❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ❈⑩■ ❚❍➚ ▼■◆❍ P❍×❒◆● ✣➲ t➔✐ ◆Û❆ ◆❍➶▼ ❈Õ❆ ❑➐ ❉➚ ✣×❮◆● ❈❖◆● P❍➃◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ✣❸■ ❙➮ ❱⑨ ▲Þ ❚❍❯❨➌❚ ❙➮ ❇➻♥❤ ✣à♥❤ ✲ ◆➠♠ ✷✵✷✷ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ❈⑩■ ❚❍➚ ▼■◆❍ P❍×❒◆● ✣➲ t➔✐ ◆Û❆ ◆❍➶▼ ❈Õ❆ ❑➐ ❉➚ ✣×❮◆● ❈❖◆● P❍➃◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè ✈➔ ỵ tt số số õ ữớ ữợ ỗ ự ổ ①✐♥ ❝❛♠ ✤♦❛♥ ❧✉➟♥ ✈➠♥ ◆û❛ ♥❤â♠ ❝õ❛ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ❧➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ✈➔ tê♥❣ ủ t ữủ tỹ ữợ sỹ ữợ ỗ ự tr÷í♥❣ ✣↕✐ ❤å❝ ❚❤➠♥❣ ▲♦♥❣✳ ◆❤ú♥❣ ♣❤➛♥ sû ❞ư♥❣ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ✤÷đ❝ ♥➯✉ rã tr♦♥❣ ♣❤➛♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✈➔ tr➼❝❤ ❞➝♥ ❝ö t❤➸ tr♦♥❣ q✉→ tr➻♥❤ t❤➸ ❤✐➺♥ ♥ë✐ ❞✉♥❣✳ ❚æ✐ ①✐♥ ❝❤à✉ ❤♦➔♥ t♦➔♥ tr→❝❤ ♥❤✐➺♠ ♥➳✉ ❝â sü ❦❤æ♥❣ tr✉♥❣ t❤ü❝ ✈➲ ❝→❝ t❤ỉ♥❣ t✐♥ sû ❞ư♥❣ tr♦♥❣ q✉→ tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚→❝ ❣✐↔ ❈→✐ ❚❤à ▼✐♥❤ P❤÷ì♥❣ ▲í✐ ❝↔♠ ì♥ ❚r♦♥❣ q✉→ tr➻♥❤ ①➙② ❞ü♥❣ ✤➲ ❝÷ì♥❣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽✱ tỉ✐ ✤➣ ♥❤➟♥ ✤÷đ❝ r➜t ♥❤✐➲✉ sü ✤ë♥❣ ✈✐➯♥✱ ❦❤✉②➳♥ ❦❤➼❝❤ ✈➔ ❣✐ó♣ ✤ï ✤➸ ❝â t❤➸ t❤✉➟♥ ❧đ✐ ✤↕t ✤÷đ❝ ❦➳t q✉↔ ♠♦♥❣ ♠✉è♥✳ ❉♦ ✈➟②✱ tỉ✐ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ỡ qỵ ổ ỡ ♣❤á♥❣ ❜❛♥ ❝õ❛ ♣❤á♥❣ ✣➔♦ t↕♦ s❛✉ ✤↕✐ ❤å❝ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ◗✉② ◆❤ì♥ ✤➣ ❧✉ỉ♥ ❧✉ỉ♥ t❤❡♦ ❞ã✐ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ tr♦♥❣ s✉èt q✉→ tr➻♥❤ tæ✐ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ ✤➙②✳ ❍ì♥ ❤➳t ❧➔ ❧í✐ tr s s qỵ ổ ố qỵ ổ ❣✐↔♥❣ ✈✐➯♥ t❤➾♥❤ ❣✐↔♥❣ ✤➣ trü❝ t✐➳♣ ❣✐↔♥❣ ❞↕② ❝→❝ ❝❤✉②➯♥ ✤➲✱ ❣✐ó♣ tỉ✐ ❝â ✤÷đ❝ ♥➲♥ t↔♥❣ ❦✐➳♥ t❤ù❝ ✈ú♥❣ ❝❤➢❝ ✤➸ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ✣➦❝ ❜✐➺t✱ tổ ữủ tọ sỹ tr qỵ t ỡ s s t ỗ ự ữợ trỹ t tổ ữủ ỡ sỹ ữợ t t ✈➔ ❝❤✉ ✤→♦✱ ❧✉æ♥ s➤♥ s➔♥❣ ❧➢♥❣ ♥❣❤❡ ✈➔ ❞➝♥ t tổ ú ữợ tr ự ❝õ❛ ♠➻♥❤✳ ❇➯♥ ❝↕♥❤ ✤â✱ tỉ✐ ❝ơ♥❣ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ ❤é trđ ✈➔ ❦❤✉②➳♥ ❦❤➼❝❤ tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ✈✐➳t ❧✉➟♥ ✈➠♥ ♥➔②✳ ❱ỵ✐ t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ❝á♥ ❤↕♥ ❝❤➳✱ t❤ü❝ t✐➵♥ ❝æ♥❣ t→❝ tr♦♥❣ ❤♦➔♥ ❝↔♥❤ ✤↕✐ ❞à❝❤ ❈♦✈✐❞✲✶✾ ❣➙② ♥❤✐➲✉ trð ♥❣↕✐✱ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sõt tổ rt ữủ ỵ õ õ t tứ qỵ ổ ỗ ✈➔ ❜↕♥ ❜➧✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ◗✉② ◆❤ì♥✱ t❤→♥❣ ✵✽ ♥➠♠ ✷✵✷✷ ❚→❝ ❣✐↔ ❈→✐ ❚❤à ▼✐♥❤ P❤÷ì♥❣ ▼ö❝ ❧ö❝ ❉❛♥❤ ♠ö❝ ❝→❝ ❦➼ ❤✐➺✉✱ ❝❤ú ✈✐➳t t➢t ▼ð ✤➛✉ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝ì sð ✶✳✶ ❱➔♥❤ ❝→❝ ❝❤✉é✐ ❧ô② t❤ø❛ ❤➻♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❈→❝ ❜➜t ❜✐➳♥ ❝õ❛ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶ ❚❤❛♠ sè ❤â❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷ ❇ë✐ ❣✐❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✸ ❇➜t ❜✐➳♥ ▼✐❧♥♦r✱ ❜➜t ❜✐➳♥ ❉❡❧t❛ ✈➔ ❜➜t ❜✐➳♥ ❑❛♣♣❛ ✶✳✷✳✹ ◆û❛ ♥❤â♠ ❝õ❛ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤÷ì♥❣ ✷ ✣à♥❤ ❧➼ ♥û❛ ♥❤â♠ ✷✳✶ ❈→❝ ❦➳t q✉↔ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ỷ õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ❇➜t ❜✐➳♥ ▼✐❧♥♦r ✈➔ ♥û❛ ♥❤â♠ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ✷✳✸✳✶ ◆❤➙♥ tû ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❝ü❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✷ ❑➳t q✉↔ ❝❤➼♥❤ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ✶ ✷ ✷ ✺ ✺ ✽ ✶✶ ✷✶ ✷✷ ✷✷ ✷✺ ✸✶ ✸✶ ✸✸ ❈❤÷ì♥❣ ✸ ❈ỉ♥❣ t❤ù❝ ▼✐❧♥♦r ♠ð rë♥❣ ✸✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✺✷ ✸✳✶ ●✐↔✐ ❦➻ ❞à ❝õ❛ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✷ ❇➔✐ t♦→♥ ♠ð rë♥❣ tø ❝æ♥❣ t❤ù❝ ▼✐❧♥♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ▼ët sè ❦➼ ❤✐➺✉ ✈✐➳t t➢t char(K) ord f (t) gcd(a, b) f, g P1 h.o.t degy ψ → ✣➦❝ sè ❝õ❛ tr÷í♥❣ K✳ ❈➜♣ ❝õ❛ ộ ụ tứ ởt f (t) ìợ ợ ♥❤➜t ❝õ❛ a ✈➔ b✳ ■✤➯❛♥ s✐♥❤ ❜ð✐ f, g ✳ ❑❤ỉ♥❣ ❣✐❛♥ ①↕ ↔♥❤ ♠ët ❝❤✐➲✉ tr➯♥ tr÷í♥❣ K✳ õ ỡ trữợ õ y tr♦♥❣ ❝❤✉é✐ ❧ơ② t❤ø❛ ψ✳ ❚♦➔♥ ❝➜✉✳ ✣ì♥ ❝➜✉✳ ✐ ▼Ð ✣❺❯ ✣➲ t➔✐ ✧◆û❛ ♥❤â♠ ❝õ❛ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣✧ ❧➔ ♠ët ✤➲ t➔✐ ❝ê ✤✐➸♥✱ ♥❤÷♥❣ ✈➝♥ ✤❛♥❣ t❤✉ ❤ót sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝✱ ✤➦❝ ❜✐➺t ❧➔ ♥❤ú♥❣ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ sè ❞÷ì♥❣✳ ✧◆û❛ ♥❤â♠ ❝õ❛ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣✧ ❧➔ ♥û❛ ♥❤â♠ ❝õ❛ Z ✤÷đ❝ ①→❝ ✤à♥❤ ❤♦➔♥ t♦➔♥ ❜ð✐ ♠ët ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ f tr♦♥❣ K[[x; y]]✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ t❤ỉ♥❣ q✉❛ ❜ë✐ ❣✐❛♦ ✭✐♥t❡rs❡❝t✐♦♥ ♠✉❧t✐♣❧✐❝✐t②✮✱ ♠ët ❜➜t ❜✐➳♥ ✤↕✐ sè q✉❛♥ trå♥❣ tr♦♥❣ ✣↕✐ sè✱ ❍➻♥❤ ❤å❝ ✤↕✐ sè✱ ▲➼ t❤✉②➳t ❦➻ ❞à✳ ❈❤ó♥❣ ❧➔ ♥❤ú♥❣ t❤ỉ♥❣ t✐♥ tê ❤đ♣ ✧✤ì♥ ❣✐↔♥✧ ❞ị♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❜➜t ❜✐➳♥ q✉❛♥ trå♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ♥❤÷ ▼✐❧♥♦r ♥✉♠❜❡r (µ)✱ ❉❡❧t❛ ✐♥✈❛r✐❛♥ts (δ)✱ ❩❡t❛ ❢✉♥❝t✐♦♥ (ζ)✱✳✳✳▼ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ✤➲ t➔✐ ❧➔ ❤➺ t❤è♥❣ ❧↕✐ ♥❤ú♥❣ ❦➳t q✉↔ q✉❛♥ trå♥❣ ❧✐➯♥ q✉❛♥ ✤➳♥ ♥û❛ ♥❤â♠ ❝õ❛ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣✱ ①➙② ❞ü♥❣ ❝ỉ♥❣ t❤ù❝ ▼✐❧♥♦r tợ ữợ qt t ❧✐➯♥ q✉❛♥✳ ❉♦ ✈➟②✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t ợ ữỡ tự ỡ s P ợ ữỡ ♥➔② ✤÷đ❝ ❤➺ t❤è♥❣ ❧↕✐ tø ❬✻❪ ❝ị♥❣ ♠ët sè ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ t→❝ ❣✐↔✱ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ t❤❛♠ ❦❤↔♦ ❝ơ♥❣ ✤÷đ❝ tr➼❝❤ ❞➝♥ ❝ư t❤➸✳ ✶✳✶ ❱➔♥❤ ❝→❝ ❝❤✉é✐ ❧ơ② t❤ø❛ ❤➻♥❤ t❤ù❝ ❈❤♦ K ❧➔ ♠ët tr÷í♥❣ ✤â♥❣ ✤↕✐ sè✳ ❱➔♥❤ ❝→❝ ❝❤✉é✐ ❧ô② t❤ø❛ ❤➻♥❤ t❤ù❝ ♥ ❜✐➳♥ tr➯♥ tr÷í♥❣ K✱ K[[x1, , xn]]✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ K[[x1 , , xn ]] = cα xα | cα ∈ K, α = (α1 , , αn ) , xα = xα1 xαnn f= α∈Nn ❝ị♥❣ ✈ỵ✐ ❤❛✐ ♣❤➨♣ t♦→♥ ” + ” ✈➔ ”.” ♥❤÷ s❛✉✿ cα x α + • α∈Nn cα x α = α∈Nn cα x α • α∈Nn (cα + cα )xα , α∈Nn cα x α = α∈Nn cα cβ xα+β ✈ỵ✐ α + β = (α1 + β1, , αn + βn) α,β∈Nn ▼é✐ ❤↕♥❣ tû cαxα, cα = tr♦♥❣ ❜✐➸✉ ❞✐➵♥ ❝õ❛ f ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤ì♥ t❤ù❝ ❝➜♣ | α | ✈ỵ✐ | α |= α1 + + αn ✣❛ t❤ù❝ f ✤÷đ❝ ❣å✐ ❧➔ t❤✉➛♥ ♥❤➜t ♥➳✉ f ❧➔ tê♥❣ ❝õ❛ ❝→❝ ✤ì♥ t❤ù❝ ❝ị♥❣ ❝➜♣✳ ✣ì♥ t❤ù❝ ❝â ❝➜♣ ♥❤ä ♥❤➜t tr♦♥❣ ❜✐➸✉ ❞✐➵♥ ❝õ❛ f ✤÷đ❝ ❣å✐ ❧➔ ✤ì♥ t❤ù❝ ❞➝♥ ✤➛✉ ❝õ❛ f ✈➔ ♥➳✉ ✤ì♥ t❤ù❝ ❞➝♥ ✤➛✉ ❝õ❛ f ❝â ❤➺ sè ❜➡♥❣ ✶ t❤➻ f ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤❛ t❤ù❝ ♠♦♥✐❝✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❱ỵ✐ f ∈ K[[x1, , xn]]✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❜ë✐ ❝õ❛ f ✱ ❦➼ ❤✐➺✉ ❧➔ mt(f )✱ ♥❤÷ s❛✉ mt(f ) = {|α| = α1 + + αn | cα = 0} ❉➵ t❤➜② mt(f.g) = mt(f )+mt(g), ✈ỵ✐ ♠å✐ f, g ∈ K[[x1, , xn]] ❚ø ✤â t❛ ❝â ♠ët sè ❦➳t q✉↔ ❝ì ❜↔♥ s❛✉✳ ✷ ✸ ▼➺♥❤ ✤➲ ✶✳✷✳ ❈❤♦ f ∈ K[[x1, , xn]] ✈➔ m = x1 , , xn ❧➔ ✐✤➯❛♥ s✐♥❤ ❜ð✐ x1, , xn ❝õ❛ K[[x1, , xn]]✳ ❑❤✐ ✤â ✐✱ mt(f ) > ⇔ f ∈ m ✐✐✱ mt(f ) = ⇔ f ❦❤↔ ♥❣❤à❝❤✱ tù❝ ❧➔ tỗ t f K[[x1, , xn]] s f.f = ự ú ỵ r f ∈ m t❤➻ f = n fk xk ✈ỵ✐ fk ∈ f ∈ K[[x1, , xn]] k=1 ♥➯♥ mt(f ) > ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ mt(f ) > t❤➻ f = cα xα ∈ m P❤→t ❜✐➸✉ i, ❞➵ 0=α∈Nn ❞➔♥❣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❇➙② ❣✐í ❣✐↔ sû f K[[x1, , xn]] õ tỗ t↕✐ f −1 ∈ K[[x1, , xn]] s❛♦ ❝❤♦ f.f −1 = ⇒ mt(f.f −1 ) = mt(1) = ⇒ mt(f ) + mt(f −1 ) = ⇒ mt(f ) = mt(f −1 ) = ◆❣÷đ❝ ❧↕✐✱ ✈ỵ✐ ♠é✐ f ∈ K[[x1, , xn]] t❛ ✈✐➳t f = f0 + f1 + = fi i≥0 tr♦♥❣ ✤â fi ❧➔ ❝→❝ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ i✳ ❚❛ s➩ ①➙② ❞ü♥❣ g = go + g1 + = gi t❤ä❛ ♠➣♥ f g = ❜➡♥❣ ❝→❝❤ q✉② ♥↕♣ ♥❤÷ i≥0 s❛✉✿ ❱➻ mt(f ) = ♥➯♥ f0 = 0✳ ❈❤å♥ g0 = f0−1✳ ❚❛ ❝â f0g0 = mod m ●✐↔ sû ✤➣ ①➙② ❞ü♥❣ ✤÷đ❝ g0, g1, , gn s❛♦ ❝❤♦ (f0 + + fn )(g0 + + gn ) = mod mn+1 ⇔ f0 g0 = k ❈❤å♥ gn+1 −1 = f0 n+1 fi gn+1−i fi gk−i = 0, ∀1 ≤ k ≤ n i=0 ✱ t❛ ❝â i=1 n+1 n+1 fi gn+1−i = f0 gn+1 + i=0 fi gn+1−i = i=1 ✹ ❚ø ❣✐↔ t❤✐➳t q✉② ♥↕♣ t❛ ❝â f0 g0 = k fi gk−i = 0, ∀1 ≤ k ≤ n + ⇔ (f0 + + fn+1 ) (g0 + + gn+1 ) = mod mn+2 i=0 ❉♦ ✤â (f0 + + fn)(g0 + + gn) = mod mn+1 ✈ỵ✐ ♠å✐ n ≥ 0✳ ❱➟② t❛ ①➙② ❞ü♥❣ ✤÷đ❝ g t❤ä❛ f.g = 1✱ ❤❛② f ❦❤↔ ♥❣❤à❝❤✳ ▼➺♥❤ ✤➲ ✶✳✸✳ K[[x1, , xn]] ❧➔ ♠ët ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣✱ ♥❣❤➽❛ ❧➔ K[[x1, , xn]] ❝â ❞✉② ♥❤➜t ♠ët ✐✤➯❛♥ ❝ü❝ ✤↕✐✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ m = x1, , xn ❧➔ ✐✤➯❛♥ ❝ü❝ ✤↕✐ ❞✉② ♥❤➜t ❝õ❛ K[[x1 , , xn ]]✳ rữợ t t ú ỵ r N ❝õ❛ K[[x1 , , xn ]] ❝❤ù❛ ♠ët ♣❤➛♥ tû c ∈ K ✈➔ c ❦❤→❝ ✵ t❤➻ N = K[[x1, , xn]]✳ ❚➼♥❤ ❝ü❝ ✤↕✐✿ ●✐↔ sû N ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ K[[x1, , xn]] s❛♦ ❝❤♦ m N ❑❤✐ ✤â cα xα , c0 = ∃f ∈ N : f ∈ /m⇒f = α∈Nn cα x α = c0 + f = c0 + α∈Nn ,α=0 ❚❛ t❤➜② f2 ∈ m ⇒ f2 ∈ N ⇒ c0 = f − f2 ∈ N ✳ ❉♦ ✤â N = K[[x1, , xn]]✳ ❱➟② m ❧➔ ♠ët ✐✤➯❛♥ ❝ü❝ ✤↕✐✳ ❚➼♥❤ ❞✉② ♥❤➜t✿ ●å✐ N ❧➔ ♠ët ỹ K[[x1, , xn]] sỷ tỗ t↕✐ f ∈ N ♠➔ f ∈ / m✱ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷ t❤➻ f ❦❤↔ ♥❣❤à❝❤ ✳ ❑❤✐ ✤â = f.f −1 ∈ N ⇒ N = K[[x1 , , xn ]] ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ ❝ü❝ ✤↕✐ ❝õ❛ N ✳ ❱➟② N ⊂ m✳ ❱➻ N ❧➔ ❝ü❝ ✤↕✐ ♥➯♥ N = m✱ ✤✐➲✉ ♥➔② ❝❤ù♥❣ tä m ❧➔ ✐✤➯❛♥ ❝ü❝ ✤↕✐ ❞✉② ♥❤➜t ❝õ❛ K[[x1 , , xn ]]✳ ❱ỵ✐ AutK (K[[x1, , xn]]) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ tü ✤➥♥❣ ❝➜✉ ✤↕✐ sè tr➯♥ K[[x1, , xn]]✱ t❛ ✤à♥❤ ♥❣❤➽❛ ♠ët sè q✉❛♥ ❤➺ ❣✐ú❛ ❝→❝ ❝❤✉é✐ ❧ơ② t❤ø❛ tr♦♥❣ K[[x1, , xn]] ♥❤÷ s❛✉✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❈❤♦ f, g ∈ K[[x1, , xn]]✱ f, g ✤÷đ❝ ❣å✐ ❧➔ ✐✱ t÷ì♥❣ ✤÷ì♥❣ ♣❤↔✐✱ f r g tỗ t AutK (K[[x1, , xn]]) t❤ä❛ ♠➣♥ f = φ(g) ✸✽ P❤➨♣ ❜✐➳♥ ✤ê✐ ❝❤➦t f ❝õ❛ f ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉ ∼ v) ✐✱ ◆➳✉ m < n t❤➻ f (u, v) = f (uv, vm ∼ ✐✐✱ ◆➳✉ m ≥ n t❤➻ f (u, v) = f (u,unuv) ∼ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ ♥➳✉ m < n t❤➻ ∼ ∼ f = αum + βv n−m + h.o.t ⇒ mt f = {m, n − m} ≤ {m, n} = mt(f ) ◆❤➟♥ ①➨t tr➯♥ ✤÷đ❝ ♣❤→t tr✐➸♥ t❤➔♥❤ ✣à♥❤ ❧➼ ❣✐↔✐ ữ s ợ f K[[x, y]] t q tỗ t ởt f1, f2, , fN ∈ K[[x, y]] t❤ä❛ ♠➣♥ ✐✱ fi+1 ❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝❤➦t ❝õ❛ fi✱ ✈ỵ✐ ♠å✐ i < N ✳ ✐✐✱ fN trì♥✱ ♥❣❤➽❛ ❧➔ mt(f ) = f1, f2, , fN tỗ t↕✐ tr♦♥❣ ✤à♥❤ ❧➼ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❣✐↔✐ ❦➻ ❞à ❝õ❛ f✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❛♠ ❦❤↔♦ ❬✽✱ tr✳✷✼❪✳ ▼➺♥❤ ✤➲ ✸✳✷✳ ❈❤♦ f = f0, f1, , fN ❧➔ ♠ët ❣✐↔✐ ❦➻ ❞à ❝õ❛ f ✳ ❑❤✐ ✤â ✐✱ ✐✐✱ δ(f ) = mt(f ) (mt(f ) − 1) + δ (f1 ) N −1 δ(f ) = i=0 mt(fi ) (mt(fi ) − 1) ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❛♠ ❦❤↔♦ ❬✻❪✱ ❈❤÷ì♥❣ ✶✱ ▼➺♥❤ ✤➲ ✸✳✸✹✳ ▼➦t ❦❤→❝✱ ✈ỵ✐ f ❜➜t ❦❤↔ q✉② t❤➻ f ❝ơ♥❣ ❜➜t ❦❤↔ q✉②✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû ∼ ord (f (x, 0)) = m < n = ord (f (0, y)) , t❛ ❝â f = αum + βvn−m + h.o.t f q t ữủ ữợ ❞↕♥❣ ∼ ∼ ∼ f (u, v) = g(u, v).h(u, v), ✸✾ tr♦♥❣ ✤â g ❜➜t ❦❤↔ q✉② ✈➔ h ❦❤æ♥❣ ❦❤↔ ♥❣❤à❝❤ t❤➻ t❛ ❣å✐ (x(t), y(t)) ❧➔ ♠ët t❤❛♠ sè ❤â❛ ❝õ❛ g✳ ❑❤✐ ✤â ∼ ord x(t) = ord g(0, y) < ord f (0, y) = n − m ✈➔ ∼ ord y(t) = ord g(x, 0) < ord f (x, 0) = m ✣➦t α(t) = x(t) y(t) ✈➔ β(t) = y(t)✱ t❛ ❝â f (α(t), β(t)) = f (x(t), y(t)) [y(t)]n = ▼➔ ord α(t) = ord x(t) + ord y(t) < n ✈➔ ord β(t) = ord