✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ P❍❸▼ ◆●➴❈ ❍❖❆ ▼❐❚ ❙➮ ❉❸◆● ❈Õ❆ ✣➚◆❍ ▲Þ ❘■❚❚ ❱⑨ Ù◆● ❉Ö◆● ❱⑨❖ ❱❻◆ ✣➋ ❉❯❨ ◆❍❻❚ ▲❯❾◆ ⑩◆ ❚■➌◆ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✽ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ P❍❸▼ ◆●➴❈ ❍❖❆ ▼❐❚ ❙➮ ❉❸◆● ❈Õ❆ ✣➚◆❍ ▲Þ ❘■❚❚ ❱⑨ Ù◆● ❉Ư◆● ❱⑨❖ ❱❻◆ ✣➋ ❉❯❨ ◆❍❻❚ ◆❣➔♥❤✿ ❚♦→♥ ●✐↔✐ t➼❝❤ ▼➣ sè✿ ✾ ✹✻ ✵✶ ✵✷ ▲❯❾◆ ⑩◆ ❚■➌◆ ❙➒ ❚❖⑩◆ ❍➴❈ ✶✳ ❚❙✳ ❱ô ❍♦➔✐ ❆♥ ✷✳ ●❙ ❚❙❑❍✳ ❍➔ ❍✉② ữớ ữợ ✐ ▲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 ữ ✈➔♦ ❧✉➟♥ →♥✳ ❈→❝ ❦➳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❤✉➟♥ ❧ñ✐ ❣✐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 ❧á♥❣ 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 ❘✐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 ❘✐tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ỵ t❤ù ❤❛✐ ❝õ❛ ❘✐tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✷✳ ✣à♥❤ 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 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ❈❤÷ì♥❣ 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❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✐✈ ❉❛♥❤ ♠ö❝ ❝→❝ ỵ ỳ t tt ã Bi U RSM ✿ s♦♥❣ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭❜✐✲✉♥✐q✉❡ r❛♥❣❡ s❡t ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✮ • Ef (S)✿ ♥❣❤à❝❤ ↔♥❤ ❝õ❛ t➟♣ ❙ q✉❛ ❤➔♠ ❢✱ ❝â t➼♥❤ ❜ë✐ • E f (S)✿ ♥❣❤à❝❤ ↔♥❤ ❝õ❛ t➟♣ ❙ q✉❛ ❤➔♠ ❢✱ ❦❤ỉ♥❣ t➼♥❤ ❜ë✐ • gcd✿ ữợ ợ t rtst sr ã M(K) : trữớ tr K ã O(1) ữủ ã O(r) ữủ ổ ũ ợ ũ ợ r r + ã o(r) ✤↕✐ ❧÷đ♥❣ ✈ỉ ❝ị♥❣ ❜➨ ❜➟❝ ❝❛♦ ❤ì♥ r ❦❤✐ r → +∞✳ • U RSE ✿ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♥❣✉②➯♥ ✭✉♥✐q✉❡ r❛♥❣❡ s❡t ❢♦r ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✮ • U RSM ✿ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭✉♥✐q✉❡ r❛♥❣❡ s❡t ❢♦r rr ts ỵ t ỵ ỡ ỵ tt số t ❜✐➸✉ r➡♥❣ ♠å✐ sè ♥❣✉②➯♥ n ≥ ✤➲✉ ❜✐➸✉ t ữợ t số tố ❝â ❞↕♥❣ mk n = pm pk , ✈ỵ✐ k ≥ 1, ð ✤â ❝→❝ t❤ø❛ sè ♥❣✉②➯♥ tè p1 , , pk ✤æ✐ ♠ët ♣❤➙♥ ❜✐➺t ✈➔ ❝→❝ sè ♠ơ t÷ì♥❣ ù♥❣ m1 ≥ 1, , mk ≥ ✤÷đ❝ ①→❝ ✤à♥❤ ♠ët ❝→❝❤ ❞✉② ♥❤➜t t❤❡♦ n tt ữớ t tữỡ tỹ ỵ ♥➔② ✤è✐ ✈ỵ✐ ❝→❝ ✤❛ t❤ù❝✳ ✣➸ ♠ỉ t↔ ❦➳t q✉↔ ❝õ❛ ❘✐tt✱ t❛ ❦➼ ❤✐➺✉ M(C) ✭t÷ì♥❣ ù♥❣✱ A(C)✮ ❧➔ t➟♣ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭♥❣✉②➯♥✮ tr➯♥ C ✈➔ ❦➼ ❤✐➺✉ L(C) ❧➔ t➟♣ ❝→❝ ✤❛ t❤ù❝ ❜➟❝ 1✳ ✣➦t E, F ❧➔ ❝→❝ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ M(C)✱ ❦❤✐ ✤â ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ F (z) ✤÷đ❝ ổ t ữủ tr Eì F ❜➜t ❦ý ❝→❝❤ ✈✐➳t t❤➔♥❤ ♥❤➙♥ tû F (z) = ϕ1 ◦ ϕ2 (z) ✈ỵ✐ ϕ1 (z) ∈ E ✈➔ ϕ2 (z) ∈ F ✤➲✉ ❦➨♦ t❤❡♦ ❤♦➦❝ ϕ1 ❧➔ t✉②➳♥ t➼♥❤ ❤♦➦❝ ϕ2 ❧➔ t✉②➳♥ t➼♥❤✳ ◆➠♠ ✶✾✷✷✱ ❘✐tt ự ỵ s ỵ ỵ tự t tt F t ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ C[z] \ L(C) ◆➳✉ ♠ët ✤❛ t❤ù❝ F (z) ❝â ❤❛✐ ❝→❝❤ ♣❤➙♥ t➼❝❤ ❦❤→❝ ♥❤❛✉ t tự ổ t ữủ tr Fì F ✿ F = ϕ1 ◦ ϕ2 ◦ · · · ϕr = ψ1 ◦ ψ2 ◦ · · · ψs , t❤➻ r = s, ✈➔ ❜➟❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ψi ❧➔ ❜➡♥❣ ✈ỵ✐ ❜➟❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ϕi ♥➳✉ ❦❤æ♥❣ t➼♥❤ ✤➳♥ t❤ù tü ①✉➜t ❤✐➺♥ ❝õ❛ ❝❤ó♥❣✳ ❈ơ♥❣ tr♦♥❣ ❬✹✻❪✱ ❘✐tt ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ỵ s ỵ ỵ tự ❘✐tt✮✳ ●✐↔ sû r➡♥❣ ϕ, α, ψ, β ∈ C[x] \ C t❤ä❛ ♠➣♥ ϕ ◦ α = ψ ◦ β ✈➔ gcd(deg(ϕ); deg(ψ)) = gcd(deg(α); deg(β)) = ❑❤✐ õ tỗ t t t lj C[x] s❛♦ ❝❤♦ (l1 ◦ ϕ ◦ l2 , l2−1 ◦ α ◦ l3 , l1 ◦ ψ ◦ l2 , l4−1 ◦ β ◦ l3 ) ❝â ♠ët tr♦♥❣ ❝→❝ ❞↕♥❣ (Fn , Fm , Fm , Fn ) ❤♦➦❝ ✷ (xn , xs h(xn ), xs h(x)n , xn ), ð ✤â m, n > ❧➔ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉✱ s > ❧➔ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ✈ỵ✐ n, ✈➔ h ∈ C[x] \ xC[x], lj−1 ❧➔ ❤➔♠ ♥❣÷đ❝ ❝õ❛ lj ✱ Fn , Fm ❧➔ ❝→❝ tự ứ õ t ữợ t sè ✤➣ ❝â r➜t ♥❤✐➲✉ ❝→❝ t→❝ ❣✐↔ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♣❤➨♣ ♣❤➙♥ t➼❝❤ ❝→❝ ✤❛ t❤ù❝ tr♦♥❣ ❝→❝ ✤à♥❤ ỵ tt ữ st r ❲❤❛♣❧❡s ❬✶✺❪✱ ❍✳❚✳❊♥❣str♦♠ ❬✶✻❪✱ ❍✳▲❡✈✐ ❬✸✼❪✱✳✳✳✳ ❈❤➥♥❣ ❤↕♥✱ ♥➠♠ ✶✾✹✶ ❊♥❣str♦♠ ❬✶✻❪ ✈➔ ♥➠♠ ✶✾✹✷ ▲❡✈✐ ❬✸✼❪ ✤➣ ❝❤ù♥❣ tä r ỵ ỏ ú tr ởt trữớ ❜➜t ❦ý ✤➦❝ sè ❦❤ỉ♥❣✳ ❚r➯♥ ♣❤÷ì♥❣ ❞✐➺♥ ❣✐↔✐ t➼❝❤✱ t t r ỵ tự tt ổ t↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ϕ(α) = ψ(β)✱ ð ✤â ϕ, α, ψ, β ❧➔ ❝→❝ ✤❛ t❤ù❝ ✈➔ ❜➟❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ❧➔ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉✳ ❘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♦ ❬✷✵❪✱ ❍➔ ❍✉② ❑❤♦→✐✲❈✳❈✳❨❛♥❣ ❬✸✹❪✱ P ỵ r ữỡ tr ❤➔♠ ❧✐➯♥ q✉❛♥ ♠➟t t❤✐➳t ✤➳♥ ✈➜♥ ✤➲ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤✲♠ët ù♥❣ ❞ư♥❣ ❝õ❛ ỵ tt ố tr ❞✉② ♥❤➜t ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❧➛♥ ✤➛✉ t✐➯♥ ❜ð✐ ự ữủ r ợ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f ✈➔ g tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✱ ♥➳✉ ❝❤ó♥❣ ❝â ❝❤✉♥❣ ♥❤❛✉ ↔♥❤ ♥❣÷đ❝ ✭❦❤ỉ♥❣ t➼♥❤ ❜ë✐✮ ❝õ❛ ✺ ✤✐➸♠ ♣❤➙♥ ❜✐➺t t❤➻ f = g ỵ ú õ ♥❤❛✉ ↔♥❤ ♥❣÷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 ❝ù✉ ❝❤õ ②➳✉ ✈➔ ✤➣ ❝â r➜t ♥❤✐➲✉ ❦➳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❝ ♠ð rë♥❣ s❛♥❣ ởt ỵ tt t õ ❧➔ ①❡♠ ①➨t t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝õ❛ ❝→❝ tự ữớ t ữợ ữợ ự ✤➣ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❦➳t q✉↔ ♥ê✐ t✐➳♥❣ r➡♥❣ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ f tr➯♥ tr÷í♥❣ sè ♣❤ù❝ C ❦❤ỉ♥❣ ♥❤➟♥ ❣✐→ trà ✵ ✈➔ ✤↕♦ ❤➔♠ ❜➟❝ k ✸ f ợ k số ữỡ ổ ♥❤➟♥ ❣✐→ trà ✶ t❤➻ f ❧➔ ❤➔♠ ❤➡♥❣✳ ❍❛②♠❛♥ ❝ơ♥❣ ✤÷❛ r❛ ❣✐↔ t❤✉②➳t s❛✉✳ ●✐↔n t❤✉②➳t ❍❛②♠❛♥✳ ❬✷✷❪ ◆➳✉ ♠ët ❤➔♠ ♥❣✉②➯♥ f t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (z)f (z) = ợ n số ữỡ ✈➔ ✈ỵ✐ ♠å✐ z ∈ C t❤➻ f ❧➔ ❤➔♠ ❤➡♥❣✳ ●✐↔ t❤✉②➳t ♥➔② ✤➣ ✤÷đ❝ ❝❤➼♥❤ ❍❛②♠❛♥ ❦✐➸♠ tr❛ ợ n > ữủ tr ợ n ≥ 1✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✈➔ ❝→❝ ✈➜♥ q t ởt ữợ ự ✤÷đ❝ ❣å✐ ❧➔ sü ❧ü❛ ❝❤å♥ ❝õ❛ ❍❛②♠❛♥✳ ❈ỉ♥❣ tr➻♥❤ q trồ tú ữợ ự tở ❨❛♥❣✲❍✉❛ ❬✺✶❪✱ ❤❛✐ 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 ữủ t t ổ tự ụ ợ ❝→❝ ❤➺ 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❛ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♥❤➟♥ tr t ữợ ự ♥➔②✱ ♥➠♠ ✷✵✵✷ ❈✳ ❨✳ ❋❛♥❣ ✈➔ ▼✳ ▲✳ ❋❛♥❣ ❬✶✽❪ ✤➣ ❝❤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✱ ✈➜♥ ✤➲ ♥❤➟♥ ❣✐→ trà ❝ô♥❣ ữủ t ố ợ tự s ❝→❝ ❤➔♠ ♥❣✉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 ❜➟❝ ❤ú✉ ❤↕♥✳ t t tứ ỵ tự tt P õ ỵ tữ t ữủ ❤❛✐ t➟♣ ❝♦♠♣❛❝t ✤è✐ ✈ỵ✐ ❤❛✐ ✤❛ t❤ù❝✳ ➷♥❣ ✤➣ t➻♠ ✤÷đ❝ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❤❛✐ ✤❛ t❤ù❝ f1 , f2 ✈➔ ❤❛✐ t➟♣ ❝♦♠♣❛❝t K1 , K2 t❤ä❛ ♠➣♥ f11 (K1 ) = f21 (K2 ) ứ ỵ t❤ù ❤❛✐ ❝õ❛ ❘✐tt ✈➔ ❦➳t q✉↔ ❝õ❛ ❋✳P❛❦♦✈✐❝❤ ♥â✐ tr ú tổ õ t t ỵ t❤ù ❤❛✐ ❝õ❛ ❘✐tt ❝â 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❤ỉ♥❣ q✉❛ ✤✐➲✉ ❦✐➺♥ ↔♥❤ ♥❣÷đ❝ ❝õ❛ t➟♣ ❤ñ♣ ✤✐➸♠✳ ❚ø ♥❤➟♥ ①➨t ♥➔② ✈➔ ❝→❝ ❦➳t q✉↔ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✭①❡♠ ❬✹❪✱ ❬✸✹❪✱ ❬✹✹❪✮ ♥➯✉ tr➯♥✱ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ ✤÷đ❝ ✤➦t r❛ tü ♥❤✐➯♥ ữ s t sỹ tữỡ tỹ ỵ ❘✐tt ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✳ ❳❡♠ ①➨t ✈➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠✱ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ữợ õ ỵ tự ❘✐tt✳ ❳❡♠ ①➨t ✈➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠✱ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ t➼❝❤ q ✲s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ ữợ õ ỵ tự tt ✷✳ ❱➜♥ ✤➲ ✸✳ ❚ø ✤â✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t ởt số ỵ tt ự ❞ö♥❣ ✈➔♦ ✈➜♥ ✤➲ ❞✉② ♥❤➜t✧ ✤➸ ❣✐↔✐ q✉②➳t ❝→❝ ự tr ỗ tớ õ ❧➔♠ ♣❤♦♥❣ ♣❤ó t❤➯♠ ❝→❝ ❦➳t q✉↔ ✈➔ ù♥❣ ❞ư♥❣ ỵ tt t t ởt số 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 trữớ ủ ự p ữợ õ ỵ tt t ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ủ ự p ữợ õ ỵ tt Pữỡ ổ ự ỷ ỵ tữỡ tỹ ❝õ❛ ❝❤ó♥❣ ❝ị♥❣ ✈ỵ✐ ❝→❝ ❦✐➸✉ ❇ê ✤➲ ❇♦r❡❧ ❝õ❛ ỵ tt ố tr ữỡ tr➻♥❤ ❤➔♠✳ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ♥➔② t÷ì♥❣ tü ♥❤÷ ữỡ tr tr ỵ ữỡ tỹ 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✳ rữớ ủ l ợ 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✉② r❛ f = ❚r÷í♥❣ ❤đ♣ ✸✳ ❚❛ tê♥❣ ❤đ♣ ❝→❝ ❦➳t q✉↔ ❝õ❛ ỵ tr ỵ s ỵ f, g ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ K, n, m ❧➔ l ✈ỵ✐ ln+m = 1, ❤♦➦❝ g n+m f = hg ✈ỵ✐ h = ♥➳✉ ♠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) ♥❤➟♥ ❦❤æ♥❣ t➼♥❤ ❜ë✐✳ ❤❛✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ q, c ∈ K, |q| = 1✳ ❑❤✐ ✤â f = ✶✳ ✷✳ ✸✳ ỵ tự tt ✤➲ ❞✉② ♥❤➜t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ✈➔ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ❈❤ó♥❣ tỉ✐ t❤✐➳t ữủ ởt t q tữỡ tỹ ỵ tự ❤❛✐ ❝õ❛ ❘✐tt ❧➔ ❜ê ✤➲ s❛✉✳ ❇ê ✤➲ ✸✳✸✳✶✳ ❈❤♦ q, c ∈ K ✈ỵ✐ |q| = 1, n, m, d, k số 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 ✤â A = (C n )(k) = C n−k P, B = (Dn )(k) = Dn−k 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 Cn Dn ✣➦t F = ,G = ❉♦ C, D ❦❤→❝ ❤➡♥❣✱ t❛ ♥❤➟♥ ✤÷đ❝ 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 ✤➲ ✸✳✸✳✶ ữủ ự ỵ s ✤➙② ✤➣ ✤÷đ❝ ❝ỉ♥❣ ❜è tr♦♥❣ ❜➔✐ ❜→♦ ❬✽❪✱ ❝→❝ ỵ t ởt số t q ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ t➼❝❤ q−s❛✐ ♣❤➙♥ ❞↕♥❣ (f nm (z)f nd (qz + c))(k) 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 ✈➔ (g nm (z)g nd (qz + c))(k) ♥❤➟♥ t➼♥❤ ❝↔ ❜ë✐✱ 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 n (k) n−k n (k) n−k ✤â A = (C ) = C P, B = (D ) = D Q ❈❤ó þ 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), N1 (r, 1 ) + N1,(2 (r, ) = N2 (r, ) B B B n (k) n (k) ❑❤✐ ✤â✱ →♣ ❞ư♥❣ ❇ê ✤➲ ✸✳✶✳✹ ✤è✐ ✈ỵ✐ (C ) ✱ (D ) 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 ỹ A t C(a) = ợ à01 (a) ≥ A n + k ≥ ✈➔ ❞♦ ❇ê ✤➲ ✸✳✶✳✹ t❛ ❝â N1 (r, C) = N1 (r, f m f d (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 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❛ ❝ô♥❣ ❝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 ) ≤ T (r, A) + O(1), P (n − 2k)(m − d)T (r, g) + kN (r, D) + N (r, ) ≤ T (r, B) + O(1) Q (n − 2k)(m − d)T (r, f ) + kN (r, C) + N (r, ✾✵ ❑➳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) k(m + d) + 16 ❱➻ n ≥ 2k + ✱ t❛ ❣➦♣ ♠➙✉ t❤✉➝♥✳ m−d +N (r, ❚r÷í♥❣ ❤đ♣ ✷✳ (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❛ tứ 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)) ♥❤➟♥ ❦❤æ♥❣ t➼♥❤ ❜ë✐✱ t❤➻ f = hg ✈ỵ✐ hn(m+d) = 1✱ h ∈ K ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ sû ❞ö♥❣ ❝→❝ ❦➼ ❤✐➺✉ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ✣à♥❤ ỵ õ ố ✈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 T (r, A) ≤ N2 (r, A) + N2 (r, ✾✶ 1 )) + N1 (r, A) + N1 (r, ) − log r + O(1) B A ❑➳t ❤ñ♣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ❝â 2(N1 (r, B) + N1 (r, T (r, A) + T (r, B) ≤ 2(N2 (r, A) + N2 (r, 1 ) + N2 (r, B) + N2 (r, )) A B 1 ) + N1 (r, B) + N1 (r, )) − log r + O(1) A B ❚ø tữỡ tỹ ữ rữớ ủ ỵ ✸✳✸✳✷ ✈➔ ❞♦ ❇ê ✤➲ ✸✳✶✳✺ t❛ ❝â +3(N1 (r, A) + N1 (r, (n − 2k)(m − d)(T (r, f ) + T (r, g)) + k(N (r, C) + N (r, D)) +N (r, 1 ) + N (r, ) ≤ T (r, A) + T (r, B) + O(1); P Q ✭✸✳✶✾✮ 1 ) + N (r, )+ P Q k(m + d)(T (r, f ) + T (r, g)) + k(N1 (r, C) + N1 (r, D)) + 3(N1 (r, A)+ T (r, A) + T (r, B) ≤ 16(T (r, f ) + T (r, g)) + N (r, N1 (r, 1 ) + N1 (r, B) + N1 (r, )) − log r + O(1); A B ✭✸✳✷✵✮ ) ≤ 2T (r, f )+ A k(m + d + 2)T (r, f ) + O(1); N1 (r, A) ≤ 2T (r, f ) + O(1), N1 (r, 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 (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 ❞ö♥❣ ❝→❝ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ ❝→❝ ❚r÷í♥❣ ❤đ♣ ✷ ✈➔ ✸ ❝õ❛ ỵ ụ õ t f = hg ợ hn(m+d) = ỵ ữủ ự ♠✐♥❤✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t n ≥ 2k + ❚r÷í♥❣ ❤đ♣ ✷ ✈➔ ❚r÷í♥❣ ❤đ♣ ✸ ✳ tờ ủ t q ỵ tr ỵ s 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 ♥❤➟♥ t➼♥❤ ❝↔ ❜ë✐❀ ✷✳ 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) ♥❤➟♥ ❦❤æ♥❣ t➼♥❤ ❜ë✐✳ ❑➳t ❧✉➟♥ ❝õ❛ ❈❤÷ì♥❣ ✸ ❚r♦♥❣ ❈❤÷ì♥❣ ✸✱ ❝❤ó♥❣ tỉ✐ ✤➣ t❤✐➳t ❧➟♣ ✤÷đ❝ ❝→❝ ❇ê ✤➲ ✸✳✷✳✶✱ ✸✳✸✳✶ ♥❤÷ 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 õ tr õ ỵ ú ỵ r t q ữ ❝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❡♠ t tữỡ tỹ ỵ tt ố ✈ỵ✐ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ✤❛ 