➦t ❦❤→❝✱ ❞♦ C ❦❤→❝ ❤➡♥❣ sè✱ t❛ ♥❤➟♥ ✤÷đ❝ T (r, C) ≥ log r + O(1) ❱➟② k(m + d) + 16 > (n−2k−1) log r+ log r ≤ O(1)✳ ❚ø ✤➙② ✈➔ ❞♦ n ≥ 2k+ m−d 2k + t❛ ❣➦♣ ♠ët ♠➙✉ t❤✉➝♥✳ ❱➟② s ≡ 0✳ ❉♦ ✈➟②✱ C n = Dn ✈➔ C = eD, f m (z)f d (qz + c) = eg m (z)g d (qz + c) ✈ỵ✐ f f (qz + c) en = 1✳ ✣➦t h = ✳ ●✐↔ sû h ❦❤→❝ ❤➡♥❣✳ ❑❤✐ ✤â h(qz + c) = g g(qz + c) ❦❤→❝ ❤➡♥❣ ✈➔ T (r, h(qz + c)) = T (r, h) + O(1), hm = mT (r, h) = T (r, hm ) + O(1) = T r, e , hd (qz + c) e + O(1) hd (qz + c) = dT (r, h(qz + c)) + O(1) = dT (r, h) + O(1) ❙✉② r❛ (m − d)T (r, h) = O(1) ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t m > d, h ❦❤→❝ ❤➡♥❣✳ ❱➟② h ❧➔ ❤➡♥❣✳ ❉♦ f m (z)f d (qz + c) = eg m (z)g d (qz + c), en = t❛ ❦➳t ❧✉➟♥ r➡♥❣ f = hg ✈ỵ✐ hm+d = e, hn(m+d) = ❱➟② ❇ê ✤➲ ✸✳✸✳✶ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ỵ f g ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ K, q, c ∈ K, |q| = 1, ❝❤♦ n, m, d, k ✱ ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ k(m + d) + 16 ❦✐➺♥ m > d ≥ 1, n ≥ 2k + ◆➳✉ (f nm (z)f nd (qz + c))(k) ✈➔ m−d nm nd (k) (g (z)g (qz + c)) ♥❤➟♥ ✶ ❈▼✱ t❤➻ f = hg ✈ỵ✐ hn(m+d) = 1✱ h ∈ K ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t A = (f nm (z)f nd (qz + c))(k) , B = (g nm (z)g nd (qz + c))(k) ✱ B A C = f m (z)f d (qz + c), D = g m (z)(g d (qz + c), P = n−k , Q = n−k ❑❤✐ C D ✤â A = (C n )(k) = C n−k P, B = (Dn )(k) = Dnk Q ú ỵ r N1 (r, A) + N1,(2 (r, A) = N2 (r, A), 1 ) + N1,(2 (r, ) = N2 (r, ), A A A N1 (r, B) + N1,(2 (r, B) = N2 (r, B), 1 N1 (r, ) + N1,(2 (r, ) = N2 (r, ) B B B ❑❤✐ ✤â✱ →♣ ❞ö♥❣ ❇ê ✤➲ ✸✳✶✳✹ ✤è✐ ✈ỵ✐ (C n )(k) ✱ (Dn )(k) t❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉✳ N1 (r, ❚r÷í♥❣ ❤đ♣ ✶✳ T (r, A) ≤ N2 (r, A) + N2 (r, 1 ) + N2 (r, B) + N2 (r, ) − log r + O(1), A B T (r, B) ≤ N2 (r, A) + N2 (r, 1 ) + N2 (r, B) + N2 (r, ) − log r + O(1) A B ✭✸✳✶✽✮ ❚❛ t❤➜② r➡♥❣ ♥➳✉ a ❧➔ ♠ët ❝ü❝ ✤✐➸♠ ❝õ❛ A✱ t❤➻ C(a) = ợ à01 (a) A n + k ≥ ✈➔ ❞♦ ❇ê ✤➲ ✸✳✶✳✹ t❛ ❝â N1 (r, C) = N1 (r, f f (qz + c)) ≤ N1 (r, f ) + N1 (r, f (qz + c)) + O(1) ≤ T (r, f ) + T (r, f (qz + c)) + O(1) = 2T (r, f ) + O(1)✳ ❚÷ì♥❣ tü✱ N1 (r, C1 ) ≤ 2T (r, f ) + O(1) ❉♦ ✤â✱ t❤❡♦ ❇ê ✤➲ ✸✳✶✳✺ t❛ ❝â m d N2 (r, A) = 2N1 (r, C) ≤ 4T (r, f ) + O(1), 1 1 ) ≤ N2 (r, n−k ) + N (r, ) = 2N1 (r, ) + N (r, ) A C P C P ≤ 4T (r, f ) + N (r, ) + O(1) P ≤ 4T (r, f ) + k(m + d)T (r, f ) + kN1 (r, C) + O(1) N2 (r, ✽✾ ❚÷ì♥❣ tü✱ t❛ ❝ơ♥❣ ❝â N2 (r, B) ≤ 4T (r, g) + O(1), 1 ) ≤ 4T (r, g) + N (r, ) + O(1) B Q ≤ 4T (r, g) + k(m + d)T (r, g) + kN1 (r, D) + O(1) N2 (r, ❚ø ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✈➔ ✭✸✳✶✽✮ t❛ ❝â T (r, A) ≤ 8(T (r, f ) + T (r, g)) + N (r, 1 ) + N (r, ) − log r + O(1) P Q ≤ 8(T (r, f )+T (r, g))+k(m+d)T (r, f )+kN1 (r, C)+N (r, )−log r+O(1), Q 1 ) + N (r, ) − log r + O(1) P Q ≤ 8(T (r, f )+T (r, g))+k(m+d)T (r, g)+kN1 (r, D)+N (r, )−log r+O(1), P 1 T (r, A) + T (r, B) ≤ 16(T (r, f ) + T (r, g)) + N (r, ) + N (r, ) P Q +k(m + d)(T (r, f ) + T (r, g)) + k(N1 (r, C) + N1 (r, D)) − log r + O(1) T (r, B) ≤ 8(T (r, f ) + T (r, g)) + N (r, ❉♦ ❇ê ✤➲ ✸✳✶✳✺ t❛ ❝â (n − 2k)(m − d)T (r, f ) + kN (r, C) + N (r, ) ≤ T (r, A) + O(1), P (n − 2k)(m − d)T (r, g) + kN (r, D) + N (r, ) ≤ T (r, B) + O(1) Q ❑➳t ❤ñ♣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤➙②✱ t❛ ♥❤➟♥ ✤÷đ❝ (n − 2k)(m − d)(T (r, f ) + T (r, g)) + k(N (r, C) + N (r, D))+ N (r, 1 ) + N (r, ) P Q ≤ (k(m + d) + 16)(T (r, f ) + T (r, g)) + k(N1 (r, C) + N (r, D)) + N (r, ) P ) − log r + O(1), Q [(n − 2k)(m − d) − (k(m + d) + 16)](T (r, f ) + T (r, g)) + log r ≤ O(1) +N (r, ✾✵ ❱➻ n ≥ 2k + ❚r÷í♥❣ ❤đ♣ ✷✳ k(m + d) + 16 ✱ t❛ ❣➦♣ ♠➙✉ t❤✉➝♥✳ m−d (f nm (z)f nd (qz + c))(k) (g nm (z)g nd (qz + c))(k) = (C n )(k) (Dn )(k) = ✣÷đ❝ s✉② r❛ tø ❇ê ✤➲ ✸✳✸✳✶✳✶✳ ❚r÷í♥❣ ❤đ♣ ✸✳ (f nm (z)f nd (qz + c))(k) = (g nm (z)g nd (qz + c))(k) , (C n )(k) = (Dn )(k) ❑➳t ❧✉➟♥ ❝õ❛ tr÷í♥❣ ❤đ♣ ♥➔② ✤÷đ❝ s✉② r❛ tø ❇ê ✤➲ ✸✳✸✳✶✳✷✳ ❱➟② ỵ ữủ ự ỵ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ K, q, c ∈ K, |q| = 1, ✈➔ ❝❤♦ n, m, d, k ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ 2k(2m + 2d + 3) + 28 m > d ≥ 1, n ≥ 2k + ◆➳✉ (f nm (z)f nd (qz + c))(k) m−d nm nd (k) ✈➔ (g (z)g (qz + c)) ♥❤➟♥ ✶ ■▼✱ t❤➻ f = hg ✈ỵ✐ hn(m+d) = 1✱ h ∈ K ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ sû ❞ö♥❣ ❝→❝ tr ự ỵ õ →♣ ❞ư♥❣ ❇ê ✤➲ ✸✳✶✳✻ ✤è✐ ✈ỵ✐ (C n )(k) ✱ (Dn )(k) t❛ ①❡♠ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉✳ ❚r÷í♥❣ ❤đ♣ ✶✳ 1 ) + N2 (r, B) + N2 (r, )+ A B 1 2(N1 (r, A) + N1 (r, )) + N1 (r, B) + N1 (r, ) − log r + O(1), A B 1 T (r, B) ≤ N2 (r, A) + N2 (r, ) + N2 (r, B) + N2 (r, )+ A B 1 2(N1 (r, B) + N1 (r, )) + N1 (r, A) + N1 (r, ) − log r + O(1) B A ❑➳t ❤ñ♣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ❝â T (r, A) ≤ N2 (r, A) + N2 (r, 1 ) + N2 (r, B) + N2 (r, )) A B 1 +3(N1 (r, A) + N1 (r, ) + N1 (r, B) + N1 (r, )) − log r + O(1) A B ❚ø ✤➙②✱ t÷ì♥❣ tü ữ rữớ ủ ỵ ❇ê ✤➲ ✸✳✶✳✺ t❛ ❝â T (r, A) + T (r, B) ≤ 2(N2 (r, A) + N2 (r, (n − 2k)(m − d)(T (r, f ) + T (r, g)) + k(N (r, C) + N (r, D)) ✾✶ 1 ) + N (r, ) ≤ T (r, A) + T (r, B) + O(1); ✭✸✳✶✾✮ P Q 1 T (r, A) + T (r, B) ≤ 16(T (r, f ) + T (r, g)) + N (r, ) + N (r, )+ P Q k(m + d)(T (r, f ) + T (r, g)) + k(N1 (r, C) + N1 (r, D)) + 3(N1 (r, A)+ +N (r, 1 ) + N1 (r, B) + N1 (r, )) − log r + O(1); A B N1 (r, A) ≤ 2T (r, f ) + O(1), N1 (r, ) ≤ 2T (r, f )+ A k(m + d + 2)T (r, f ) + O(1); N1 (r, B) ≤ 2T (r, g) + O(1), N1 (r, ) ≤ 2T (r, g) + k(m + d + 2)T (r, g) + O(1) B ❉♦ ✭✸✳✶✾✮✱ ✭✸✳✷✵✮✱ ✭✸✳✷✶✮ t❛ ❝â N1 (r, ✭✸✳✷✵✮ ✭✸✳✷✶✮ (n−2k)(m−d)(T (r, f )+T (r, g)) ≤ 16(T (r, f )+T (r, g))+k(m+d)(T (r, f ) +T (r, g))+3[2T (r, f )+2T (r, f )+k(m+d+2)T (r, f )+2T (r, g)+2T (r, g) +k(m + d + 2)T (r, g)] − log r + O(1) = (28 + 2k(2m + 2d + 3))(T (r, f ) + T (r, g)) − log r + O(1) ❚ø ✤â✱ s✉② r❛ [(n−2k)(m−d)−(28+2k(2m+2d+3))](T (r, f )+T (r, g))+4 log r ≤ O(1) 28 + 2k(2m + 2d + 3) ✳ m−d ❚❛ sû ❞ư♥❣ ❝→❝ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ ❝→❝ ❚r÷í♥❣ ❤đ♣ ỵ ụ õ ❦➳t ❧✉➟♥ f = hg ✈ỵ✐ hn(m+d) = ✣à♥❤ ỵ ữủ ự t ợ ❣✐↔ t❤✐➳t n ≥ 2k + ❚r÷í♥❣ ❤đ♣ ✷ ✈➔ rữớ ủ ỵ f g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ K, q, c ∈ K, |q| = 1, ✈➔ ❝❤♦ n, m, d, k ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ m > d ≥ ❑❤✐ ✤â f = gh ✈ỵ✐ hn(m+d) = 1, h ∈ K ♥➳✉ ♠ët tr♦♥❣ ❤❛✐ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿ ✶✳ n ≥ 2k + k(m+d)+16 ✈➔ (f nm (z)f nd (qz + c))(k) ✈➔ (g nm (z)g nd (qz + c))(k) m−d ♥❤➟♥ ✶ ❈▼❀ ✷✳ n ≥ 2k + 2k(2m+2d+3)+28 ✈➔ (f nm (z)f nd (qz + c))(k) ✈➔ (g nm (z)g nd (qz + m−d c))(k) ♥❤➟♥ ✶ ■▼✳ ✾✷ ❑➳t ❧✉➟♥ ❝õ❛ ❈❤÷ì♥❣ ✸ ❚r♦♥❣ ❈❤÷ì♥❣ ✸✱ ❝❤ó♥❣ tỉ✐ ✤➣ t❤✐➳t ❧➟♣ ✤÷đ❝ ❝→❝ ❇ê ✤➲ ữ ỵ tt tự ❤❛✐ ❝❤♦ t➼❝❤ q ✲s❛✐ ♣❤➙♥ ❞↕♥❣ f n f m (qz + c) ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ s❛✐ ♣❤➙♥ ❞↕♥❣ (f nm f nd (qz + c))(k) , ð ✤â q, c ∈ K ✈ỵ✐ |q| = ✈➔ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ K ❈❤ó♥❣ tỉ✐ ❝ơ♥❣ ✤➣ t❤✐➳t ❧➟♣ ✤÷đ❝ ❤❛✐ ❦➳t q✉↔ ✈➲ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ t➼❝❤ q ✲s❛✐ ♣❤➙♥ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ s❛✐ ♣❤➙♥ ❝â ❞↕♥❣ tr➯♥✱ ✤â ❧➔ ỵ ú ỵ r t q✉↔ ♥➔② ❝❤÷❛ ❝â tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝✳ ✾✸ ❑➳t ❧✉➟♥ ✈➔ ❦✐➳♥ ♥❣❤à ▲✉➟♥ →♥ ♥❣❤✐➯♥ ❝ù✉ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝ ✈➔ p✲❛❞✐❝✱ ①❡♠ ①➨t ❝→❝ t÷ì♥❣ tü ❝õ❛ ❝→❝ ✤à♥❤ ỵ tt ố ợ t ố ợ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ t➼❝❤ q✲s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣ ❆❝s✐♠❡t✳ ◆❤ú♥❣ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ →♥ ✶✳ ❚❤✐➳t ữủ ởt ỵ tữỡ tỹ ỵ tự tt ỵ ởt ỵ tữỡ tỹ ỵ tự t tt ỵ t ✤÷đ❝ ♠ët ❦➳t q✉↔ ✈➲ Bi − U RSM ✭✣à♥❤ ỵ ởt t q U RSM ỵ ✶✳✸✳✸✮✱ ♠ët ❦➳t q✉↔ ✈➲ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t tự ỵ t ỵ tự tt ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✈❡❝✲tì ❝→❝ ❤➔♠ ♥❣✉②➯♥ tr➯♥ ♠ët trữớ ổst ỵ ỵ t ❧➟♣ ✤÷đ❝ ❜❛ ❦➳t q✉↔ ✈➲ ❱➜♥ ✤➲ ❞✉② ♥❤➜t tự ỵ q ỵ t ữủ ỵ tt tự t q ✲s❛✐ ♣❤➙♥ ❞↕♥❣ f n f m (qz + c) ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ s❛✐ ♣❤➙♥ ❞↕♥❣ (f nm f nd (qz + c))(k) , ð ✤â q, c ∈ K ✈ỵ✐ |q| = ✈➔ f ❧➔ ❤➔♠ tr K, ợ K ởt trữớ ổst ✭❇ê ✤➲ ✸✳✷✳✶✱ ❇ê ✤➲ ✸✳✸✳✶✮✳ ❚❤✐➳t ❧➟♣ ✤÷đ❝ ❤❛✐ ❦➳t q✉↔ ✈➲ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ t➼❝❤ q ✲s❛✐ ♣❤➙♥ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ s❛✐ ♣❤➙♥ ❝â tr ỵ ỳ t✐➳♣ tö❝ ♥❣❤✐➯♥ ❝ù✉✳ ✶✳ ❚✐➳♣ tö❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ tữỡ tỹ ỵ tt ①→❝ ✤à♥❤ ✈➔ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝ ✈➔ ♣✲❛❞✐❝✳ ✷✳ ❚✐➳♣ tö❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ù♥❣ ❞ö♥❣ ỵ tt t ❤➔♠ ✈➔ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t✳ ✾✹ ❉❛♥❤ ♠ö❝ ❈æ♥❣ tr➻♥❤ ❝õ❛ t→❝ ❣✐↔ ✤➣ ❝æ♥❣ ❜è ❧✐➯♥ q✉❛♥ ✤➳♥ ✤➲ t➔✐ ✶✳ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✵✽✮✱ ✧ ❆♥ ❛♥❛❧♦❣✉❡ ♦❢ ▼❛s♦♥✬s t❤❡♦r❡♠ ❢♦r ♣✲ ❛❞✐❝ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s ✐♥ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✧✱ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥❝❡✱ ◆❳❇ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ❍➔ ◆ë✐✱ ✺✸✭✶✮✱ ♣♣✳ ✶✷✲✷✶✳ ✷✳ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✷✮✱ ✧❆ ✈❡rs✐♦♥ ♦❢ t❤❡ ❍❛②♠❛♥ ❝♦♥❥❡❝t✉r❡ ❢♦r ♣✲❛❞✐❝ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❞✐❢❢❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧s✧✱ ■♥t❡r✲ r❛❝t✐♦♥s ❜❡t✇❡❡♥ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ❛♥❛❧②s✐s✱ ❙❝✐✳ ❚❡❝❤♥✐❝s P✉❜❧✳❍♦✉s❡✱ ❍❛♥♦✐✱ ♣♣✳ ✶✺✷✲✶✻✶✳ ✸✳ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛✱ ❛♥❞ ❍❛ ❍✉② ❑❤♦❛✐ ✭✷✵✶✼✮✱ ✧❱❛❧✉❡ s❤❛r✐♥❣ ♣r♦❜❧❡♠s ❢♦r ❞✐❢❢❡r❡♥t✐❛❧ ❛♥❞ ❞✐❢❢❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ♠❡r♦✲ ♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ✐♥ ❛ ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ❢✐❡❧❞✧✱ ♣✲❆❞✐❝ ◆✉♠❜❡rs✱ ❯❧✲ tr❛♠❡tr✐❝ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✉♠❡ ✾✱ ■ss✉❡ ✶✱ ♣♣✳ ✶✕✶✹✳ ✹✳ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✼✮✱ ✧❖♥ t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦❜❧❡♠ ♦❢ ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②✲ ♥♦♠✐❛❧s✧✱ ❆♥♥❛❧❡s ❯♥✐✈✳❙❝✐✳❇✉❞❛♣❡st✱ ❙❡❝t✳ ❈♦♠♣✱ ✹✻✱ ♣♣✳✷✽✾✲✸✵✷✳ ✺✳ ❍❛ ❍✉② ❑❤♦❛✐✱ ❱✉ ❍♦❛✐ ❆♥✱ ❛♥❞ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✼✮✱ ✧❖♥ ❢✉♥❝✲ t✐♦♥❛❧ ❡q✉❛t✐♦♥s ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❆r❝❤✐✈ ❞❡r ▼❛t❤❡♠❛t✐❦✱ ❙♣r✐♥❣❡r ■♥t❡r♥❛t✐♦♥❛❧ P✉❜❧✐s❤✐♥❣✱ ❱♦❧✉♠❡ ✶✵✾✱ ■ss✉❡ ✻✱ ♣♣ ✺✸✾✕✺✹✾✳ ✻✳ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛✱ ❛♥❞ ❍❛ ❍✉② ❑❤♦❛✐ ✭❙✉❜♠✐t✮✱ ✧❯♥✐q✉❡✲ ♥❡ss t❤❡♦r❡♠s ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②✲ ♥♦♠✐❛❧✧✳ ✾✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❳✉➙♥ ▲❛✐ ✭✷✵✶✼✮✱ ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠ ❦❤✐ ❤❛✐ ✤↕♦ ❤➔♠ ❝ò♥❣ ♥❤➟♥ ♠ët t➟♣✱ ▲✉➟♥ →♥ ❚✐➳♥ sÿ ❚♦→♥ ❤å❝✱ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❚❤→✐ ◆❣✉②➯♥✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❬✷❪ ▲➯ ◗✉❛♥❣ ◆✐♥❤ ✭✷✵✶✼✮✱ ❱➲ ①→❝ ✤à♥❤ ❤➔♠ ✈➔ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ q✉❛ ✤✐➲✉ ❦✐➺♥ ↔♥❤ ♥❣÷đ❝ ❝õ❛ t➟♣ ❤đ♣ ✤✐➸♠✱ ▲✉➟♥ →♥ ❚✐➳♥ sÿ ❚♦→♥ ❤å❝✱ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❚❤→✐ ◆❣✉②➯♥✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❚❛ ❚❤✐ ❍♦❛✐ ❆♥✱ ◆❣✉②❡♥ ❚❤✐ ◆❣♦❝ ❉✐❡♣ ✭✷✵✶✸✮✱ ✧●❡♥✉s ♦♥❡ ❢❛❝t♦rs ♦❢ ❝✉r✈❡ ❞❡❢✐♥❡❞ ❜② s❡♣❛r❛t❡❞ ✈❛r✐❛❜❧❡ ♣♦❧②♥♦♠✐❛❧✧✱ ❏✳ ◆✉♠❜❡r ❚❤❡♦r②✱ ✶✸✸ ✭✽✮✱ ♣♣✳ ✷✻✶✻✲✷✻✸✹✳ ❬✹❪ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✷✮✱ ✧❆ ✈❡rs✐♦♥ ♦❢ t❤❡ ❍❛②♠❛♥ ❝♦♥✲ ❥❡❝t✉r❡ ❢♦r ♣✲❛❞✐❝ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❞✐❢❢❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧s✧✱ ■♥t❡rr❛❝✲ t✐♦♥s ❜❡t✇❡❡♥ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ❛♥❛❧②s✐s✱ ❙❝✐✳ ❚❡❝❤♥✐❝s P✉❜❧✳❍♦✉s❡✱ ❍❛♥♦✐✱ ♣♣✳ ✶✺✷✲✶✻✶✳ ❬✺❪ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✼✮✱ ✧❖♥ t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦❜❧❡♠ ♦❢ ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②✲ ♥♦♠✐❛❧s✧✱ ❆♥♥❛❧❡s ❯♥✐✈✳❙❝✐✳❇✉❞❛♣❡st✱ ❙❡❝t✳ ❈♦♠♣✱ ✹✻✱ ♣♣✳✷✽✾✲✸✵✷✳ ❬✻❪ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛✱ ❛♥❞ ❍❛ ❍✉② ❑❤♦❛✐ ✭✷✵✶✼✮✱ ✧❱❛❧✉❡ s❤❛r✐♥❣ ♣r♦❜❧❡♠s ❢♦r ❞✐❢❢❡r❡♥t✐❛❧ ❛♥❞ ❞✐❢❢❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ♠❡r♦✲ ♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ✐♥ ❛ ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ❢✐❡❧❞✧✱ ♣✲❆❞✐❝ ◆✉♠❜❡rs✱ ❯❧✲ tr❛♠❡tr✐❝ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✉♠❡ ✾✱ ■ss✉❡ ✶✱ ♣♣✳ ✶✕✶✹✳ ❬✼❪ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛✱ ❛♥❞ ❍❛ ❍✉② ❑❤♦❛✐ ✭❙✉❜♠✐t✮✱ ✧❯♥✐q✉❡✲ ♥❡ss t❤❡♦r❡♠s ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②✲ ♥♦♠✐❛❧✧✳ ❬✽❪ ❱✉ ❍♦❛✐ ❆♥ ❛♥❞ ▲❡ ◗✉❛♥❣ ◆✐♥❤ ✭✷✵✶✻✮✱ ✧❖♥ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❋❡r♠❛t✲❲❛r✐♥❣ t②♣❡ ❢♦r ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ✈❡❝t♦r✐❛❧ ❡♥t✐r❡ ❢✉♥❝✲ t✐♦♥s✧✱ ❇✉❧❧✳❑♦r❡❛♥ ▼❛t❤✳❙♦❝✱ ✺✸✭✹✮✱ ♣♣✳ ✶✶✽✺✲✶✶✾✻✳ ✾✻ ❬✾❪ ❇❡③✐✈✐♥ ❏✳ P✳✱ ❇♦✉ss❛❢ ❑✳ ❛♥❞ ❊s❝❛ss✉t ❆✳ ✭✷✵✶✷✮✱ ✧❩❡r♦s ♦❢ t❤❡ ❞❡r✐✈❛✲ t✐✈❡ ♦❢ ❛ ♣✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥✧✱ ❇✉❧❧✳ ❙❝✐✳ ▼❛t❤➨♠❛t✐q✉❡s✱ ✶✸✻✭✽✮✱ ♣♣✳ ✽✸✾✕✽✹✼✳ ❬✶✵❪ ❇❤♦♦s♥✉r♠❛t❤ ❙✉❜❤❛s ❙✳ ❛♥❞ ❉②❛✈❛♥❛❧ ❘❡♥✉❦❛❞❡✈✐ ❙✳ ✭✷✵✵✼✮✱ ✧❯♥✐q✉❡♥❡ss ❛♥❞ ✈❛❧✉❡✲s❤❛r✐♥❣ ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❈♦♠♣✉t❡rs ❛♥❞ ▼❛t❤❡♠❛t✐❝s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ✺✸✱ ♣♣✳ ✶✶✾✶✲✶✷✵✺✳ ❬✶✶❪ ❇♦✉ss❛❢ ❑✳ ✱ ❊s❝❛ss✉t ❆✳✱ ❖❥❡❞❛ ❏✳ ✭✷✵✶✷✮✱ ✧p✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝✲ t✐♦♥s (f )( ) P (f ), (g)( ) P (g) s❤❛r✐♥❣ ❛ s♠❛❧❧ ❢✉♥❝t✐♦♥✧✱ ❇✉❧❧✳ ❙❝✐✳ ♠❛t❤✱ ✶✸✻✱ ♣♣✳ ✶✼✷✲✷✵✵✳ ❬✶✷❪ ❇♦✉t❛❜❛❛ ❆✳ ✭✶✾✾✵✮✱ ✧❚❤✬❡♦r✐❡ ❞❡ ◆❡✈❛♥❧✐♥♥❛ ♣✲❛❞✐q✉❡✧✱ ▼❛♥✉s❝r✐♣t❛ ▼❛t❤✱ ✻✼✱ ♣♣✳ ✷✺✶✲✷✻✾✳ ❬✶✸❪ ❚✳ ❉✐♥❤ ✭✷✵✵✷✮✱ ✧❊♥s❡♠❜❧❡s ❞✬✉♥✐❝✐t❡✬ ♣♦✉r ❧❡s ♣♦❧♥♦♠❡s✧✱ ❊r❣♦❞✐❝ ❚❤❡♦r② ❉②♥❛♠ ❙②st❡♠s✱ ✷✷✭✶✮✱ ♣♣✳ ✶✼✶✲✶✽✻✳ ❬✶✹❪ ❚✳ ❉✐♥❤ ✭✷✵✵✺✮✱ ✧❉✐str✐❜✉t✐♦♥ ❞❡s ♣r➨✐♠❛❣❡s ❡t ❞❡s ♣♦✐♥ts ♣➨r✐♦❞✐q✉❡s ❞✬✉♥❡ ❝♦rr❡s♣♦♥❞❛♥❝❡ ♣♦❧②♥♦♠✐❛❧❡✧✱ ❇✉❧❧✳ ❙♦❝✳ ▼❛t❤✳ ❋r❛♥❝❡✱ ✶✸✸✭✸✮✱ ♣♣✳ ✸✻✸✕✸✾✹✳ ❬✶✺❪ ❊s❝❛ss✉t ❆✳ ❛♥❞ ❖❥❡❞❛ ❏✳ ✭✷✵✶✹✮✱ ✧❚❤❡ ♣✲❛❞✐❝ ❍❛②♠❛♥ ❈♦♥❥❡❝t✉r❡ ✇❤❡♥ ♥❂✷✧✱ ❈♦♠♣❧❡① ❱❛r✐❛❜❧❡ ❛♥❞ ❊❧❧✐♣t✐❝ ❊q✉❛t✐♦♥s✱ ✺✾✭✶✵✮✱ ♣♣✳ ✶✹✺✶✲✶✹✺✻✳ ❬✶✻❪ ❊s❝❛ss✉t ❆✳ ✭✷✵✶✺✮✱ ✧❱❛❧✉❡ ❉✐str✐❜✉t✐♦♥ ✐♥ ♣✲❛❞✐❝ ❆♥❛❧②s✐s✧✱ ❲♦r❧❞ ❙❝✐✳ P✉❜❧✳ ❈♦✳ Pt❡✱ ▲t❞✳ ❙✐♥❣❛♣♦r❡✳ ❬✶✼❪ ❋❛♥❣ ❈✳ ❨✳ ❛♥❞ ❋❛♥❣ ▼✳ ▲✳ ✭✷✵✵✷✮✱ ✧❯♥✐q✉❡♥❡ss ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝✲ t✐♦♥s ❛♥❞ ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧s✧✱ ❈♦♠♣✉t✳ ▼❛t❤✳ ❆♣♣❧✱ ✹✹ ✱ ♣♣✳ ✻✵✼✲ ✻✶✼✳ ❬✶✽❪ ❋❛♥❣ ▼✳ ▲✳ ✭✷✵✵✷✮✱ ✧❯♥✐q✉❡♥❡ss ❛♥❞ ✈❛❧✉❡✲s❤❛r✐♥❣ ♦❢ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✧✱ ❈♦♠♣✉t✳ ▼❛t❤✳ ❆♣♣❧✱ ✹✹✱ ♣♣✳ ✽✷✸✲✽✸✶✳ ❚❤❡♦r② ❆♣♣❧✱ ✸✼✭✶✲✹✮✱ ♣♣✳ ✶✽✺✲✶✾✸✳ ❬✶✾❪ ❋✉❥✐♠♦t♦ ❍✳ ✭✷✵✵✵✮✱ ✧❖♥ ✉♥✐q✉❡♥❡ss ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s s❤❛r✐♥❣ ❢✐♥✐t❡ s❡ts✧✱ ❆♠❡r✳ ❏✳▼❛t❤✱ ✶✷✷✭✻✮✱ ♣♣✳ ✶✶✼✺ ✲ ✶✷✵✸✳ ❬✷✵❪ ❍❛②♠❛♥ ❲✳❑✳ ✭✶✾✻✹✮✱ ✧▼❡r♦♠♦r♣❤✐❝ ❋✉♥❝t✐♦♥s✧✱ ❖①❢♦r❞ ▼❛t❤❡♠❛t✲ ✐❝❛❧ ▼♦♥♦❣r❛♣❤s ❈❧❛r❡♥❞♦♥ Pr❡ss✱ ❖①❢♦r❞✳ ❬✷✶❪ ❍❛②♠❛♥ ❲✳❑✳ ✭✶✾✻✼✮✱ ✧❘❡s❡❛r❝❤ ♣r♦❜❧❡♠s ✐♥ ❋✉♥❝t✐♦♥ ❚❤❡♦r②✧✱ ❚❤❡ ❆t❤❧♦♥❡ Pr❡ss ❯♥✐✈❡rs✐t② ♦❢ ▲♦♥❞♦♥✱ ▲♦♥❞♦♥✳ ❬✷✷❪ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✵✽✮✱ ✧ ❆♥ ❛♥❛❧♦❣✉❡ ♦❢ ▼❛s♦♥✬s t❤❡♦r❡♠ ❢♦r ♣✲❛❞✐❝ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s ✐♥ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✧✱ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥❝❡✱ ◆❳❇ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ❍➔ ◆ë✐✱ ✺✸✭✶✮✱ ♣♣✳ ✶✷✲✷✶✳ ✾✼ ❬✷✸❪ ❍✉ P✳ ❈✳ ❛♥❞ ❨❛♥❣ ❈✳ ❈✳ ✭✶✾✾✾✮✱ ✧ ❆ ✉♥✐q✉❡ r❛♥❣❡ s❡t ❢♦r ♣✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ✇✐t❤ ✶✵ ❡❧❡♠❡♥ts✧✱ ❆❝t❛ ▼❛t❤✳ ❱✐❡t♥❛♠✐❝❛✳✱ ✷✹✱ ♣♣✳ ✾✺✲✶✵✽✳ ❬✷✹❪ ❍✉ P✳ ❈✳ ❛♥❞ ❨❛♥❣ ❈✳ ❈✳ ✭✷✵✵✵✮✱ ✧▼❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ♦✈❡r ♥♦♥✲ ❆r❝❤✐♠❡❞❡❛♥ ❢✐❡❧❞s✧✱ ❑❧✉✇❡r✳ ❬✷✺❪ ❍❛ ❍✉② ❑❤♦❛✐ ✭✶✾✽✸✮✱ ✧❖♥ ♣✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❉✉❦❡ ▼❛t❤✳ ❏✳✱ ✺✵✱♣♣✳ ✻✾✺✲✼✶✶✳ ❬✷✻❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ❱✉ ❍♦❛✐ ❆♥ ✭✷✵✵✸✮✱ ✧❱❛❧✉❡ ❞✐str✐❜✉t✐♦♥ ❢♦r ♣✲❛❞✐❝ ❤②♣❡rs✉r❢❛❝❡s✧✱ ❚❛✐✇❛♥❡s❡ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✼✭✶✮✱ ♣♣✳ ✺✶✲✻✼✳ ❬✷✼❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ❱✉ ❍♦❛✐ ❆♥ ✭✷✵✶✶✮✱ ✧❱❛❧✉❡ ❞✐str✐❜✉t✐♦♥ ♣r♦❜❧❡♠ ❢♦r ♣✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✧✱ ❆♥♥✳ ❋❛❝✳ ❙❝✳ ❚♦✉❧♦✉s❡✱ ❱♦❧✳ ❳❳ ✭◆♦✳ ❙♣❡❝✐❛❧✮✱ ♣♣✳✶✸✺✲✶✹✾✳ ❬✷✽❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ❱✉ ❍♦❛✐ ❆♥ ✭✷✵✶✷✮✱ ✧ ❱❛❧✉❡ s❤❛r✐♥❣ ♣r♦❜❧❡♠ ❢♦r p✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐❢❢❡r❡♥❝❡ ♦♣❡r❛t♦rs ❛♥❞ ❞✐❢✲ ❢❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧s✧✱ ❯❦r❛♥✐❛♥ ▼❛t❤✳ ❏✳✱ ✻✹✭✷✮✱ ♣♣✳ ✶✹✼✲✶✻✹✳ ❬✷✾❪ ❍❛ ❍✉② ❑❤♦❛✐✱ ❱✉ ❍♦❛✐ ❆♥✱ ❛♥❞ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✼✮✱ ✧❖♥ ❢✉♥❝✲ t✐♦♥❛❧ ❡q✉❛t✐♦♥s ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❆r❝❤✐✈ ❞❡r ▼❛t❤❡♠❛t✐❦✱ ❙♣r✐♥❣❡r ■♥t❡r♥❛t✐♦♥❛❧ P✉❜❧✐s❤✐♥❣✱ ❱♦❧✉♠❡ ✶✵✾✱ ■ss✉❡ ✻✱ ♣♣ ✺✸✾✕✺✹✾✳ ❬✸✵❪ ❍❛ ❍✉② ❑❤♦❛✐✱ ❱✉ ❍♦❛✐ ❆♥ ❛♥❞ ◆❣✉②❡♥ ❳✉❛♥ ▲❛✐ ✭✷✵✶✷✮✱ ✧❱❛❧✉❡ s❤❛r✐♥❣ ♣r♦❜❧❡♠ ❛♥❞ ❯♥✐q✉❡♥❡ss ❢♦r p✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❆♥♥✳ ❯♥✐✈✳ ❙❝✐✳ ❇✉❞❛♣❡st✳✱ ❙❡❝t✳ ❈♦♠♣✳✱ ✸✽✱ ♣♣✳ ✼✶✲✾✷✳ ❬✸✶❪ ❍❛ ❍✉② ❑❤♦❛✐✱ ❱✉ ❍♦❛✐ ❆♥ ❛♥❞ ◆❣✉②❡♥ ❳✉❛♥ ▲❛✐ ✭✷✵✶✼✮✱ ✧❱❛❧✉❡✲ s❤❛r✐♥❣ ❛♥❞ ❯♥✐q✉❡♥❡ss ♣r♦❜❧❡♠s ❢♦r ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧s ✐♥ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✧✱ ❈♦♠♣❧❡① ❱❛r✐❛❜❧❡s ❛♥❞ ❊❧❧✐♣t✐❝ ❊q✉❛t✐♦♥s✱ ♣♣ ✶✲✶✼✳ ❬✸✷❪ ❍❛ ❍✉② ❑❤♦❛✐✱ ❱✉ ❍♦❛✐ ❆♥ ❛♥❞ ▲❡ ◗✉❛♥❣ ◆✐♥❤ ✭✷✵✶✹✮✱ ✧❯♥✐q✉❡✲ ♥❡ss t❤❡♦r❡♠s ❢♦r ❤♦❧♦♠♦r♣❤✐❝ ❝✉r✈❡s ✇✐t❤ ❍②♣❡rs✉r❢❛❝❡s ♦❢ ❋❡r♠❛t✲ ❲❛r✐♥❣ t②♣❡✧✱ ❈♦♠♣❧❡① ❆♥❛❧✳ ❖♣❡r✳ ❚❤❡♦r②✱ ❱♦❧✳✽✱ ◆♦✳✸✱ ♣♣✳ ✺✾✶✲✼✾✵✳ ❬✸✸❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ❚❛ ❚❤✐ ❍♦❛✐ ❆♥ ✭✷✵✵✶✮✱ ✧❖♥ ✉♥✐q✉❡♥❡ss ♣♦❧②✲ ♥♦♠✐❛❧s ❛♥❞ ❇✐✲❯❘❙ ❢♦r ♣✲❛❞✐❝ ▼❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❏✳ ◆✉♠❜❡r ❚❤❡♦r②✱ ✽✼✱ ♣♣✳ ✷✶✶✲✷✷✶✳ ❬✸✹❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ▼❛✐ ❱❛♥ ❚✉ ✭✷✵✵✹✮✱ ✧♣✲❛❞✐❝ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ❚❤❡♦r❡♠✧✱ ■♥t❡r♥❛t✳ ❏✳ ▼❛t❤✱ ✻✭✶✾✾✺✮✱ ♣♣✳ ✼✶✾✲✼✸✶✳ ❬✸✺❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ❨❛♥❣ ❈✳ ❈✳✱ ✧❖♥ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥ P (f ) = Q(g)✧✱ ❆❞✈❛♥❝❡s ✐♥ ❈♦♠♣❧❡① ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥✱ ❱❛❧✉❡ ❉✐s✲ tr✐❜✉t✐♦♥ ❚❤❡♦r② ❛♥❞ ❘❡❧❛t❡❞ ❚♦♣✐❝s✱ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡rs✱ ❉♦r❞r❡❝❤t✱ ❇♦st♦♥✱ ▲♦♥❞♦♥✱ ♣♣✳ ✷✵✶✲✷✵✽✳ ✾✽ ❬✸✻❪ ▲❛❤✐r✐ ■✳ ✭✶✾✾✼✮ ✱ ✧❯♥✐q✉❡♥❡ss ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛s ❣♦✈❡r♥❡❞ ❜② t❤❡✐r ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧s✧✱ ❨♦❦♦❤❛♠❛ ▼❛t❤✳ ❏✳✱ ✹✹✱ ♣♣✳ ✶✹✼✲ ✶✺✻✳ ❬✸✼❪ ▲❛✐♥❡ ■✳ ❛♥❞ ❨❛♥❣ ❈✳ ❈✳ ✭✷✵✵✼✮✱ ✧❱❛❧✉❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❞✐❢❢❡r❡♥❝❡ ♣♦❧②✲ ♥♦♠✐❛❧s✧✱ Pr♦❝✳ ❏❛♣❛♥✳ ❆❝❛❞✳✱ ❙❡r✳ ❆✱ ✽✸✭✽✮✱ ♣♣✳✶✹✽✲✶✺✶✳ ❬✸✽❪ ▲✐ P✳ ❛♥❞ ❨❛♥❣ ❈✳❈✳ ✭✷✵✵✹✮✱ ✧❙♦♠❡ ❋✉rt❤❡r ❘❡s✉❧ts ♦♥ t❤❡ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥ P (f ) = Q(g)✧✱ ❱❛❧✉❡ ❉✐str✐❜✉t✐♦♥ ❚❤❡♦r② ❛♥❞ ❘❡❧❛t❡❞ ❚♦♣✲ ✐❝s✱ ❆❞✈❛♥❝❡❞ ❈♦♠♣❧❡① ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥✱ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝✱ ❇♦st♦♥✱ ▼❆✱ ❱♦❧✳✸✱ ♣♣✳ ✷✶✾✲✷✸✶✳ ❬✸✾❪ ▲✐✉ ❑✳✱ ▲✐✉ ❳✳✱ ❈❛♦ ❚✳ ❇✳ ✭✷✵✶✶✮✱ ✧❱❛❧✉❡ ❞✐str✐❜✉t✐♦♥ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ❞✐❢❢❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧s✧✱ ❆❞✈✳ ❉✐❢❢❡r❡♥❝❡ ❊q✉✳✱ ❛rt✐❝❧❡ ■❉✷✸✹✷✶✺✱ ✶✷♣♣✳ ❬✹✵❪ ❑✳ ▼❛s✉❞❛ ❛♥❞ ❏✳ ◆♦❣✉❝❤✐ ✭✶✾✾✻✮✱ ✧❆ ❝♦♥str✉❝t✐♦♥ ♦❢ ❤②♣❡r❜♦❧✐❝ ❤②✲ ♣❡rs✉r❢❛❝❡ ♦❢ P N (C)✧✱ ▼❛t❤✳ ❆♥♥✳✱ ✸✵✹ ✱♣♣✳ ✸✸✾✲✸✻✷✳ ✶✽✱ ♣♣✳ ✹✸✼✲✹✺✵✳ ❬✹✶❪ ❖❥❡❞❛ ❏✳ ✭✷✵✵✽✮✱ ✧❍❛②♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ✐♥ ❛ p✲❛❞✐❝ ❢✐❡❧❞✧✱ ❚❛✐✇❛♥❡s❡ ❏✳ ▼❛t❤✳ ✶✷✭✾✮✱ ♣♣✳ ✷✷✾✺✲✷✸✶✸✳ ❬✹✷❪ ❖str♦✈s❦✐✐ ■✳✱ P❛❦♦✈✐t❝❤ ❋✳✱ ❩❛✐❞❡♥❜❡r❣ ▼✳ ✭✶✾✾✻✮✱ ✧❆ r❡♠❛r❦ ♦♥ ❝♦♠✲ ♣❧❡① ♣♦❧②♥♦♠✐❛❧s ♦❢ ❧❡❛st ❞❡✈✐❛t✐♦♥✧✱ ■♥t❡r♥❛t✳ ▼❛t❤✳ ❘❡s✳ ◆♦t✐❝❡s✱ ✶✹✱ ♣♣✳ ✻✾✾✕✼✵✸✳ ❬✹✸❪ P❛❦♦✈✐❝❤ ❋✳ ✭✷✵✵✽✮✱ ✧ ❖♥ ♣♦❧②♥♦♠✐❛❧s s❤❛r✐♥❣ ♣r❡✐♠❛❣❡s ♦❢ ❝♦♠♣❛❝t s❡ts✱ ❛♥❞ r❡❧❛t❡❞ q✉❡st✐♦♥s✧✱ ●❡♦♠✳ ❋✉♥❝t✳ ❆♥❛❧✱ ✶✽✭✶✮✱ ♣♣✳ ✶✻✸✲✶✽✸ ✳ ❬✹✹❪ P❛❦♦✈✐❝❤ ❋✳ ✭✷✵✶✵✮✱ ✧❖♥ t❤❡ ❡q✉❛t✐♦♥ P (f ) = Q(g), ✇❤❡r❡ P, Q ❛r❡ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ f, g ❛r❡ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✧✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳✱ ✶✸✷✭✻✮✱ ♣♣✳ ✶✺✾✶✲✶✻✵✼✳ ❬✹✺❪ ◗✐ ❳✳ ❈✳✱ ❨❛♥❣ ▲✳ ❩✳✱ ▲✐✉ ❑✳ ✭✷✵✶✵✮✱ ✧❯♥✐q✉❡♥❡ss ❛♥❞ ♣❡r✐♦❞✐❝✐t② ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❝♦♥❝❡r♥✐♥❣ t❤❡ ❞✐❢❢❡r❡♥❝❡ ♦♣❡r❛t♦r✧✱ ❈♦♠♣✳ ▼❛t❤✳ ❆♣♣❧✳✱ ✻✵✭✻✮✱ ♣♣✳ ✶✼✸✾✲✶✼✹✻✳ ❬✹✻❪ ❘✐tt ❏✳ ✭✶✾✷✷✮✱ ✧Pr✐♠❡ ❛♥❞ ❝♦♠♣♦s✐t❡ ♣♦❧②♥♦♠✐❛❧s✧✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✷✸✭✶✮✱ ♣♣✳ ✺✶✲✻✻✳ ❬✹✼❪ ❘✉ ▼✳ ✭✷✵✵✶✮✱ ✧❆ ♥♦t❡ ♦♥ ♣✲❛❞✐❝ ◆❡✈❛♥❧✐♥♥❛ t❤❡♦r②✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✷✾✱ ♣♣✳ ✶✷✻✸✲✶✷✻✾✳ ❬✹✽❪ ❙✐✉ ❨✳❚✳✱ ❨❡✉♥❣ ❙✳❑✳ ✭✶✾✾✼✮✱ ✧❉❡❢❡❝ts ❢♦r ❛♠♣❧❡ ❞✐✈✐s♦rs ♦❢ ❆❜❡❧✐❛♥ ✈❛✲ r✐❡t✐❡s✱ ❙❝❤✇❛r③ ❧❡♠♠❛✱ ❛♥❞ ❤②♣❡r❜♦❧✐❝ ❤②♣❡rs✉r❢❛❝❡s ♦❢ ❧♦✇ ❞❡❣r❡❡s✧✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳✱ ✶✶✾✱ ♣♣✳ ✶✶✸✾✲✶✶✼✷✳ ✾✾ ❬✹✾❪ ❨❛♥❣ ❈✳ ✭✶✾✼✽✮✱ ✧❖♣❡♥ ♣r♦❜❧❡♠ ✐♥ ❈♦♠♣❧❡① ❛♥❛❧②s✐s✧✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❙✳❯✳◆✳❨✳❇r♦❝❦♣♦rt ❈♦♥❢✳ ♦♥ ❈♦♠♣❧❡① ❋✉♥❝t✐♦♥ ❚❤❡♦r②✱ ❏✉♥❡ ✼✕✾✱ ✶✾✼✻✱ ❊❞✐t❡❞ ❜② ❙❛♥❢♦r❞ ❙✳ ▼✐❧❧❡r✳ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ▼❛r❝❡❧ ❉❡❦❦❡r✱ ■♥❝✳✱ ◆❡✇ ❨♦r❦✲❇❛s❡❧✱ ❱♦❧✳✸✻✳ ❬✺✵❪ ❨❛♥❣ ❈✳❈✳ ✭✶✾✼✻✮✱ ✧❖♥ t✇♦ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✱ ✇❤✐❝❤ t♦❣❡t❤❡r ✇✐t❤ t❤❡✐r ❢✐rst ❞❡r✐✈❛t✐✈❡s ❤❛✈❡ t❤❡ s❛♠❡ ③❡r♦s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✺✻✱ ♣♣✳ ✶✲✻✳ ❬✺✶❪ ❨❛♥❣ ❈✳❈✳ ❛♥❞ ❍✉❛ ❳✳❍✳ ✭✶✾✾✼✮✱ ✧❯♥✐q✉❡♥❡ss ❛♥❞ ✈❛❧✉❡✲s❤❛r✐♥❣ ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❆♥♥✳ ❆❝❛❞✳ ❙❝✐✳ ❋❡♥♥✳ ▼❛t❤✳✱ ✷✷✱ ♣♣✳ ✸✾✺✲ ✹✵✻✳ ❬✺✷❪ ❨✐ ❍✳ ❳✳ ✭✶✾✾✵✮✱ ✧❆ q✉❡st✐♦♥ ♦❢ ❈✳ ❈✳ ❨❛♥❣ ♦♥ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✧✱ ❑♦❞❛✐ ▼❛t❤✳ ❏✳✱ ✶✸✱ ♣♣✳ ✸✾✲✹✻✳ ❬✺✸❪ ❨✐ ❍✳ ❳✳ ✭✶✾✾✺✮✱ ✧❚❤❡ ✉♥✐q✉❡ r❛♥❣❡ s❡ts ♦❢ ❡♥t✐r❡ ♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❈♦♠♣❧❡① ❱❛r✐❛❜❧❡s ❚❤❡♦r② ❆♣♣❧✳✱ ✷✽✱ ♣♣✳ ✶✸✲✷✶✳ ❬✺✹❪ ❩❤❛♥❣ ❳✳❨✳✱ ▲✐♥ ❲✳❈✳ ✭✷✵✵✽✮✱ ✧❯♥✐q✉❡♥❡ss ❛♥❞ ✈❛❧✉❡✲s❤❛r✐♥❣ ♦❢ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✹✸✱ ♣♣✳ ✾✸✽✲✾✺✵✳ ...ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM PHẠM NGỌC HOA MỘT SỐ DẠNG CỦA ĐỊNH LÝ RITT VÀ ỨNG DỤNG VÀO VẤN ĐỀ DUY NHẤT Ngành: Toán Giải tích Mã số: 46 01 02 LUẬN ÁN TIẾN SĨ TOÁN HỌC Người hướng