y(t) < m ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ ♣❤ê ❞ư♥❣ ✤è✐ ✈ỵ✐ ♠ët t❤❛♠ sè ❤â❛ ❝õ❛ f ✳ ❱➟② ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝❤➦t ❝õ❛ f ❜➜t ❦❤↔ q✉② ❧✉æ♥ ❧✉æ♥ ❜➜t ❦❤↔ q✉②✳ ∼ ✸✳✷ ❇➔✐ t♦→♥ ♠ð rë♥❣ tø ❝æ♥❣ t❤ù❝ ▼✐❧♥♦r ❈→❝ ❦➳t q✉↔ tr♦♥❣ ♠ư❝ ♥➔② ✤÷đ❝ t→❝ ❣✐↔ ✤å❝✱ ❤✐➸✉✱ tr➻♥❤ ❜➔② ✈➔ ❣✐↔✐ t❤➼❝❤ ❧↕✐ ♠ët sè ✈➜♥ ✤➲ tø ❬✽❪✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✸✳ ❈❤♦ f t❤✉ ❣å♥ ✈ỵ✐ ♣❤➙♥ t➼❝❤ f = f1 fr ✳ ✣➦❝ sè p = char(K) ≥ ✤÷đ❝ ❣å✐ ❧➔ ✐✱ tèt t❤❡♦ ♥❣❤➽❛ ❜ë✐ ✤è✐ ✈ỵ✐ f ✱ ❦➼ ❤✐➺✉ ♠✲❣♦♦❞✱ ♥➳✉ mt(fi) ≡ 0(mod p) ✈ỵ✐ ♠å✐ i = 1, , r ✐✐✱ tèt t❤❡♦ ♥❣❤➽❛ ❜ë✐ ❣✐❛♦ ✤è✐ ✈ỵ✐ f ✱ ❦➼ ❤✐➺✉ ✐♠✲❣♦♦❞✱ ♥➳✉ ✈ỵ✐ ♠å✐ i = 1, , r t❛ ❝â i0 (fi , x) ≡ 0(mod p) ❤♦➦❝ i0 (fi , y) ≡ 0(mod p) ✐✐✐✱ tèt t❤❡♦ ♥❣❤➽❛ ❜ë✐ ❣✐❛♦ ♣❤↔✐ ✤è✐ ✈ỵ✐ f tỗ t g ∼r f s❛♦ ❝❤♦ p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ g ú ỵ r tr ✈ỵ✐ ✤➦❝ sè ✵✱ ♥❣❤➽❛ ❧➔ p = ❧➔ ♠✲❣♦♦❞✱ ✐♠✲❣♦♦❞✱ ✐♠✲❣♦♦❞ ♣❤↔✐ ✤è✐ ✈ỵ✐ ♠å✐ f t❤✉ ❣å♥✳ ❍ì♥ ♥ú❛ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝→❝ ✤à♥❤ ♥❣❤➽❛ tr➯♥ t❤➸ ❤✐➺♥ ♥❤÷ s❛✉ ✧♠✲❣♦♦❞✧ ⇒ ✧✐♠✲❣♦♦❞✧ ⇒ ✧✐♠✲❣♦♦❞ ♣❤↔✐✧✳ ❱ỵ✐ char(f ) = p ≥ 0✱ t❛ s➩ ✤à♥❤ ♥❣❤➽❛ ❜➜t ❜✐➳♥ ●❛♠♠❛ ❝õ❛ ♠ët ❦➻ ❞à f ∈ K[[x, y]] t ỗ tớ tr t q✉↔ ❝â ✤÷đ❝ ❣✐ú❛ ❜➜t ❜✐➳♥ ●❛♠♠❛ ✈➔ ❝→❝ ❜➜t ❜✐➳♥ ✤➣ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛✳ ✹✵ ✣à♥❤ ♥❣❤➽❛ ✸✳✹✳ ❈❤♦ f ∈ K[[x, y]] t❤✉ ❣å♥✳ ❙è ∼γ x,y (f ) ✭❤♦➦❝ ∼γ(f ) ♥➳✉ {x, y} ✤➣ ✤÷đ❝ ❝❤➾ rã✮ ❝õ❛ f ✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉ ∼ ∼ • γ(x) := 0; γ(y) := 0✳ • ◆➳✉ f ❧➔ ❜➜t ❦❤↔ q✉② ✈➔ t❤✉➟♥ t✐➺♥ ✭♥❣❤➽❛ ❧➔ i0 (f, x), i0 (f, y) < +∞✮ t❤➻ ∼ γ(f ) := {i0 (f, fx ) − i0 (f, y) + 1, i0 (f, fy ) − i0 (f, x) + 1} • ◆➳✉ f = f1 fr , t❤➻ r ∼ ∼ γ(fi ) + γ(f ) = i=1 i0 (fi , fj ) − r + j=i ✣à♥❤ ♥❣❤➽❛ ✸✳✺✳ ❇➜t ❜✐➳♥ ●❛♠♠❛ ❝õ❛ ❦➻ ❞à f t❤✉ ❣å♥✱ ❦➼ ❤✐➺✉ γ(f )✱ ❧➔ ❣✐→ trà ∼ ♥❤ä ♥❤➜t t❤✉ ✤÷đ❝ q✉❛ t➜t ❝↔ ❝→❝ ♣❤➨♣ ✤ê✐ tå❛ ✤ë X, Y ✳ ❚ø ✤à♥❤ ♥❣❤➽❛ t❛ t❤➜②✱ γ(f ) ≤ ∼γ(f )✱ ợ tự r s ữủ tr tr t q s ỡ ỳ tỗ t↕✐ g ∼r f s❛♦ ❝❤♦ γ(f ) = ∼ ∼ γ(g)✳ ▼➦t ❦❤→❝✱ sè γ ♣❤ö t❤✉ë❝ ✈➔♦ ♣❤➨♣ ❝❤å♥ tå❛ ✤ë✱ ♥❣❤➽❛ ❧➔ ♥â ❦❤æ♥❣ ❜➜t ❜✐➳♥ ✤è✐ ợ q tữỡ ữỡ ợ f = x3 + x4 + y5 ✈➔ ∼ ∼ g = (x + y)3 + (x + y)4 + y ✱ t❛ ❝â f ∼r g ♥❤÷♥❣ γ(f ) = = γ(g) = ❚✉② ♥❤✐➯♥ t❛ s➩ t❤➜② ∼γ(f ) = ∼γ(g), ∀g ∼c f ♥➳✉ char(K) = p ❧➔ ♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ✳ ▼➦t ❦❤→❝✱ ✈ỵ✐ ∼ ∼ ∼ u ❦❤↔ ♥❣❤à❝❤✱ γ(u) = ✈➔ γ(u.f ) = γ(f )✳ ❱➟② sè γ ❧➔ t ố ợ q tữỡ ữỡ t ❚❛ ♥❤➢❝ ❧↕✐ ♠ët sè ❦➳t q✉↔ ✈➲ ❝→❝ ❜➜t trữợ ố q ỳ ú ợ t ú ỵ ◆➳✉ f t❤✉ ❣å♥ ✈ỵ✐ ♣❤➙♥ t➼❝❤ f = f1 fr t❤➻ γ X,Y (f ) r i0 (f, g) = ∀g ∈ K[[x, y]]; i0 (fi , g) , i=1 r 2δ(f ) = 2δ (fi ) + i=1 r κ(f ) = i0 (fi , fj ) ; j=i κ (fi ) + i=1 i0 (fi , fj ) j=i ▼➺♥❤ ✤➲ ✸✳✼✳ ❱ỵ✐ f ∈ K[[x, y]] t❤✉ ❣å♥✱ t❛ ❧✉ỉ♥ ❝â ∼ γ(f ) ≤ γ(f ) ≤ κ(f ) − mt(f ) + ✈ỵ✐ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ p ❧➔ ♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ✳ ✹✶ ❈❤ù♥❣ ♠✐♥❤✳ ✣→♥❤ ❣✐→ γ(f ) ≤ ∼γ(f ) ❤✐➸♥ ♥❤✐➯♥ t❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ ❝á♥ ữủ t t ữợ ữ s ữợ ✶✿ ●✐↔ sû f ❧➔ ❜➜t ❦❤↔ q✉②✱ t❛ ❝â mt(f ) = {i0(f, x), i0(f, fy )}✳ ❉♦ ✤â ∼ γ(f ) = {i0 (f, fx ) − i0 (f, y) + 1; i0 (f, fy ) − i0 (f, x) + 1} ≤ {i0 (f, fx ) − mt(f ) + 1; i0 (f, fy ) − mt(f ) + 1} = {i0 (f, fx ) , i0 (f, fy )} − mt(f ) + = κ(f ) − mt(f ) + ●✐↔ sû p ❧➔ ♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ✳ ●å✐ (x(t), y(t)) ❧➔ ♠ët t❤❛♠ sè ❤â❛ ❝õ❛ f ✳ ●✐↔ t❤✐➳t r➡♥❣ m = mt(f ) = ord x(t) ≤ ord y(t) ❱➻ f (x(t), y(t)) = ♥➯♥ fx (x(t), y(t)) x (t) + fy (x(t), y(t)) y (t) = ❉♦ ✤â ord (fx (x(t), y(t)) x (t)) = ord (fy (x(t), y(t)) y (t)) ⇒ i0 (f, fx ) + ord x (t) = i0 (f, fy ) + ord y (t) ⇒ i0 (f, fx ) − ord y (t) = i0 (f, fy ) − ord