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 ❧➟♣ ✤÷đ❝ ♠ët ỵ tữỡ tỹ ỵ tự tt ỵ ởt ỵ tữỡ tỹ ỵ tự t tt ỵ t ữủ ởt t q Bi U RSM ỵ ởt t q U RSM ỵ ởt t q✉↔ ✈➲ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝❤♦ ✤❛ t❤ù❝ ỵ t ỵ tự tt ✈➔ ✈❡❝✲tì ❝→❝ ❤➔♠ ♥❣✉②➯♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✭✣à♥❤ ỵ ỵ t ữủ ❦➳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 ✤➲ ✸✳✸✳✶✮✳ ❚❤✐➳t ❧➟♣ ✤÷đ❝ ❤❛✐ ❦➳t q✉↔ ✈➲ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ t➼❝❤ q ✲s❛✐ ♣❤➙♥ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ s❛✐ ♣❤➙♥ ❝â ❞↕♥❣ tr➯♥ ✭✣à♥❤ ỵ ỳ t tử ❝ù✉✳ ✶✳ ❚✐➳♣ tư❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t÷ì♥❣ tü ❝õ❛ ỵ tt ❱➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝ ✈➔ ♣✲❛❞✐❝✳ ✷✳ ❚✐➳♣ tö❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ù♥❣ ❞ö♥❣ ❝õ❛ ❤❛✐ ✤à♥❤ ỵ tt t t ①→❝ ✤à♥❤ ❞✉② ♥❤➜t✳ ✾✹ ❉❛♥❤ ♠ư❝ ❈ỉ♥❣ tr➻♥❤ ❝õ❛ t→❝ ❣✐↔ ✤➣ ❝æ♥❣ ❜è ❧✐➯♥ q✉❛♥ ✤➳♥ ✤➲ t➔✐ ✶✳ ❬✺❪ ❱✉ ❍♦❛✐ ❆♥✱ 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❤↕♠ ◆❣å❝ ❍♦❛✱ ◆❣✉②➵♥ ❳✉➙♥ ▲❛✐ ✭✷✵✶✽✮✱ ✧❱➜♥ ✤➲ ♥❤➟♥ ❣✐→ trà ✈➔ ❞✉② ♥❤➜t ❝õ❛ t♦→♥ tû s❛✐ ♣❤➙♥ ✈➔ t➼❝❤ s❛✐ ♣❤➙♥ ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣ ❆r❝❤✐♠❡❞❡s✧✱ ❚↕♣ ❝❤➼ ❑❤♦❛ ❤å❝ ❈ỉ♥❣ ♥❣❤➺ ❱✐➺t ◆❛♠✱ ❱♦❧✉♠❡ ✻✵✱ ◆✉♠❜❡r ✻✱ ♣♣ ✶✲✹✳ ✻✳ ❬✼❪ ❱✉ ❍♦❛✐ ❆♥ ❛♥❞ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✽✮✱ ✧❆ ♥❡✇ ❝❧❛ss ♦❢ ✉♥✐q✉❡ r❛♥❣❡ s❡ts ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❆♥♥❛❧❡s ❯♥✐✈✳❙❝✐✳❇✉❞❛♣❡st✱ ❙❡❝t✳ ❈♦♠♣✱ ✹✼✱ ♣♣✳✶✵✾✲✶✶✻✳ ✾✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ P❤↕♠ ◆❣å❝ ❍♦❛✱ ◆❣✉②➵♥ ❳✉➙♥ ▲❛✐ ✭✷✵✶✽✮✱ ✧❱➜♥ ✤➲ ♥❤➟♥ ❣✐→ trà ✈➔ ❞✉② ♥❤➜t ❝õ❛ t♦→♥ tû s❛✐ ♣❤➙♥ ✈➔ t➼❝❤ s❛✐ ♣❤➙♥ ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣ ❆r❝❤✐♠❡❞❡s✧✱ ❚↕♣ ❝❤➼ ❑❤♦❛ ❤å❝ ❈æ♥❣ ♥❣❤➺ ❱✐➺t ◆❛♠✱ ❱♦❧✉♠❡ ✻✵✱ ◆✉♠❜❡r ✻✱ ♣♣ ✶✲✹✳ ❬✷❪ ◆❣✉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❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✽✮✱ ✧❆ ♥❡✇ ❝❧❛ss ♦❢ ✉♥✐q✉❡ r❛♥❣❡ s❡ts ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥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✉❡ ✶✱ ♣♣✳ ✶✕✶✹✳ ✾✻ ❬✾❪ ❱✉ ❍♦❛✐ ❆♥ ❛♥❞ ▲❡ ◗✉❛♥❣ ◆✐♥❤ ✭✷✵✶✻✮✱ ✧❖♥ ❢✉♥❝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❤✱ ✻✼✱ ♣♣✳ ✷✺✶✲✷✻✾✳ ❬✶✹❪ ❈♦st❡✲❘♦② ▼✳❋✳ ✭✶✾✾✵✮✱✧❆ ♥♦t❡ ♦♥ ❘✐tt✬s t❤❡♦r❡♠ ♦♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧s✧✱ ❏♦✉r♥❛❧ ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ❆❧❣❡❜r❛✱ ✻✽✱ ♣♣✳ ✷✾✸✲✷✾✻✳ ❬✶✺❪ ❉♦r❡② ❋✳ ❛♥❞ ❲❤❛♣❧❡s ●✳ ✭✶✾✼✷✮✱✧Pr✐♠❡ ❛♥❞ ❝♦♠♣♦s✐t❡ ♣♦❧②♥♦♠✐❛❧s✧✱ ❏✳ ❆❧❣❡❜r❛✱ ✷✽✱ ♣♣✳ ✽✽✲✶✵✶✳ ❬✶✻❪ ❊♥❣str♦♠ ❍✳❚✳ ✭✶✾✹✶✮✱✧P♦❧②♥♦♠✐❛❧ s✉❜st✐t✉t✐♦♥s✧✱ ❆♠❡r✳❏✳▼❛t❤✱ ✻✸✱ ♣♣✳ ✷✹✾✲✷✺✺✳ ❬✶✼❪ ❊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❤✳ ❆♣♣❧✱ ✹✹✱ ♣♣✳ ✽✷✸✲✽✸✶✳ ❬✷✵❪ ❋✉❥✐♠♦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✳ ❈✳ ❛♥❞ ❨❛♥❣ ❈✳ ❈✳ ✭✶✾✾✾✮✱ ✧ ❆ ✉♥✐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 ❑❤♦❛✐ ❛♥❞ ▼❛✐ ❱❛♥ ❚✉ ✭✷✵✵✹✮✱ ✧♣✲❛❞✐❝ ◆❡✈❛♥❧✐♥♥❛✲❈❛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✳ ❆✱ ✽✸✭✽✮✱ ♣♣✳✶✹✽✲✶✺✶✳ ✾✽ ❬✸✼❪ ▲❡✈✐ ❍✳ ✭✶✾✹✷✮✱✧❈♦♠♣♦s✐t❡ ♣♦❧②♥♦♠✐❛❧s ✇✐t❤ ❝♦❡❢❢✐❡♥ts ✐♥ ❛♥ ❛r❜✐tr❛r② ❢✐❡❧❞ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ ③❡r♦✧✱ ❆♠❡r✳❏✳▼❛t❤✱ ✻✹✱ ♣♣✳ ✸✽✾✲✹✵✵✳ ❬✸✽❪ ▲✐ 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✉❡♥❡ss ❛♥❞ ✈❛❧✉❡✲s❤❛r✐♥❣ ♦❢ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✹✸✱ ♣♣✳ ✾✸✽✲✾✺✵✳ ... ❱➜♥ ✤➲ ✸✳ ❚ø ✤â✱ ❝❤ó♥❣ tổ t ởt số ỵ ❘✐tt ✈➔ ù♥❣ ❞ö♥❣ ✈➔♦ ✈➜♥ ✤➲ ❞✉② ♥❤➜t✧ ✤➸ qt ự tr ỗ t❤í✐ ❣â♣ ♣❤➛♥ ❧➔♠ ♣❤♦♥❣ ♣❤ó t❤➯♠ ❝→❝ ❦➳t q✉↔ ự ỵ tt t t ởt số ỵ tữỡ tỹ ỵ tt ố ợ ♣❤➙♥ ❤➻♥❤... ❞ö♥❣ ❇ê ✤➲ ✶✳✷✳✸ ❝❤♦ ✭✶✳✶✾✮ t❛ s✉② r❛ tỗ t số ổ h s = hϕ, tù❝ ❧➔ Rr−1 ◦ · · · ◦ R1 ◦ f = hDs−1 ◦ · · · ◦ D1 ◦ g ❚✐➳♣ tư❝ ♥❤÷ ✈➟②✱ t❛ t❤➜② r➡♥❣ tỗ t số t ổ s R1 ◦ f = tDs−r+1 ◦ · · · ◦ D1... t❤ä❛ ♠➣♥ ✐✳ ▼é✐ Iv ✤➲✉ ❝❤ù❛ ➼t ♥❤➜t ❝❤➾ sè❀ ✐✐✳ ❱ỵ✐ j, i ∈ Iv ; t❛ ❝â fi = cij fj , ð ✤â cij số ổ ỵ ❘✐tt ✤è✐ ✈ỵ✐ ❝→❝ ✤❛ t❤ù❝ ❦✐➸✉ ❋❡r♠❛t✲❲❛r✐♥❣ ❝õ❛ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❳➨t ❝→❝ ✤❛ t❤ù❝ t❤✉➛♥