x (t) ▼➔ ord x (t) = ord x(t) − ✈➔ ord y (t) ≥ ord y(t) − ♥➯♥ i0 (f, fx ) − ord y(t) ≥ i0 (f, fy ) − ord x(t) = i0 (f, fy ) − m s✉② r❛ ∼ γ(f ) = i0 (f, fy ) − m + ❑❤✐ ✤â ✈➔ i0 (f, fy ) ≤ i0 (f, fx) ∼ γ(f ) = i0 (f, fy ) − m + = {i0 (f, fx ) , i0 (f, fy )} − m + = κ(f ) mt(f ) + ữợ f t❤✉ ❣å♥ ✈ỵ✐ ♣❤➙♥ t➼❝❤ f = f1 fr ✱ tr♦♥❣ ✤â fi ❧➔ ❜➜t ❦❤↔ q✉② ✈ỵ✐ ♠å✐ i = 1, r ❙û ❞ư♥❣ ❦➳t q✉↔ ❝â ✤÷đ❝ tø ữợ ú ỵ t õ r γ(f ) = ∼ γ (fi ) + i=1 r ≤ i0 (fi , fj ) − r + j=i κ (fi ) − mt (fi ) + + i=1 i0 (fi , fj ) − r + j=i ✹✷ r = i=1 r i0 (fi , fj ) − κ (fi ) + mt (fi ) + i=1 j=i = κ(f ) − mt(f ) + ❉➵ t❤➜② ✤➥♥❣ t❤ù❝ ❝ô♥❣ ①↔② r❛ ❦❤✐ p ❧➔ ♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ✳ ❚✐➳♣ t❤❡♦ t❛ ❝❤ù♥❣ ♠✐♥❤ γ(f ) = ∼γ(f ) ♥➳✉ p ❧➔ ♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f t g tữỡ ữỡ ợ f t❤ä❛ γ(f ) = ∼γ(g)✳ ❱➻ g ∼r f ⇒ g ∼c f ⇒ mt(f ) = mt(g) ♥➯♥ p ❝ơ♥❣ ❧➔ ♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ g✳ ❚❤❡♦ ❦➳t q✉↔ ❝õ❛ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ t❛ ✤÷đ❝ ∼ γ(g) = κ(g) − mt(g) + ✈➔ ∼γ(f ) = κ(f ) − mt(f ) + ▼➔ sè ❑❛♣♣❛ ❝ô♥❣ ❧➔ ♠ët t ố ợ q tữỡ ữỡ t ✈➟② ∼ ∼ γ(f ) = γ(g) = κ(g) − mt(g) + = κ(f ) − mt(f ) + = γ(f ) ❚❛ ❦➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤ t↕✐ ✤➙②✳ ❈❤ù♥❣ ♠✐♥❤ ❝õ❛ ♠➺♥❤ ✤➲ tr➯♥ ❝ô♥❣ ✤➣ ❝❤➾ r❛ r➡♥❣ sè ∼γ ❧➔ ❜➜t ❜✐➳♥ ✤è✐ ✈ỵ✐ q✉❛♥ tữỡ ữỡ t tr ợ ộ ụ t❤ø❛ ♠➔ p ❧➔ ♠✲❣♦♦❞✱ ❝ư t❤➸ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉ ▼➺♥❤ ✤➲ ✸✳✽✳ ❈❤♦ f ∈ K[[x, y]] t❤✉ ❣å♥ t❤ä❛ ♠➣♥ p ❧➔ ♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ✳ ❑❤✐ ✤â ∼ ∼ γ(f ) = γ(g) ✈ỵ✐ ♠å✐ g c f rữợ ợ t q t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❜ê ✤➲ ❝❤♦ ♣❤➨♣ s♦ s→♥❤ sè ●❛♠♠❛ q✉❛ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝❤➦t✳ ❇ê ✤➲ ✸✳✾✳ ❈❤♦ f ∈ K[[x, y]] ❜➜t ❦❤↔ q✉② ❝â i0(f, x) = i0(f, y) := m ●å✐ g t❤✉ë❝ K[[x, y]] s❛♦ ❝❤♦ f (x, y) = g(x, x y) ợ (, ) P1 ữợ t✐➳♣ t✉②➳♥ ❝õ❛ f ✳ ❑❤✐ ✤â ✐✱ m = i0(g, x) < i0(g, y) ✐✐✱ ∼γ(f ) ≥ ∼γ(g) ✐✐✐✱ ◆➳✉ ✤➦❝ sè p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ g ữ ổ ố ợ f t (f ) > γ(g) ✹✸ ❈❤ù♥❣ ♠✐♥❤✳ ✐✱ ❳➨t ♣❤➨♣ ✤ê✐ tå❛ ✤ë φ ✤÷đ❝ ❝❤♦ ❜ð✐ φ(x) = x ✈➔ φ(y) = αx−βy ❚❛ ❝â i0 (g, x) = i0 (φ(g), φ(x)) = i0 (g(αx − βy, y), x) = i0 (f, x) = m, ✈➔ i0(g, y) = i0 (φ(g), φ(y)) = i0 (g(αx − βy, y), αx − βy) = i0(f, αx−βy) > m ✐✐✱ ●å✐ (x(t), y(t)) ❧➔ ♠ët t❤❛♠ sè ❤â❛ ❝õ❛ f ✳ ❑❤✐ ✤â X(t) = x(t) Y (t) = αx(t) − βy(t) ❧➔ ♠ët t❤❛♠ sè ❤â❛ ❝õ❛ g✳ ❱➻ f (x, y) = g(x, αx − βy) ♥➯♥ fx (x, y) = gx (x, αx − βy) + αgy (x, αx − βy) fy (x, y) = −βgy (x, αx − βy) ❱➔ ✈➻ t❤➳ fx (x(t), y(t)) = gx (X(t), Y (t)) + αgy (X(t), Y (t)) fy (x(t), y(t)) = −βgy (X(t), Y (t)) ❚❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉ • ◆➳✉ i0 (f, fx ) ≥ i0 (f, fy ) t❤➻ ∼ γ(f ) = {i0 (f, fx ) − i0 (f, y) + 1, i0 (f, fy ) − i0 (f, x) + 1} = i0 (f, fy ) − m + = i0 (g, gy ) − i0 (g, x) + ∼ ≥ γ(g) • ◆➳✉ i0 (f, fx) < i0 (f, fy ) s✉② r❛ ordfx (x(t), y(t)) < ordfy (x(t), y(t)) = ordgy (X(t), Y (t)) , t❛ ❝â ordfx (x(t), y(t)) = ordgx (X(t), Y (t)) < ordgy (X(t), Y (t)) , tù❝ ❧➔ i0 (f, fx) = i0 (g, gx) < i0 (g, gy ) ❉♦ ✤â ∼ γ(g) = {i0 (g, gx ) − i0 (g, y) + 1, i0 (g, gx ) − i0 (g, y) + 1} = i0 (g, gx ) − i0 (g, y) + < i0 (f, fx ) − i0 (g, x) + = i0 (f, fx ) − m + ∼ = γ(f ) ✹✹ ✐✐✐✱ ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ♠➺♥❤ ✤➲ ✐✐✱ ♥➳✉ i0 (f, fx) < i0 (f, fy ) t❤➻ ∼γ(f ) > ∼γ(g) ❚❛ ❝❤➾ ❝➛♥ ①➨t tr÷í♥❣ ❤đ♣ i0 (f, fx) ≥ i0 (f, fy ) ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ t❛ ❝â ∼ γ(f ) = i0 (g, gy ) − i0 (g, x) + ❱➻ p ❦❤ỉ♥❣ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ♥➯♥ i0(g, x) = m ≡ (modp)✳ ▼➔ p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ g ♥➯♥ i0(g, y) ≡ 0(modp) ❉♦ ✤â✱ ordY (t) = ordY (t) − = i0 (g, y) − ✈➔ ordX (t) > ordX(t) − = i0 (g, x) − ▼➦t ❦❤→❝✱ g (X(t), Y (t)) = ⇒ X (t).gx (X(t), Y (t)) + Y (t).gy (X(t), Y (t)) = ⇒ ⇒ ordX (t) + ordgx (X(t), Y (t)) = ordY (t) + ordgy (X(t), Y (t)) i0 (g, gx ) − ordY (t) = i0 (g, gy ) − ordX (t) ❈ị♥❣ ✈ỵ✐ ❦➳t q✉↔ tr➯♥ t❛ ✤÷đ❝ i0 (g, gx ) − i0 (g, y) < i0 (g, gy ) − i0 (g, x) ❑❤✐ ✤â✱ ∼ γ(g) = {i0 (g, gx ) − i0 (g, y) + 1, i0 (g, gx ) − i0 (g, y) + 1} = i0 (g, gx ) − i0 (g, y) + < i0 (g, gy ) − i0 (g, x) + ∼ = γ(f ) ❑➳t q✉↔ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳ ❇ê ✤➲ ✸✳✶✵✳ ❈❤♦ f ∈ K[[x, y]] ❜➜t ❦❤↔ q✉② ✈➔ ∼f ❧➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝❤➦t ❝õ❛ f ✳ ❑❤✐ ✤â ∼ ∼ ∼ γ(f ) ≥ m2 − m + γ f ✈ỵ✐ m = mt(f ) ◆➳✉ i0(f, x) = i0(f, y) t❤➻ ✐✱ ✐✐✱ ∼ ∼ ∼ γ(f ) = m2 − m + γ f ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ∼ p ✹✺ ❈❤ù♥❣ ♠✐♥❤✳ ✐✱ ◆➳✉ i0(f, x) = +∞ ❤♦➦❝ i0(f, y) = +∞ t❤➻ f = x.u ❤♦➦❝ f = y.v ✈ỵ✐ u, v ♥➔♦ ✤â ❦❤↔ ♥❣❤à❝❤ ✭✈➻ f ❜➜t ❦❤↔ q✉②✮ ✈➔ ❜ê ✤➲ ❤✐➸♥ ♥❤✐➯♥ ∼ ✤ó♥❣✳ ❇➙② ❣✐í ❣✐↔ sû i0(f, x) < i0(f, y) < +∞ ✈➔ f (u, uv) = umf (u, v) ❑❤✐ ✤â ∼ ∼ fx (u, uv) + vfy (u, uv) = mum−1 f (u, v) + um f u (u, v) ✈➔ ∼ ufy (u, uv) = um f v (u, v) ❙✉② r❛ ∼ ∼ xfx (x, y) + yfy (x, y) = mum f (u, v) + um uf u (u, v) ✈➔ ∼ yfy (x, y) = um v f v (u, v) , tr♦♥❣ ✤â x = u ✈➔ y = uv✳ ∼ ●å✐ (u(t), v(t)) ❧➔ ♠ët t❤❛♠ sè ❤â❛ ❝õ❛ f ✳ ❑❤✐ ✤â x(t) = u(t) ✈➔ y(t) = u(t)v(t) ❧➔ ♠ët t❤❛♠ sè ❤â❛ ❝õ❛ f ✳ ❚❛ ❝â ∼ x(t)fx (x(t), y(t)) + y(t)fy (x(t), y(t)) = u(t)m u(t)f u (u(t), v(t)) ✈➔ ∼ y(t)fy (x(t), y(t)) = u(t)m v(t)f v (u(t), v(t)) ❱ỵ✐ (α : β) ∈ P1 t❛ ❝â αx(t)fx (x(t), y(t)) + (α + β)y(t)fy (x(t), y(t)) ∼ ∼ = u(t)m αu(t)f u (u(t), v(t)) + βv(t)f v (u(t), v(t)) ▲➜② ❝➜♣ ❝õ❛ ❤❛✐ ✈➳ ✤➥♥❣ tự tr ữ ỵ ord u(t) = ord x(t) = i0(f, x) = m✱ ❝ị♥❣ ✈ỵ✐ t➼♥❤ ❝❤➜t ❝õ❛ ❜ë✐ ❣✐❛♦ ✤è✐ ✈ỵ✐ t❤❛♠ sè ❤â❛ t❛ ❝â ∼ ∼ ∼ i0 (f, αxfx + (α + β)yfy ) = m2 + i0 f , αuf u + βv f v ❇➯♥ ❝↕♥❤ ✤â i0 (f, x) + i0 (f, y) = ord x(t) + ord y(t) = ord u(t) + ord u(t) + ord v(t) ∼ ∼ = m + i0 f , u + i0 f , v ✹✻ ▼➔ f t❤✉➟♥ t✐➺♥ ♥➯♥ ∼ γ(f ) = i0 (f, αxfx + (α + β)yfy ) − i0 (f, x) − i0 (f, y) + ∼ ∼ ∼ ∼ ∼ = m2 − m + i0 f , αuf u + βv f v − i0 f , u − i0 f , v + ∼ ∼ = m2 − m + γ f ✐✐✱ ❑❤➥♥❣ ✤à♥❤ ❝â ✤÷đ❝ tø ❤❛✐ ✤➥♥❣ t❤ù❝ ∼ i0 (f, x) = ord x(t) = ord u(t) = i0 f , u ✈➔ ∼ ∼ i0 (f, y) = ord y(t) = ord u(t) + ord v(t) = i0 f , u + i0 f , v ❇➙② ❣✐í t❛ ①➨t tr÷í♥❣ ❤đ♣ i0(f, x) = i0(f, y) ( : ) ữợ t t ❝õ❛ f ✈➔ g ∈ K[[x, y]] t❤ä❛ ♠➣♥ f (x, y) = g(x, αx − βy) ❚❤❡♦ ❇ê ✤➲ ✸✳✾ t❛ ❝â ∼ ∼ i0 (g, x) < i0 (g, y) ✈➔ γ(f ) < γ(g) ❚ø ❝❤ù♥❣ ♠✐♥❤ ✐✱ t❛ ❝â ∼ ∼ ∼ γ(g) = m2 − m + γ g , ✈ỵ✐ ∼g ❧➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝❤➦t ❝õ❛ g✳ ∼ ▼➦t ❦❤→❝✱ f t↕✐ ✤✐➸♠ (β, α) ❧➔ trị♥❣ ✈ỵ✐ ∼g t↕✐ ✤✐➸♠ (1 : 0) ❚❤➟t ✈➟②✱ ∼ f u, u v + f (u, v) = um ❉♦ ✤â✱ ∼γ(f ) ≥ ∼γ(g) ≥ m2 − m + ∼γ ∼ g α β = g (u, −βuv) ∼ ∼ = g(u, v) um ∼ ∼ = m2 − m + γ f ú ỵ r số t ụ õ t q tữỡ tü ✤÷đ❝ ♣❤→t ❜✐➸✉ tr♦♥❣ ▼➺♥❤ ✤➲ ✸✳✷✳ ❚✉② ♥❤✐➯♥ tữỡ tỹ ố ợ số ❧➔ ♠ët ❝➙✉ ❤ä✐✳ ❚❛ ❝ị♥❣ ✤➳♥ ✈ỵ✐ ❦➳t q✉↔ ❝❤➼♥❤ ✤÷đ❝ ♣❤→t ❜✐➸✉ ❝❤♦ ❜➜t ❜✐➳♥ ●❛♠♠❛ tr♦♥❣ ❤❛✐ ✤à♥❤ ❧➼ s❛✉ ✳ ✣à♥❤ ❧➼ ✸✳✶✶✳ ❈❤♦ f ∈ K[[x, y]] t❤✉ ❣å♥✳ ❑❤✐ ✤â ∼ γ(f ) ≥ 2δ(f ) − r(f ) + ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✤➦❝ sè p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ t✐➳♥ ❤➔♥❤ ❝❤ù♥❣ t ữợ ữợ t f ❧➔ ❜➜t ❦❤↔ q✉② ✈➔ s➩ q✉② ♥↕♣ t❤❡♦ sè δ(f ) r➡♥❣ ∼γ(f ) ≥ 2δ(f ), ✹✼ ỗ tớ r (f ) = 2(f ) ✈➔ ❝❤➾ ❦❤✐ p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ✳ ◆➳✉ δ(f ) = t❤➻ mt(f ) = 1✱ s✉② r❛ ∼γ(f ) = = 2δ(f ), ①↔② r❛ ✤➥♥❣ t❤ù❝ ✤ó♥❣ ✈ỵ✐ ♣❤→t ❜✐➸✉ ❝õ❛ ✤à♥❤ ❧➼✳ ●✐↔ sû δ(f ) > ✈➔ ✤à♥❤ ❧➼ ✤ó♥❣ ✈ỵ✐ ♠å✐ g ❜➜t ❦❤↔ q✉② t❤ä❛ ♠➣♥ δ(g) < (f ) ú ỵ r (f ) > t m = mt(f ) ≥ ✈➔ sû ❞ö♥❣ ❦➳t q✉↔ ❝õ❛ ▼➺♥❤ ✤➲ ✸✳✷ t❛ ❝â δ(f ) = ∼ ∼ m(m − 1) +δ f >δ f ( ❞♦ m ≥ 2) ❙û ❞ö♥❣ ❣✐↔ t❤✐➳t q✉② f ũ ợ t ữủ ∼ ∼ ∼ ∼ γ(f ) ≥ m2 − m + γ f ✭✤♣❝♠✮ ∼ ≥ m2 − m + 2δ f = 2δ(f ) ❇➙② ❣✐í t❛ ❣✐↔ sû p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ✳ ❚❛ ❝➛♥ ❝❤➾ r❛ ✤➥♥❣ t❤ù❝ ∼γ(f ) = 2δ(f ) ◆➳✉ i0(f, x) = i0(f, y) t❤➻ p ❝ô♥❣ ❧➔ ✐♠✲ ❣♦♦❞ ✤è✐ ✈ỵ✐ f tø ❇ê ✤➲ ✸✳✶✵✳ ❚❤❡♦ ∼ ∼ ∼ ❣✐↔ t❤✐➳t q✉② ♥↕♣ t❛ ❝â γ f = 2δ f ✳ ❑❤✐ ✤â ∼ • ∼ ∼ ∼ γ(f ) = m2 − m + γ f ( ∼ t❤❡♦ ❇ê ✤➲ ✸✳✶✵) = m2 − m + 2δ f = 2δ(f ) • ( t❤❡♦ ▼➺♥❤ ✤➲ ✸✳✷) ◆➳✉ i0(f, x) = i0(f, y) t❤➻ ❞♦ p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ♥➯♥ i0 (f, x) = i0 (f, y) = m ≡ (mod p) ●å✐ g ∈ K[[x, y]] s❛♦ ❝❤♦ f (x, y) = g (x, αx − βy) ✈ỵ✐ (β : α) ❧➔ ữợ t t f õ t ✤➲ ✸✳✾ t❛ ❝â i0(g, x) < i0(g, y)✳ ❉♦ ✤â t❤❡♦ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ tr÷í♥❣ ❤đ♣ tr➯♥ t❤➻ ∼γ ∼g = 2δ ∼g ✈➔ t❤❡♦ ❇ê ✤➲ ✸✳✶✵ t❤➻ ∼ ∼ ∼ γ(g) = m2 − m + γ g ✳ ❈✉è✐ ❝ò♥❣ ❞♦ g ∼c f ♥➯♥ ∼ ∼ γ(f ) = γ(g) ∼ ∼ = m2 − m + γ g ∼ = m2 − m + 2δ g = 2δ(g) = 2δ(f ) ✹✽ ❱➟② ♥➳✉ p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f t❤➻ ∼γ(f ) = 2δ(f ) ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ p ❦❤ỉ♥❣ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f t❤➻ ∼γ(f ) > δ(f )✳ ❚❤➟t ✈➟②✱ ✈➻ p ❦❤ỉ♥❣ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ♥➯♥ m ≥ p✱ ✈➔ ❞♦ ✤â δ(f ) ≥ p(p − 1) , ❞➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ m = p ✈➔ δ(f ) = ∼ • ◆➳✉ δ(f ) = p(p 2− 1) ✱ t❛ ✈✐➳t f = fp + fp+1 + ✈ỵ✐ fp = (αx − βy)p, β = ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ α = P❤↔♥ ❝❤ù♥❣ r➡♥❣ α = t❛ ❝â i0(f, y) > p = i0(f, x) ▼➔ i0(f, y) ≡ (mod p) ♥➯♥ i0(f, y) ≥ 2p ❉♦ ✤â ∼ mt(f ) = {i0 (f, x), i0 (f, y) − i0 (f, x)} ≥ p > ✣✐➲✉ ♥➔② ❝â ♠➙✉ t❤✉➝♥ ❞♦ δ(f ) = t❤➻ mt(f ) = ❱➟② α = ❚✐➳♣ tö❝ ❧➜② g ∈ K[[x, y]] ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✸✳✾✱ t❛ ❝â p = i0(g, x) < i0(g, y) ❍ì♥ ♥ú❛ ∼ mt( g) = ✈➻ ∼ ∼ p(p − 1) p(p − 1) ∼ ∼ = δ(f ) = δ(g) = + δ( g) ⇒ δ( g) = 2 ▼➔ i0(∼g, u) = p > ♥➯♥ i0(∼g, v) = ●å✐ (u(t), v(t)) ❧➔ ♠ët t❤❛♠ sè ❤â❛ ❝õ❛ ∼ g ✱ ❦❤✐ ✤â x(t) = u(t) ✈➔ y(t) = u(t).v(t) ❧➔ ♠ët t❤❛♠ sè ❤â❛ ❝õ❛ g✳ ❚❛ ❝â ∼ i0 (g, y) = ord y(t) = ord (u(t).v(t)) = ord x(t) + ord v(t) = i0 (g, x) + i0 ( g, x) ⇒i0 (g, y) = p + ≡ 0(mod p), ❤❛② p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ g ❙û ❞ư♥❣ ❦➳t q✉↔ ❝õ❛ tr÷í♥❣ ❤đ♣ tr➯♥ t❛ ✤÷đ❝ ∼γ(g) ≥ 2δ(g) = 2δ(f )✳ ▼➦t ❦❤→❝✱ t❤❡♦ ❇ê ✤➲ ✸✳✾✱ ∼γ(f ) > ∼γ(g) ❱➟② ∼ γ(f ) > 2δ(f ) • ◆➳✉ δ(f ) > p(p 2− 1) ✈➔ i0(f, x) = i0(f, y) t❤➻ t❤❡♦ ❇ê ✤➲ ✸✳✶✵✱ p ❧➔ ❦❤ỉ♥❣ ∼ ∼ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ❙û ❞ư♥❣ ❣✐↔ t❤✐➳t q✉② ♥↕♣ ❝❤♦ f t❛ ✤÷đ❝ ∼ ∼ ∼ γ(f ) = m(m − 1) + γ(f ) ∼ > m(m − 1) + 2δ(f ) = 2δ(f ) ✹✾ ❚r÷í♥❣ ❤đ♣ i0(f, x) = i0(f, y)✱ ❧➜② g ∈ K[[x, y]] ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✸✳✾✳ ◆➳✉ p ❦❤ỉ♥❣ ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ g✱ ✈➔ i0(g, x) = i0(g, y)✱ t❤❡♦ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ t❛ ✤÷đ❝ ∼γ(g) > 2δ(g) ✈➔ ❞♦ ✤â ∼ ∼ γ(f ) ≥ γ(g) > 2δ(g) = 2δ(f ) ( ❞♦ ❈❤ó þ 1.20) ◆➳✉ p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ g t❤➻ t❤❡♦ ❇ê ✤➲ ✸✳✾✱ ∼γ(f ) > ∼γ(g)✳ ❈ị♥❣ ✈ỵ✐ ❣✐↔ t❤✐➳t q✉② ♥↕♣ t❛ ❝â ∼ ∼ γ(f ) > (g) 2(g) = 2(f ) ữợ ữủ tỹ t ữợ sỷ f t❤✉ ❣å♥ ✈ỵ✐ ♣❤➙♥ t➼❝❤ f = f1 fr , tr♦♥❣ ✤â fi ❜➜t ❦❤↔ q✉② ✈ỵ✐ ♠å✐ i = 1, , r✳ ❑❤✐ ✤â✱ sû ❞ö♥❣ ❦➳t q✉↔ ❝õ❛ ữợ t õ r (fi ) + γ(f ) = i=1 i0 (fi , fj ) − r + j=i r ≥ i0 (fi , fj ) − r + 2δ(fi ) + i=1 j=i = 2δ(f ) − r + ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ fi, ∀i = 1, r✱ ❤❛② p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f ✳ ✣à♥❤ ❧➼ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳ ❍➺ q✉↔ ✸✳✶✷✳ ◆➳✉ p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ f t❤➻ γ(f ) = ∼γ(f ) ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② g ❧➔ t÷ì♥❣ ✤÷ì♥❣ ♣❤↔✐ ❝õ❛ f s❛♦ ❝❤♦ γ(f ) = ∼γ(g) ❑❤✐ ✤â sû ❞ö♥❣ ú ỵ t õ ∼ ∼ γ(f ) ≥ γ(f ) = γ(g) ≥ 2δ(g) − r(g) + = 2δ(f ) − r(f ) + = γ(f ), ❉♦ ✤â γ(f ) = ∼γ(f ) ✣à♥❤ ❧➼ ✸✳✶✸✳ ❈❤♦ f ∈ K[[x, y]] t❤✉ ❣å♥✳ ❑❤✐ ✤â γ(f ) ≥ 2δ(f ) − r(f ) + ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ p ❧➔ ✐♠✲❣♦♦❞ ♣❤↔✐ ✤è✐ ✈ỵ✐ f ✳ ✺✵ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② g ❧➔ t÷ì♥❣ ✤÷ì♥❣ ♣❤↔✐ ❝õ❛ f s❛♦ ❝❤♦ γ(f ) = ∼γ(g) ✈➔ sû ú ỵ t ữủ γ(f ) = γ(g) ≥ 2δ(g) − r(g) + = 2δ(f ) − r(f ) + 1, ✈ỵ✐ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ g✳ ✣➸ ❦➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤ t❛ ❝á♥ ❝➛♥ ❝❤➾ r❛ r➡♥❣ ♥➳✉ p ❧➔ ✐♠✲❣♦♦❞ ♣❤↔✐ ✤è✐ ✈ỵ✐ f t❤➻ γ(f ) = 2δ(f ) − r(f ) + ❚❤➟t ✈➟②✱ ❣å✐ h ∈ K[[x, y]] s❛♦ ❝❤♦ h ∼r f ✈➔ p ❧➔ ✐♠✲❣♦♦❞ ✤è✐ ✈ỵ✐ h✳ ❚✐➳♣ tư❝ sû ❞ư♥❣ ✣à♥❤ ❧➼ ✸✳✶✶ ✈➔ ú ỵ t ữủ (f ) = (h) ≤ γ(h) = 2δ(h) − r(h) + = 2δ(f ) − r(f ) + ≤ γ(f ) ✣✐➲✉ ♥➔② ❝❤♦ t❤➜② γ(f ) = 2δ(f ) − r(f ) + ✈➔ t❛ ❦➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤ t↕✐ ✤➙②✳ ❑➳t ❧✉➟♥ ✈➔ ❑✐➳♥ ♥❣❤à ❚r➯♥ ✤➙② ❧➔ t♦➔♥ ❜ë ♥ë✐ ❞✉♥❣ ♠➔ t→❝ ❣✐↔ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✤÷đ❝ ✈➔ ❤➺ t❤è♥❣ ❧↕✐✱ ✈➲ ❝ì ❜↔♥ ✤➣ ❤♦➔♥ t❤➔♥❤ ữủ ởt số t r ã ✤à♥❤ ♥❣❤➽❛ ✈➔ ♥➯✉ r❛ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝→❝ ❜➜t ❜✐➳♥ ❝ê ✤✐➸♥❀ • ♠ỉ t↔ ✤÷đ❝ ❝➜✉ tró❝ ♥û❛ ♥❤â♠ ❝õ❛ ♠ët ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ✈➔ ❞ü❛ tr➯♥ ❝➜✉ tró❝ ✤â ✤÷❛ r❛ ♠ët sè q✉❛♥ ❤➺ ❣✐ú❛ ♥û❛ ♥❤â♠ ✈➔ ❝→❝ ❜➜t ❜✐➳♥ ❦❤→❝❀ • ①➙② ❞ü♥❣ ❜➜t ❜✐➳♥ ●❛♠♠❛✱ tø ✤â t ổ tự r rở ố ợ trữớ ❝â ✤➦❝ sè ❞÷ì♥❣✳ ❚✉② ♥❤✐➯♥✱ ●✐↔ t❤✉②➳t ✷✳✷✺ ✤÷đ❝ ✤➦t r❛ tr♦♥❣ q✉→ tr➻♥❤ t→❝ ❣✐↔ ♥❣❤✐➯♥ ❝ù✉ ✈➝♥ ❝❤÷❛ t➻♠ ✤÷đ❝ ❝➙✉ tr↔ ❧í✐✱ ❝ơ♥❣ ♥❤÷ t❤❛♠ ✈å♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ♥û❛ ♥❤â♠ ❝õ❛ ❝→❝ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ q✉❛ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝❤➦t ✈➝♥ ❝❤÷❛ ✤÷đ❝ ❣✐↔✐ q✉②➳t✳ ❚→❝ ❣✐↔ r➜t ữủ sỹ q t ỵ õ õ qỵ ổ õ ữủ ữợ ợ tr qt ✈➜♥ ✤➲ ♥➔②✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ❚→❝ ❣✐↔ ❈→✐ ❚❤à ▼✐♥❤ P❤÷ì♥❣ ✺✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❆✳ ❈❛♠♣✐❧❧♦✳ ❆❧❣❡❜r♦✐❞ ❈✉r✈❡ ✐♥ P♦s✐t✐✈❡ ❈❤❛r❛❝t❡r✐st✐❝✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✱ ✶✾✽✵✳ ❬✷❪ ❆✳ ❍❡❢❡③✳ ■rr❡❞✉❝✐❜❧❡ P❧❛♥❡ ❈✉r✈❡✳ ❘❡❛❧ ❛♥❞ ❈♦♠♣❧❡① ❙✐♥❣✉❧❛r✐t✐❡s✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ P✉r❡ ❛♥❞ ❆♣♣❧✳ ▼❛t❤✳ ✱ ◆❡✇ ❨♦r❦✱ ✷✵✵✸✳ ❬✸❪ ❆✳ ❙❛t❤❛②❡✱ ❏✳ ❙t❡♥❡rs♦♥✳ P❧❛♥❡✱ P♦❧②♥♦♠✐❛❧ ❈✉r✈❡s✳ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡tr② ❛♥❞ ➑t ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✱ ✶✾✾✹✳ ❬✹❪ ❊✈❡❧✐❛ ❘♦s❛ ●❛r❝➼❛ ❇❛rr♦s♦✱ ❆r❦❛❞✐✉s③ P❧♦s❦✐✳ ❆♥ ❛♣♣r♦❛❝❤ t♦ ♣❧❛♥❡ ❛❧❣❡✲ ❜r♦✐❞ ❜r❛♥❝❤❡s✱ ❘❡✈✐st❛ ▼❛t✳ ❈♦♠♣❧✉t❡♥s❡ ✷✽ ✭✷✵✶✺✮✱ ✷✷✼ ✲ ✷✺✷✳ ❬✺❪ ❊✈❡❧✐❛ ❘♦s❛ ●❛r❝➼❛ ❇❛rr♦s♦✱ ❆r❦❛❞✐✉s③ P❧♦s❦✐✳ ❚❤❡ ▼✐❧♥♦r ♥✉♠❜❡r ♦❢ ♣❧❛♥❡ ✐rr❡❞✉❝✐❜❧❡ s✐♥❣✉❧❛r✐t✐❡s ✐♥ ♣♦s✐t✐✈❡ ❝❤❛r❛❝t❡r✐st✐❝✱ ❇✉❧❧✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ✹✽ ✭✷✵✶✻✮✱ ✾✹✲✾✽✳ ❬✻❪ ●✳✲▼✳ ●r❡✉❡❧✱ ❈✳ ▲♦ss❡♥✱ ❊✳ ❙❤✉st✐♥✳ ■♥tr♦❞✉❝t✐♦♥ t♦ s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ ❞❡✲ ❢♦r♠❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✱ ✷✵✵✼✳ ỗ ự sst srts st rtrst rst srstr ỗ ự rts ♣❧❛♥❡ ❝✉r✈❡ s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ P❧✉❝❦❡r ❢♦r✲ ♠✉❧❛s ✐♥ ♣♦s✐t✐✈❡ ❝❤❛r❛❝t❡r✐st✐❝✱ ❆♥♥❛❧❡s ❞❡ ❧✬✐♥st✐t✉t ❋♦✉r✐❡r✱ ✷✵✶✻✳ ✺✷