Luận văn thạc sĩ hàm zêta tôpô của kì dị đường cong phẳng phức không suy biến

✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ❱ô ❚❤à ❱➙♥ ❍⑨▼ ❩❊❚❆ ❚➷P➷ ❈Õ❆ ❑➐ ❉➚ ✣×❮◆● ❈❖◆● P❍➃◆● P❍Ù❈ ❑❍➷◆● ❙❯❨ ❇■➌◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❍➔ ◆ë✐ ✲ ◆➠♠ ✷✵✷✵ ✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ❱ô ❚❤à ❱➙♥ ❍⑨▼ ❩❊❚❆ ❚➷P➷ ❈Õ❆ ❑➐ ❉➚ ✣×❮◆● ❈❖◆● P❍➃◆● P❍Ù❈ ❑❍➷◆● ❙❯❨ ❇■➌◆ ❈❤✉②➯♥ số ỵ tt số số ✿ ✽✹✻✵✶✵✶✳✵✹ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆●×❮■ ❍×❰◆● ị ì ◆➠♠ ✷✵✷✵ ▲í✐ ❝↔♠ ì♥ ✣➸ ❤♦➔♥ t❤➔♥❤ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❧í✐ ✤➛✉ t✐➯♥ t→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ s➙✉ s➢❝ ỵ ữớ ❈ì ✲ ❚✐♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥ ✲ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❚❤➛② ✤➣ trỹ t ữợ t tr s✉èt q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤✐➺♥ ❧✉➙♥ ✈➠♥ ♥➔②✳ ◆❣♦➔✐ r❛ t→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❈ì ✲ ❚✐♥ ❤å❝ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ✈➔ ✤â♥❣ ❣â♣ ỳ ỵ qỵ t t ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❈✉è✐ ❝ò♥❣ t→❝ ❣✐↔ ữủ ỷ ỡ t tợ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ♥❣÷í✐ t❤➙♥ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❝ê ✈ơ✱ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✷ ♥➠♠ ✷✵✶✾ ❍å❝ ✈✐➯♥ ❝❛♦ ❤å❝ ❱ơ ❚❤à ❱➙♥ ✶ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ▲í✐ ♥â✐ ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶ ✸ ✻ ✶✳✶ ●✐↔✐ ❦➻ ❞à ❝❤♦ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❈→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ●✐↔✐ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❜➡♥❣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹ ❍➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✺ ❱➼ ❞ö✿ ❑➻ ❞à f (x, y) = y − x3 t↕✐ O ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷ ❍➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ✤ì♥ ✻ ✾ ✶✼ ✷✳✶ ❑➻ ❞à ✤ì♥ A2n−1 ✭n ≥ 2✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷ ❑➻ ❞à ✤ì♥ A2n ✭n ≥ 1✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸ ❍➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❝â ♣❤➛♥ ❝❤➼♥❤ tü❛ t❤✉➛♥ ♥❤➜t ✷✻ ✸✳✶ ❑➻ ❞à y a − xb ✈ỵ✐ (a, b) = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✷ ❇✐➯♥ ◆❡✇t♦♥ ❝❤➾ ❝â ♠ët ❝↕♥❤ ❝♦♠♣➢❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✹ ❑➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ♣❤ù❝ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✸✺ ✹✳✶ P❤➨♣ ❣✐↔✐ ①✉②➳♥ ❝❤♦ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✷ ❍➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✷ ✷ ▲í✐ ♥â✐ ✤➛✉ ◆➠♠ ✶✾✾✷✱ ❉❡♥❡❢ sr t r ởt t ợ ữủ ❣å✐ ❧➔ ❤➔♠ ③❡t❛ tỉ♣ỉ✱ ❜ð✐ ✤➦❝ tr÷♥❣ ❊✉❧❡r✲P♦✐♥❝❛r➨ tỉ♣ỉ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ✭①❡♠ ❬✸❪✮✳ ◆â✐ ♠ët ❝→❝❤ ♥æ♠ ♥❛✱ ❤➔♠ ③❡t❛ tæ♣æ Zftop (s) ❝õ❛ ♠ët ✤❛ t❤ù❝ d ❜✐➳♥ ❤➺ sè ♣❤ù❝ f ❧➔ ♠ët ❤➔♠ ❤ú✉ t✛ ❝õ❛ s ❝❤ù❛ ♥❤ú♥❣ t❤ỉ♥❣ t✐♥ ✤÷đ❝ ❧➜② r❛ tø ♠ët ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à ❝õ❛ ✤❛ t↕♣ ♣❤ù❝ X0 := {x ∈ Cd | f (x) = 0}✳ ❈❤ó♥❣ t❛ ♥❤➢❝ ❧↕✐ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ Zftop (s) ♥❤÷ s❛✉✳ ❈❤♦ h : Y → (X, X0 ) ❧➔ ♠ët ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à ❝õ❛ X0 ✳ ❑❤✐ ✤â✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à✱ h : Y → X ❧➔ ♠ët →♥❤ ①↕ r✐➯♥❣ t❤❡♦ tæ♣æ ♣❤ù❝ ✭♥❣❤à❝❤ ↔♥❤ ❝õ❛ ♠ët t➟♣ ❝♦♠♣➢❝ tr♦♥❣ X ❧➔ ♠ët t➟♣ ❝♦♠♣➢❝ tr♦♥❣ Y ✮✱ Y ❧➔ ♠ët ✤❛ t↕♣ ♣❤ù❝ trì♥✱ s❛♦ ❝❤♦ →♥❤ ①↕ ❤↕♥ ❝❤➳ h : Y \ h−1 (X0 ) → X \ X0 ❧➔ ♠ët ✤➥♥❣ ❝➜✉ ❣✐ú❛ ❝→❝ ✤❛ t↕♣ ✤↕✐ sè ✈➔ s❛♦ ❝❤♦ h−1 (X0 ) ❧➔ ❤ñ♣ ❝õ❛ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ♠➔ ❝❤ó♥❣ ❤♦➦❝ ❦❤ỉ♥❣ ❣✐❛♦ ♥❤❛✉ ❤♦➦❝ ❝❤➾ ❣✐❛♦ ❤♦➔♥❤ ✭❣✐❛♦ ♥❤❛✉ ✈ỵ✐ ❜ë✐ ❣✐❛♦ ❜➡♥❣ s❛✉ ❦❤✐ ❜ä q✉❛ sè ❜ë✐ tr➯♥ ♠é✐ t❤➔♥❤ ♣❤➛♥✮✳ ❈→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ h−1 (X0 ) ❝â ❤❛✐ ❧♦↕✐✿ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ✭♥➳✉ ❝❤ó♥❣ ✤➥♥❣ ❝➜✉ ✈ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ①↕ ↔♥❤ Pd−1 C ✮✱ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ✭♥➳✉ ❝❤ó♥❣ d−1 ✤➥♥❣ ❝➜✉ ✈ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ❛❢❢✐♥❡ AC ✮✳ ●å✐ {Ei | i ∈ S} ✭✈ỵ✐ S ❧➔ ♠ët t➟♣ ❤ú✉ ❤↕♥✮ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ h−1 (X0 )✳ ●å✐ Ni ❧➔ sè ❜ë✐ ❝õ❛ f ◦ h tr➯♥ Ei ✈➔ νi − ❧➔ sè ❜ë✐ ❝õ❛ ❏❛❝♦❜✐❛♥ ♦❢ h tr➯♥ Ei ✳ ❱ỵ✐ ♠é✐ t➟♣ ❝♦♥ I ❝õ❛ S ✱ t❛ ❦➼ ❤✐➺✉ EI ❝❤♦ t➟♣ ❣✐❛♦ ❤ñ♣ EI \ Ei ✈➔ EI◦ ❝❤♦ t➟♣ j∈I Ej ✳ ❑❤✐ ✤â ❉❡♥❡❢ ✈➔ ▲♦❡s❡r ❬✸❪ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ f ♥❤÷ s❛✉ Zftop (s) = χ(EI◦ ) I⊆S i∈I i∈I , Ni s + νi tr♦♥❣ ✤â χ ❧➔ ✤➦❝ tr÷♥❣ ❊✉❧❡r✲P♦✐♥❝❛r➨ tỉ♣ỉ✳ ❚r♦♥❣ ❬✸❪✱ ❝→❝ t→❝ ❣✐↔ ❝❤➾ r❛ r➡♥❣ Zftop (s) ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ sü ❧ü❛ ❝❤å♥ ❝õ❛ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à h ❝õ❛ X0 ✳ ❍ì♥ ✸ ✹ ♥ú❛✱ ❤➔♠ ③❡t❛ tæ♣æ ♥➔② ❝á♥ ❧➔ ♠ët ❜➜t ❜✐➳♥ r➜t t❤ó ✈à✱ ❧✐➯♥ q✉❛♥ ✤➳♥ ●✐↔ t❤✉②➳t ỡ ởt tt q trồ tr ỵ tt ❦➻ ❞à ✈➔ ❍➻♥❤ ❤å❝ ✤↕✐ sè✱ ✤÷đ❝ ♣❤→t ❜✐➸✉ ❜ð✐ ♥❤➔ t♦→♥ ❤å❝ ◆❤➟t ❇↔♥ ❏✉♥✲■❝❤✐ ■❣✉s❛ ♥❤ú♥❣ ♥➠♠ ✶✾✽✵✮✳ ◆➳✉ x ❧➔ ♠ët ✤✐➸♠ ❝õ❛ X0 ✱ t❛ ✤à♥❤ ♥❣❤➽❛ Sx := {i ∈ S | h(Ei ) = x}✳ ❑❤✐ ✤â ❤➔♠ ③❡t❛ tỉ♣ỉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ f t↕✐ ✤✐➸♠ x ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉ top Zf,x (s) = χ(EI◦ ) I⊆S,I∩Sx =∅ i∈I Ni s + νi ●✐↔ t❤✉②➳t ✤ì♥ ✤↕♦ ✭♣❤✐➯♥ ❜↔♥ tỉ♣ỉ✮ ❦➳t ♥è✐ ❝→❝ ❝ü❝ ❝õ❛ ❤➔♠ ③❡t❛ tỉ♣ỉ ✈ỵ✐ t q trồ ỵ tt ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❝→❝ ❤➔♠ ③❡t❛ top tæ♣æ✱ ♠å✐ ❝ü❝ ❝õ❛ Zftop (s) ✈➔ Zf,x (s) ✤➲✉ ❝â ❞↕♥❣ − Nνii ✈ỵ✐ i ∈ S ✳ ❚✉② ♥❤✐➯♥✱ ❜➡♥❣ r➜t ♥❤✐➲✉ ✈➼ ❞ư✱ ♥❣÷í✐ t❛ ♥❤➟♥ t❤➜② r➡♥❣ ❝â r➜t ♥❤✐➲✉ sè ❤ú✉ t✛ − Nνii ✭✈ỵ✐ top (s)✳ ❉♦ ✤â✱ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ●✐↔ i ∈ S ✮ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❝ü❝ ❝õ❛ Zftop (s) ✈➔ Zf,x top top (s) ❧➔ ♠ët ❜➔✐ t♦→♥ r➜t t❤✉②➳t ✤ì♥ ✤↕♦✱ ❜➔✐ t♦→♥ t➻♠ ❝ü❝ ❝õ❛ Zf (s) ✈➔ Zf,x q✉❛♥ trå♥❣✳ ◆â✐ ❝❤✉♥❣✱ ❝❤♦ ✤➳♥ ♥❛②✱ ❜➔✐ t♦→♥ ♥➔② ❝❤÷❛ ✤÷đ❝ ❣✐↔✐ tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✳ ❈â r➜t ♥❤✐➲✉ ❜➔✐ ❜→♦ ✈✐➳t ✈➲ ❤➔♠ ③❡t❛ tæ♣æ ✈➔ ●✐↔ t❤✉②➳t ✤ì♥ ✤↕♦ ❝❤♦ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ✭tù❝ ❧➔ ❝❤♦ tr÷í♥❣ ❤đ♣ d = 2✮✱ ❝❤➥♥❣ ❤↕♥ ❬✼❪✱ ❬✶✶❪✱ ❬✶✵❪✱ ❬✹❪✱ ❬✻❪✳ ❚r÷í♥❣ ❤đ♣ d = 2✱ ●✐↔ t❤✉②➳t ✤ì♥ ✤↕♦ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ❬✼❪✱ ❬✶✵❪✱ ❬✻❪✳ ◆â✐ r✐➯♥❣✱ ♣❤÷ì♥❣ ♣❤→♣ tr♦♥❣ ❬✻❪ ❞ü❛ tr➯♥ ♥❤ú♥❣ ❤✐➸✉ ❜✐➳t q✉❛♥ trå♥❣ ✈➲ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥✱ ♣❤➨♣ ❣✐↔✐ ①✉②➳♥✱ ❤➻♥❤ ❤å❝ ①✉②➳♥ ✈➔ t❤→♣ ❣✐↔✐ ❚s❝❤✐r♥❤❛✉s❡♥ ❝❤♦ ❦➻ ữớ ữủ ợ t trữợ õ tr♦♥❣ ❬✽❪✱ ❬✾❪✱ ❬✶❪✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ t➻♠ ❤✐➸✉ ♣❤÷ì♥❣ ♣❤→♣ t➼♥❤ ❤➔♠ ③❡t❛ tỉ♣ỉ ✤à❛ ♣❤÷ì♥❣ tr♦♥❣ ❜➔✐ ❜→♦ ❝õ❛ ❚❤÷í♥❣ ✈➔ ❍÷♥❣ ❬✻❪✳ ▼➦❝ ❞ị ❜➔✐ ❜→♦ ❬✻❪ ✤➲ ❝➟♣ ✤➳♥ ❤➔♠ ③❡t❛ tæ♣æ ❝❤♦ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ tê♥❣ q✉→t✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤ó♥❣ tỉ✐ ❝❤➾ ✤å❝ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❝❤♦ ❝→❝ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ tø ❝→❝ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ♥❤÷ ❦➻ ❞à ✤ì♥ ✤➳♥ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥✳ ▲✉➟♥ ✈➠♥ ❝❤✐❛ t❤➔♥❤ ❜è♥ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ ✈➔ ❤➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣✳ ❈❤÷ì♥❣ ✷ ❝❤♦ ❝→❝ t➼♥❤ t♦→♥ ❝ư t❤➸ ✈➲ ❣✐↔✐ ❦➻ ❞à ✈➔ ❤➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ✤ì♥✳ ❈❤÷ì♥❣ ✸ tr➻♥❤ ❜➔② ✈➲ ❤➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❝â ♣❤➛♥ ❝❤➼♥❤ ❧➔ ♠ët ✤❛ t❤ù❝ tü❛ t❤✉➛♥ ♥❤➜t✳ ❈✉è✐ ❝ò♥❣✱ tr♦♥❣ ❈❤÷ì♥❣ ✹✱ ❝❤ó♥❣ tỉ✐ ❦❤↔♦ s→t ❦➻ ❞à ✺ ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥✱ tø ✤â ♠æ t↔ ❤➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ♥➔② t❤æ♥❣ q✉❛ ✤❛ ❞✐➺♥ ◆❡✇t♦♥ ❝õ❛ ♥â✳ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ●✐↔✐ ❦➻ ❞à ❝❤♦ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ●✐↔✐ ởt ổ q trồ ỵ tt ❦➻ ❞à ♥â✐ r✐➯♥❣ ✈➔ ❤➻♥❤ ❤å❝ ✤↕✐ sè ♥â✐ ❝❤✉♥❣✳ ◆â ❝❤♦ ♣❤➨♣ ❝❤✉②➸♥ t❤æ♥❣ t✐♥ ❤➻♥❤ ❤å❝ ❝õ❛ ✤✐➸♠ ❦➻ ❞à t❤➔♥❤ ❝→❝ t❤ỉ♥❣ t✐♥ tê ❤đ♣ ♥❤÷ ♠ët tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ s➢♣ ①➳♣ s✐➯✉ ♣❤➥♥❣✱ ❝❤♦ ♣❤➨♣ ♥❣❤✐➯♥ ❝ù✉ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✤❛ t❤ù❝ t❤ỉ♥❣ q ự ữỡ tr ỡ tự ỹ tỗ t↕✐ ❝õ❛ ❣✐↔✐ ❦➻ ❞à tr➯♥ tr÷í♥❣ ✤➦❝ sè ✵ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ❍✐r♦♥❛❦❛✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ t❛ ❝❤➾ ✤➲ ❝➟♣ ✤➳♥ ❣✐↔✐ ❦➻ ❞à ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ♣❤ù❝✳ ●å✐ O ❧➔ ❣è❝ tå❛ ✤ë ❝õ❛ C2 ✳ ❚❛ s➩ ❦➼ ❤✐➺✉ ❜ð✐ C{x, y} ✈➔♥❤ ❝→❝ ❝❤✉é✐ ❧✉ÿ t❤ø❛ ❤❛✐ ❜✐➳♥ ❤➺ sè ♣❤ù❝ ❤ë✐ tö tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ O tr♦♥❣ C2 ✳ ❳➨t ♠ët ♣❤➛♥ tû f (x, y) ❝õ❛ C{x, y}✳ ✣➦t C = {(x, y) ∈ C2 | f (x, y) = 0}✳ ✣✐➸♠ O ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❦➻ ❞à ❝õ❛ C ✭❤♦➦❝ ❝õ❛ f ✮ ♥➳✉ f (O) = ∂f ∂f (O) = (O) = ∂x ∂y ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ f (x, y) ❧➔ ♠ët ❝❤✉é✐ ❧ô② t❤ø❛ ❤ë✐ t❤ö tr➯♥ ♠ët ❧➙♥ ❝➟♥ W ❝õ❛ O tr♦♥❣ C2 s❛♦ ❝❤♦ O ❧➔ ♠ët ✤✐➸♠ ❦➻ ❞à ❝õ❛ f ✳ ▼ët →♥❤ ①↕ π : Y → W ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♣❤➨♣ ❣✐↔✐ tèt ❝õ❛ C t↕✐ O ✭❤❛② ❝õ❛ (C, O)✱ ❝õ❛ (f, O)✱ ❤❛② ỡ ỡ f tỗ t ởt ❧➙♥ ❝➟♥ V ❝õ❛ O s❛♦ ❝❤♦ V ⊆ W ✈➔ ❝→❝ ✤✐➲✉ s❛✉ ✤➙② ✤÷đ❝ t❤ä❛ ♠➣♥✳ ✭❛✮ Y ❧➔ ♠ët ✤❛ t↕♣ trì♥✱ tù❝ ❧➔ Y ✤÷đ❝ ♣❤õ ỗ ữỡ ộ ỗ ổ ợ (C2 ; x, y) ỗ ữủ ợ trỡ ữỡ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✼ ✭❜✮ π ❧➔ ♠ët →♥❤ ①↕ ①→❝ ✤à♥❤ ❜ð✐ ❝→❝ ❝❤✉é✐ ❧ô② t❤ø❛ ❤ë✐ tö tr➯♥ V ✭tù❝ ❧➔ ♠ët →♥❤ ①↕ ❣✐↔✐ t➼❝❤✮✱ π ❧➔ r✐➯♥❣ ✭♥❣❤à❝❤ ↔♥❤ ❝õ❛ ♠ët t➟♣ ❝♦♠♣➢❝ ❧➔ ♠ët t➟♣ ❝♦♠♣➢❝✮✱ t♦➔♥ →♥❤ ✈➔ π|π−1 (V )\π−1 (O) : π −1 (V ) \ π −1 (O) → V \ {O} ❧➔ ♠ët ✤➥♥❣ ❝➜✉ ❣✐↔✐ t➼❝❤✳ ✭❝✮ ×ỵ❝ div(π ∗ f ) := π −1 (C ∩ V ) ❝❤➾ ❝â ❦➻ ❞à ❧➔ ❝→❝ ✤✐➸♠ ❣✐❛♦ ❤♦➔♥❤ ❝õ❛ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ ♥â✱ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ♥➔② ❧➔ trì♥ tr♦♥❣ π −1 (V )✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ ❣å✐ ♠ët ♣❤➨♣ ❣✐↔✐ tèt ❝õ❛ f ❧➔ ♠ët ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à ❝õ❛ f ✳ ▼é✐ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ π−1(O) ✤÷đ❝ ❣å✐ ❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ❝õ❛ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à π✳ ▼é✐ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ ❜❛♦ ✤â♥❣ π −1 (C \ {O}) tr♦♥❣ Y ❝õ❛ π −1 (C \ {O}) ✤÷đ❝ ❣å✐ ❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ❝õ❛ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à π ✳ ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à t❛ t❤➜② ♠é✐ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ❝õ❛ π ✤➥♥❣ ❝➜✉ ✈ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ①↕ ↔♥❤ ♣❤ù❝ ♠ët ❝❤✐➲✉ P1 ✱ ♠é✐ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ợ ữớ t ự C sỷ (U ; u, v) ởt ỗ ữỡ tr Y s❛♦ ❝❤♦ tr➯♥ ✤â π ∗ f ❝â ❞↕♥❣ π ∗ f (u, v) = λ(u, v)um v n , tr♦♥❣ ✤â λ(u, v) ❦❤→❝ ✈ỵ✐ ♠å✐ (u, v) ∈ U ✳ ❑❤✐ ✤â u = ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ①→❝ ✤à♥❤ ♠ët t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E õ tr ỗ (U ; u, v) ỗ t t t ởt ỗ ự ♠ët ♣❤➛♥ ❝õ❛ E ✈➔ ①→❝ ✤à♥❤ E tr➯♥ ❜↔♥ ỗ õ ởt ữỡ tr ữỡ t t÷ì♥❣ tü✳ ❙è m ❧➔ ♠ët ❜➜t ❜✐➳♥ tr➯♥ ♠å✐ ỗ ợ E õ ữủ sè ❜ë✐ ❝õ❛ π∗f tr➯♥ E ✱ t❛ s➩ ❦➼ ❤✐➺✉ sè ♥➔② ❜ð✐ N (E)✳ ◆➳✉ Cj ❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ✈➔ ♥➳✉ f ❧➔ rót ❣å♥ ✭❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ f ✤➲✉ ❝â ❧ô② t❤ø❛ ❜➡♥❣ 1✮ t❤➻ sè ❜ë✐ tr➯♥ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ❧✉æ♥ ❧➔ N (Cj ) = 1✳ ◆➳✉ ❦➼ ❤✐➺✉ Ei ✱ ✈ỵ✐ i t❤✉ë❝ ♠ët t➟♣ ❤ú✉ ❤↕♥ S ✱ ❧➔ t➜t ❝↔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ π −1 (C)✱ ✈➔ ❦➼ ❤✐➺✉ Ni t❤❛② ❝❤♦ N (Ei )✱ t❛ ❝â div(π ∗ f ) = π −1 (C) = N i Ei iS ú ỵ r Ei õ t ởt t ♣❤➛♥ ❝→ ❜✐➺t✱ ❝ô♥❣ ❝â t❤➸ ❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ tỹ sỹ ụ tr ởt ỗ (U ; u, v) ♥❤÷ ✈➟②✱ ❤➔♠ det Jacπ ❝â ❞↕♥❣ det Jacπ (u, v) = δ(u, v)up v q , ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✽ tr♦♥❣ ✤â δ(u, v) ❦❤→❝ ✈ỵ✐ ♠å✐ (u, v) ∈ U ✳ ◆➳✉ E ởt t t tr ỗ ♥➔② ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ u = t❤➻✱ t÷ì♥❣ tü ♥❤÷ tr➯♥✱ p ❧➔ ♠ët ❜➜t ❜✐➳♥ ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ E ✱ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ỗ (E) := p + 1✳ ❚÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ❝â (νi − 1)Ei , Kπ := KY /W := div(det Jacπ ) = i∈S tr♦♥❣ ✤â νi := ν(Ei ) ✈ỵ✐ ♠å✐ i ∈ S ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✣➦t X = (x1 , x2 ; y1 : y2 ) ∈ C2 × P1 | x1 y2 = x2 y1 ❑❤✐ ✤â ♣❤➨♣ ♥ê t➙♠ O ❝õ❛ C2 ❧➔ →♥❤ ①↕ ρ : X → C2 ①→❝ ✤à♥❤ ❜ð✐ ρ(x1 , x2 ; y1 : y2 ) = (x1 , x2 ) ✣❛ t↕♣ ρ−1 (O) = (x1 , x2 ; y1 : y2 ) ∈ C2 × P1 | (x1 , x2 ) = O, x1 y2 = x2 y1 ∼ = P1 ✤÷đ❝ ❣å✐ ❧➔ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ❝õ❛ ♣❤➨♣ ♥ê ρ✳ ❚r♦♥❣ ✤à♥❤ ♥❣❤➽❛ ✈➲ ♣❤➨♣ ♥ê ρ ð tr➯♥✱ ✤❛ t↕♣ X ✤÷đ❝ ❞→♥ tø ❤❛✐ ❜↔♥ ỗ ữỡ U1 = {(x2 y1 , x2 ; y1 : 1) | x2 , y1 ∈ C} ✈➔ U2 = {(x1 , x1 y2 ; : y2 ) | x1 , y2 ∈ C} t❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ y1 y2 = ỗ t U1 ợ {(x2 , y1 ) | x2 , y1 C} ỗ ♥❤➜t U2 ✈ỵ✐ {(x1 , y2 ) | x1 , y2 ∈ C}✳ ❑❤✐ ✤â ❜✐➸✉ t❤ù❝ t÷í♥❣ ♠✐♥❤ ❝õ❛ ρ tr➯♥ U1 ❧➔ ρ(x2 , y1 ) = (x2 y1 , x2 ), ❜✐➸✉ t❤ù❝ t÷í♥❣ ♠✐♥❤ ❝õ❛ ρ tr➯♥ U2 ❧➔ ρ(x1 , y2 ) = (x1 , x1 y2 ) ỵ ❍✐r♦♥❛❦❛ →♣ ❞ư♥❣ ✈➔♦ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ♣❤ù❝ ❝â t❤➸ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿ ▼é✐ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à ❝õ❛ C t↕✐ O ❧➔ ♠ët ♣❤➨♣ ❤ñ♣ t❤➔♥❤ ❝õ❛ ♠ët sè ❤ú✉ ❤↕♥ ❝→❝ ♣❤➨♣ ♥ê ❝â ❞↕♥❣ ρ|ρ−1 (V ) : ρ−1 (V ) → V ✱ ✈ỵ✐ V ❧➔ ♠ët t➟♣ ❝♦♥ ♠ð ❝❤ù❛ O ❝õ❛ C2 ✈➔ ρ ①→❝ ✤à♥❤ ♥❤÷ tr➯♥✳ ❈❤÷ì♥❣ ✸✳ ❍➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❝â ♣❤➛♥ ❝❤➼♥❤ tü❛ t❤✉➛♥ ♥❤➜t ✷✾ top ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ ❤➔♠ ③❡t❛ tæ♣æ Zf,O (s) ❜➡♥❣ tê♥❣ ❝õ❛ ❝→❝ ♣❤➙♥ t❤ù❝ 1 −1 , , , N (T1 )s + ν(T1 ) N (Tm )s + ν(Tm ) N (P1 )s + ν(P1 ) χ(E(Ti0 ) ∩ E0 ) (s + 1)(N (Ti0 )s + ν(Ti0 )) ✈➔ m−1 i=1 χ(E(Ti ) ∩ E(Ti+1 )) (N (Ti )s + ν(Ti ))(N (Ti+1 )s + ν(Ti+1 )) top ❉♦ ✤â✱ t❤❡♦ t➼♥❤ t♦→♥ tr➯♥✱ ❤➔♠ ③❡t❛ tæ♣æ Zf,O (s) ❝õ❛ f (x, y) = y a − xb t↕✐ O ợ a, b số ữỡ tố ❝ò♥❣ ♥❤❛✉✮ ❜➡♥❣ tê♥❣ ❝õ❛ ❝→❝ ♣❤➙♥ t❤ù❝ s❛✉ ✤➙② ú ỵ d1 = cm = 1 1 , , , , as + c1 + bs + + dm abs + a + b (s + 1)(abs + a + b) i0 −1 i=1 m−1 i=i0 , (adi s + ci + di )(adi+1 s + ci+1 + di+1 ) (bci s + ci + di )(bci+1 s + ci+1 + di+1 ) ❇ð✐ ✈➻ di+1 di = − , (adi s + ci + di )(adi+1 s + ci+1 + di+1 ) adi+1 s + ci+1 +di+1 adi s + ci +di ci ci+1 = − , (bci s + ci + di )(bci+1 s + ci+1 + di+1 ) bci s + ci +di bci+1 s + ci+1 +di+1 t õ ú ỵ d1 = cm = 1✮✿ i0 −1 i=1 di0 d1 = − (adi s + ci + di )(adi+1 s + ci+1 + di+1 ) adi0 s + ci0 +di0 ad1 s + c1 +d1 = b − abs + a + b as + c1 + ✈➔ m−1 i=i0 ci0 cm = − (bci s + ci + di )(bci+1 s + ci+1 + di+1 ) bci0 s + ci0 +di0 bcm s + cm +dm = a − abs + a + b bs + + dm ❈❤÷ì♥❣ ✸✳ ❍➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❝â ♣❤➛♥ ❝❤➼♥❤ tü❛ t❤✉➛♥ ♥❤➜t ✸✵ ❉♦ ✤â top Zf,O (s) = a+b−1 (a + b − 1)s + a + b + = abs + a + b (s + 1)(abs + a + b) (s + 1)(abs + a + b) ỵ ữủ ự ợ a = ✈➔ b = t❛ trð ❧↕✐ ✈ỵ✐ ❦➻ ❞à f (x, y) = y − x3 t O ỵ t õ top Zf,O (s) = (2 + − 1)s + + 4s + = (s + 1)(2 · 3s + + 3) (s + 1)(6s + 5) ❑➳t q✉↔ ♥➔② ♣❤ị ❤đ♣ ✈ỵ✐ ▼ư❝ ✶✳✺✳ ✸✳✷ ❇✐➯♥ ◆❡✇t♦♥ ❝❤➾ ❝â ♠ët ❝↕♥❤ ❝♦♠♣➢❝ ❈❤♦ f (x, y) ∈ C{x, y} ❧➔ ♠ët ❦➻ ❞à t↕✐ O ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✤è✐ ✈ỵ✐ ✤❛ ❞✐➺♥ ◆❡✇t♦♥ Γ ❝õ❛ ♥â s❛♦ ❝❤♦ Γ ❝â ❞✉② ♥❤➜t ♠ët ❝↕♥❤ ❝♦♠♣➢❝ P1 ✳ ●å✐ (a, b) ❧➔ ✈➨❝tì ♥❣✉②➯♥ ❞÷ì♥❣ ♥❣✉②➯♥ sỡ tỡ t tữỡ ự ợ P1 ✳ ❑❤✐ ✤â f (x, y) ❝â ❞↕♥❣ s❛✉ ✤➙②✿ f (x, y) = f1 (x, y) · · · fr (x, y), ✭✸✳✶✮ tr♦♥❣ ✤â✱ ✈➻ f (x, y) ❦❤æ♥❣ s✉② ❜✐➳♥ t❤❡♦ ♥❣❤➽❛ ❝õ❛ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✱ f (x, y) ❝â ❞↕♥❣ f (x, y) = y a + ξ xb + ✭❝→❝ sè ❤↕♥❣ ❝❛♦ ❤ì♥✮, ✈ỵ✐ ≤ ≤ r✱ ξ ∈ C∗ ✈➔ ξ = = ỵ ố ✈ỵ✐ ❦➻ ❞à f (x, y) t↕✐ O ❦❤ỉ♥❣ s✉② ố ợ t õ ữủ tr♦♥❣ ✭✸✳✶✮ ✈➔ ✭✸✳✷✮✱ ❤➔♠ ③❡t❛ tỉ♣ỉ ✤÷đ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝ top Zf,O (s) = (a + b − r)s + a + b (s + 1)(rabs + a + b) ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥ ỡ q ữủ ố ợ ❤➔♠ f (x, y) ✈ỵ✐ ❝→❝ ✤➾♥❤ s❛✉ ✤➙② Ti = ci di ✈ỵ✐ ≤ i ≤ m + 1, tr♦♥❣ ✤â (c0 , d0 ) = (1, 0)✱ (cm+1 , dm+1 ) = (0, 1)✱ ✈➔ ❝→❝ ✤➾♥❤ ✤÷đ❝ s➢♣ t❤ù tü s❛♦ ❝❤♦ det(Ti , Ti+1 ) = ❈❤÷ì♥❣ ✸✳ ❍➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❝â ♣❤➛♥ ❝❤➼♥❤ tü❛ t❤✉➛♥ ♥❤➜t ✸✶ ✈ỵ✐ ♠å✐ ≤ i ≤ m✳ ●✐↔ sû Ti0 = P1 ✈ỵ✐ ♠ët i0 ∈ {1, , m}✳ ✣➦t ci ci+1 di di+1 σi = (Ti , Ti+1 ) = ✈ỵ✐ ≤ i ≤ m ❳➨t ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ ❧✐➯♥ ❦➳t ✈ỵ✐ ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥ ✤ì♥ ❤➻♥❤ ❝❤➼♥❤ q✉② Σ∗ s❛✉ ✤➙② π : X → C2 ; x, y , tr♦♥❣ ✤â X ❧➔ ♠ët ✤❛ t↕♣ ♣❤ù❝ trì♥✱ ✤÷đ❝ ❞→♥ tø ỗ ữỡ C2i ; xi , yi ✱ ≤ i ≤ m✱ t❤❡♦ ❧✉➟t ❞→♥ ✭✶✳✶✮✳ ❑❤✐ ✤â✱ ✈➻ f (x, y) ❦❤æ♥❣ s✉② ❜✐➳♥ ✤è✐ ✈ỵ✐ Γ✱ π ❧➔ ♠ët ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à ❝õ❛ f (x, y) t↕✐ O✳ ❇✐➸✉ t❤ù❝ ❝õ❛ π tr➯♥ ỗ C2i ; xi , yi c d π(xi , yi ) = Φσi (xi , yi ) = (xci i yi i+1 , xdi i yi i+1 ) tr ỗ C2i ; xi , yi ✱ ❤➔♠ π ∗ f (xi , yi ) ❝â ❜✐➸✉ ❞✐➵♥ adi+1 i π ∗ f (xi , yi ) = xad i yi bci+1 i + ξ xbc i yi + xi R (xi , yi ), tr♦♥❣ ✤â R (xi , yi ) ∈ C{xi , yi } ❧➔ tê♥❣ ❝õ❛ ❝→❝ ✤ì♥ t❤ù❝ ❝â ❝➜♣ ❝❛♦ ❤ì♥ ♣❤➛♥ adi+1 i ❝❤➼♥❤✱ ❝❤✐❛ ❤➳t ữợ ợ t xad i yi bci+1 i ✈➔ xbc i yi ✳ ❚÷ì♥❣ tü ▼ư❝ ✸✳✶✱ t❛ ①➨t t❤ù tü t÷ì♥❣ ✤è✐ ❝õ❛ ❝õ❛ ❝→❝ ✈➨❝tì trå♥❣ ♥❣✉②➯♥ Ti ✈➔ Ti0 ✳ ◆➳✉ det(Ti , Ti0 ) > 0✱ t❤➻ r ∗ π f (xi , yi ) = bci+1 −adi+1 i radi+1 xrad yi i i −adi (1 + ξ xbc yi i ) + xi R(xi , yi ) , =1 tr♦♥❣ ✤â R(xi , yi ) ∈ C{xi , yi }✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ sè ❜ë✐ ❝õ❛ π ∗ f tr➯♥ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Ti ) ❧➔ N (Ti ) = radi ◆➳✉ det(Ti0 , Ti ) ≥ 0✱ t❤➻ r ∗ π f (xi , yi ) = adi+1 −bci+1 i rbci+1 xrbc i yi i −bci (xad yi i =1 ✈ỵ✐ R(xi , yi ) ∈ C{xi , yi }✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ N (Ti ) = rbci ữỡ tỹ ợ ≤ i ≤ m✱ t❛ ❝â ν(Ti ) = ci + di + ξ ) + xi R(xi , yi ) , ❈❤÷ì♥❣ ✸✳ ❍➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❝â ♣❤➛♥ ❝❤➼♥❤ tü❛ t❤✉➛♥ ♥❤➜t P1 tữỡ ự ợ t ❝õ❛ Γ ♥➯♥ E(Ti0 ) = E(P1 ) ❧➔ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ❞✉② ♥❤➜t ❣✐❛♦ ✈ỵ✐ ❝→❝ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ❝õ❛ π ✳ P❤÷ì♥❣ tr➻♥❤ ❝õ❛ ❝→❝ t❤➔♥❤ tỹ sỹ tr ỗ ữỡ C2i ; xi0 , yi0 ❧➔ r ✭✸✳✸✮ (yi0 + ξ ) + xi0 R(xi0 , yi0 ) = =1 ❉♦ ξ1 , , ξr ❧➔ ❝→❝ sè ♣❤ù❝ ✤ỉ✐ ♠ët ❦❤→❝ ♥❤❛✉ ✭✈➔ ❦❤→❝ 0✮ ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✸✮ ❝❤ù♥❣ tä π ❝â r t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ❦❤→❝ ♥❤❛✉ ✭❦❤æ♥❣ ❣✐❛♦ ♥❤❛✉✮✳ ❑➼ ❤✐➺✉ ❝→❝ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ✤â ❧➔ E01 , , E0r ✳ ❚❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Ti0 ) = E(P1 ) ❣✐❛♦ ✈ỵ✐ t➜t ❝↔ ❝→❝ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü t↕✐ r ✤✐➸♠ ♣❤➙♥ ❜✐➺t (0, −ξ )✱ ✈ỵ✐ r tr ỗ C2i ; xi0 , yi0 ✳ ❚ø ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ❝→❝ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü t❛ ❝â N (E0 ) = 1, ν(E0 ) = 1, ≤ ≤ r ❚❛ ♠✐♥❤ ❤å❛ ✈✐➺❝ s➢♣ ①➳♣ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Ti )✱ ≤ i ≤ m✱ ✈➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü E01 , , E0r tr♦♥❣ ❧÷đ❝ ỗ s r ộ ữủ ỗ ❝→❝ t❤ỉ♥❣ sè E (N, ν) ❤♦➦❝ ✤ì♥ ❣✐↔♥ ❝❤➾ ❧➔ t➯♥ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t✳ E(T2 ) ✳✳✳ E(T1 ) (rad1 , c1 + d1 ) E(Ti0 −1 ) (radi0 −1 , ci0 −1 + di0 −1 ) ✳✳ ✳ E01 ✲ (1, 1) E02 ✲ (1, 1) E0r ✲ (1, 1) ✳✳✳ E(Ti0 +1 ) (rbci0 +1 , ci0 +1 + di0 +1 ) E(Tm ) E(P1 ) (rab, a + b) ❙❛✉ ✤➙② ❧➔ ❜↔♥❣ t➼♥❤ ✤➦❝ tr÷♥❣ ❊✉❧❡r ✭t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ▼ư❝ ✶✳✺✮✿ ❈❤÷ì♥❣ ✸✳ ❍➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❝â ♣❤➛♥ ❝❤➼♥❤ tü❛ t❤✉➛♥ ♥❤➜t ●✐❛♦ ✣➦❝ tr÷♥❣ ❊✉❧❡r E(T1 )◦ ✱ E(Tm )◦ E(Ti )◦ E(Ti0 )◦ = E(P1 )◦ −r E0◦ 1≤ ≤r E0 ∩ E0 = ∅ = E(Ti ) ∩ E(Ti+1 ) = (1 ✤✐➸♠) 1≤i≤m−1 E(Ti ) ∩ E(Tj ) = ∅ |i − j| ≥ E(Ti ) ∩ E0 = ∅ ≤ ≤ r ✱ i = i0 E(P1 ) ∩ E0 = (1 ✤✐➸♠) 1≤ ≤r ✸✸ ✣✐➲✉ ❦✐➺♥ ≤ i ≤ m − 1, i = i0 top ❚❤❡♦ t➼♥❤ t♦→♥ tr➯♥✱ ❤➔♠ ③❡t❛ tæ♣æ Zf,O (s) ❝õ❛ ❤➔♠ f (x, y) ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ✭✸✳✶✮ ✈➔ ✭✸✳✷✮ t↕✐ O ❜➡♥❣ tờ tự s ú ỵ d1 = cm = 1✮✿ 1 −r , , , ras + c1 + rbs + + dm rabs + a + b (r ❧➛♥), (s + 1)(rabs + a + b) i0 −1 i=1 m−1 i=i0 ❇ð✐ ✈➻ ❜➡♥❣ ✈➔ , (radi s + ci + di )(radi+1 s + ci+1 + di+1 ) (rbci s + ci + di )(rbci+1 s + ci+1 + di+1 ) (radi s + ci + di )(radi+1 s + ci+1 + di+1 ) di+1 di − , radi+1 s + ci+1 + di+1 radi s + ci + di (rbci s + ci + di )(rbci+1 s + ci+1 + di+1 ) ❜➡♥❣ ci+1 ci − , rbci s + ci + di rbci+1 s + ci+1 + di+1 t õ ú ỵ d1 = cm = 1✱ ci0 = a✱ di0 = b✮✿ i0 −1 i=1 b = − , (adi s + ci + di )(adi+1 s + ci+1 + di+1 ) rabs + a + b ras + c1 + ❈❤÷ì♥❣ ✸✳ m−1 i=i0 ❍➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❝â ♣❤➛♥ ❝❤➼♥❤ tü❛ t❤✉➛♥ ♥❤➜t ✸✹ a 1 = − (bci s + ci + di )(bci+1 s + ci+1 + di+1 ) rabs + a + b rbs + + dm ❉♦ ✤â top Zf,O (s) = a+b−r r (a + b − r)s + a + b + = rabs + a + b (s + 1)(rabs + a + b) (s + 1)(rabs + a + b) ỵ ữủ ự t ❦➻ ❞à f (x, y) = y − x6 + 2x7 t O ỵ a = 2✱ b = ✈➔ r = t❛ ❝â top Zf,O (s) = 3s + (2 + − 2)s + + = (s + 1)(2 · · 3s + + 3) (s + 1)(12s + 5) ỵ ❤➺ q✉↔ ❤✐➸♥ ♥❤✐➯♥ s❛✉ ✤➙②✳ ❍➺ q✉↔ ✸✳✺✳ ❈❤♦ ❦➻ ❞à f (x, y) t↕✐ O ♥❤÷ tr♦♥❣ ✣à♥❤ ỵ tt ỹ top (s) −1 ✈➔ − a+b ❝õ❛ ❤➔♠ ③❡t❛ tæ♣æ Zf,O ✳ rab ❈❤÷ì♥❣ ✹ ❑➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ♣❤ù❝ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✹✳✶ P❤➨♣ ❣✐↔✐ ①✉②➳♥ ❝❤♦ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❳➨t ❦➻ ❞à f (x, y) ∈ C{x, y} t↕✐ O ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✤è✐ ✈ỵ✐ ✤❛ ❞✐➺♥ ◆❡✇t♦♥ Γ ❝õ❛ ♥â✳ ❑❤✐ ✤â f (x, y) ❝â ❞↕♥❣ s❛✉ ✤➙② k f (x, y) = fi (x, y), ✭✹✳✶✮ i=1 ri (y + ξi xbi ) + ✭❝→❝ sè ❤↕♥❣ ❝❛♦ ❤ì♥✮, fi (x, y) = =1 tr♦♥❣ ✤â ✈ỵ✐ ♠é✐ ≤ i ≤ k t❛ ❝â ξi = 0, ✈➔ ξi = ξi ♥➳✉ = ✭✹✳✷✮ ✣➦t P1 = a1 , , Pk = b1 ak bk ❱➟② Γ ❧➔ ♠ët ✤❛ ❞✐➺♥ ◆❡✇t♦♥ ❝â ✤ó♥❣ k ❝↕♥❤ ❝♦♠♣➢❝✱ ♠é✐ ❝↕♥❤ ❝â ♠ët ✈➨❝tì ♣❤→♣ t✉②➳♥ ❧➔ ♠ët tr♦♥❣ ❝→❝ ✈➨❝tì (a1 , b1 ), , (ak , bk )✳ ●✐↔ sû P1 , , Pk ❧➔ ❝→❝ ✈➨❝tì ♥❣✉②➯♥ ❞÷ì♥❣ ♥❣✉②➯♥ s❛♦ ❝❤♦ ≥ ✈ỵ✐ ♠å✐ ≤ i ≤ k ✈➔ s❛♦ ❝❤♦ det(Pi , Pi+1 ) ≥ ✈ỵ✐ ♠å✐ ≤ i ≤ k − 1✳ ❱➻ f (x, y) ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✤è✐ ✈ỵ✐ Γ✱ t❛ ❝â t❤➸ ❣✐↔✐ ❦➻ ❞à t↕✐ O ❝õ❛ f (x, y) ❜➡♥❣ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ ❝❤➜♣ ♥❤➟♥ ữủ ố ợ f (x, y) õ ởt ❣✐↔✐ ①✉②➳♥ ❝õ❛ f (x, y) t↕✐ O✳ ❳➨t ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥ ✤ì♥ ❤➻♥❤ ❝❤➼♥❤ q✉② Σ∗ ❝❤➜♣ ♥❤➟♥ ữủ ố ợ ữỡ ữớ ❝♦♥❣ ♣❤➥♥❣ ♣❤ù❝ ❦❤æ♥❣ s✉② ❜✐➳♥ ✸✻ f (x, y) ✈ỵ✐ ❝→❝ ✤➾♥❤ s❛✉ ✤➙② Tj = cj dj ✈ỵ✐ ≤ j ≤ m + 1, tr♦♥❣ ✤â (c0 , d0 ) = (1, 0)✱ (cm+1 , dm+1 ) = (0, 1)✱ ✈➔ ❝→❝ ✤➾♥❤ ✤÷đ❝ s➢♣ t❤ù tü s❛♦ ❝❤♦ det(Tj , Tj+1 ) = ✈ỵ✐ ♠å✐ ≤ j ≤ m✳ ❉♦ Σ∗ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ✤è✐ ✈ỵ✐ f ✈➔ ❞♦ ≥ ✈ỵ✐ ♠å✐ ≤ i ≤ k ♥➯♥ t❛ ❝â {P1 , , Pk } ⊆ {T1 , , Tm }✳ ✣➦t cj cj+1 dj dj+1 σj = (Tj , Tj+1 ) = ✈ỵ✐ ≤ j ≤ m ❳➨t ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ ❧✐➯♥ ❦➳t ✈ỵ✐ ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥ ✤ì♥ ❤➻♥❤ Σ∗ s❛✉ ✤➙② π : X → C2 ; x, y , tr♦♥❣ ✤â X ❧➔ ♠ët ✤❛ t↕♣ ♣❤ù❝ trì♥✱ ✤÷đ❝ ❞→♥ tứ ỗ ữỡ (C2j ; xj , yj )✱ ≤ j ≤ m✱ t❤❡♦ ❧✉➟t ❞→♥ tự tr ỗ (C2j ; xj , yj ) ❧➔ c c d d π(xj , yj ) = Φσj (xj , yj ) = (xjj yj j+1 , xj j yj j+1 ) ❱✐➺❝ t➼♥❤ sè ❜ë✐ ν(Tj ) − ❝õ❛ π ∗ dx ∧ dy tr➯♥ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Tj ) ❧➔ ❞➵ ❞➔♥❣✱ t÷ì♥❣ tü ▼ư❝ ✸✳✶❀ t❛ t❤✉ ữủ (Tj ) = cj + dj , ợ ♠å✐ ≤ j ≤ m✳ ❇➙② ❣✐í t❛ t➼♥❤ sè ❜ë✐ ❝õ❛ π ∗ fi tr➯♥ E(Tj )✳ ❈â t➜t ❝↔ ❜❛ tr÷í♥❣ ❤đ♣ ❝➛♥ ①➨t✳ ❚❛ s➩ sû ❞ö♥❣ ♠ët sè t➼♥❤ t♦→♥ tr♦♥❣ ▼ö❝ ✸✳✷ →♣ ❞ö♥❣ ✈➔♦ ❝→❝ tr÷í♥❣ ❤đ♣ ✤â✳ ◆❤➢❝ ❧↕✐ r➡♥❣ P < Q ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ det(P, Q) > 0✳ ❚r÷í♥❣ ❤ñ♣ ✶✿ Pi ≤ Tj < Pi0 +1 ✱ tù❝ ❧➔ det(Pi0 , Tj ) ≥ ✈➔ det(Tj , Pi0 +1 ) > 0, tr♦♥❣ ✤â ≤ i0 ≤ k − 1✳ ❱➻ Pi0 ≤ Tj ♥➯♥ Pi ≤ Tj ✈ỵ✐ ♠å✐ ≤ i ≤ i0 ✳ ⑩♣ ❞ư♥❣ ▼ư❝ ✸✳✷ t❛ ❝â✱ ✈ỵ✐ ♠å✐ ≤ i ≤ i0 ✱ ri ∗ π fi (xj , yj ) = a dj −bi cj dj+1 −bi cj+1 yj +ξi rbc rbc xji i j yj i i j+1 (xj i )+xj Ri (xj , yj ) , =1 ✈ỵ✐ Ri (xj , yj ) ∈ C{xj , yj }✳ ❱➻ Tj < Pi0 +1 ♥➯♥ Tj < Pi ✈ỵ✐ ♠å✐ i0 + ≤ i ≤ k ✳ ⑩♣ ❞ư♥❣ ▼ư❝ ✸✳✷ t❛ ❝â✱ ✈ỵ✐ ♠å✐ i0 + ≤ i ≤ k ✱ ri π ∗ b cj −ai dj bi cj+1 −ai dj+1 yj rad rad fi (xj , yj ) = xji i j yj i i j+1 1+ξi xji =1 +xj Ri (xj , yj ) , ❈❤÷ì♥❣ ✹✳ ❑➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ♣❤ù❝ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✸✼ tr♦♥❣ ✤â Ri (xj , yj ) ∈ C{xj , yj }✳ ❚ø ❝→❝ ❜✐➸✉ t❤ù❝ tr➯♥ t❛ ❝â t❤➸ t❤✉ ✤÷đ❝ tự f tr ỗ (C2j ; xj , yj ) ♥❤÷ s❛✉ i0 ∗ k N (Tj ) N (Tj+1 ) yj u(xj , yj ), π ∗ fi (xj , yj ) = xj ∗ π fi (xj , yj ) · π f (xj , yj ) = i=i0 +1 i=1 tr♦♥❣ ✤â u(xj , yj ) ❧➔ ♠ët ♣❤➛♥ tû ❦❤↔ ♥❣❤à❝❤ tr♦♥❣ C{xj , yj }✱ ✈➔ i0 N (Tj ) = cj k ✭✹✳✹✮ ri ri bi + dj i=i0 +1 i=1 ◆❤➢❝ ❧↕✐ r➡♥❣✱ N (Tj ) ❝❤➼♥❤ ❧➔ sè ❜ë✐ ❝õ❛ π ∗ f tr➯♥ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Tj )✳ ❚r÷í♥❣ ❤đ♣ ✷✿ T0 ≤ Tj < P1 ✳ ✣✐➲✉ ❦✐➺♥ ♥➔② ❝â ♥❣❤➽❛ ❧➔ Tj < Pi ✈ỵ✐ ♠å✐ ≤ i ≤ k ✳ ⑩♣ ❞ư♥❣ ▼ư❝ ✸✳✷ t❛ ❝â✱ ✈ỵ✐ ♠å✐ ≤ i ≤ k ✱ ri π ∗ b cj −ai dj bi cj+1 −ai dj+1 yj rad rad fi (xj , yj ) = xji i j yj i i j+1 1+ξi xji +xj Ri (xj , yj ) , =1 tr♦♥❣ ✤â Ri (xj , yj ) ∈ C{xj , yj }✳ ❚÷ì♥❣ tü ♥❤÷ tr➯♥✱ ❞➵ ❞➔♥❣ t➼♥❤ ✤÷đ❝ sè ❜ë✐ ❝õ❛ π ∗ f tr➯♥ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Tj ) ❧➔ k N (Tj ) = dj ✭✹✳✺✮ ri i=1 ❚r÷í♥❣ ❤đ♣ ✸✿ Pk ≤ Tj ≤ Tm+1✳ ❚÷ì♥❣ tü ♥❤÷ ❚r÷í♥❣ ❤đ♣ ✷ t❛ ❝â k N (Tj ) = cj ✭✹✳✻✮ ri bi i=1 ◗✉❛♥ s→t ❝→❝ ❝æ♥❣ t❤ù❝ ✭✹✳✹✮✱ ✭✹✳✺✮ ✈➔ ✭✹✳✻✮ t❛ t❤➜②✱ ♥➳✉ ✤➦t P0 := T0 = (1, 0)t , Pk+1 := Tm+1 = (0, 1)t , ❝↔ ❜❛ tr÷í♥❣ ❤đ♣ tr➯♥ ❝ỉ♥❣ t❤ù❝ ❝õ❛ N (Tj ) ❝â t❤➸ ✤÷đ❝ ✈✐➳t ❝❤✉♥❣ ♥❤÷ s❛✉ i0 N (Tj ) = cj k+1 ri bi + dj i=0 ri , ✭✹✳✼✮ i=i0 +1 tr♦♥❣ ✤â Pi0 ≤ Tj < Pi0 +1 ✳ ●✐↔ sû Tj = Pi õ tr ỗ (C2j ; xj , yj )✱ ❜✐➸✉ t❤ù❝ ❝õ❛ π ∗ fi (xj , yj ) ✤÷đ❝ ❝❤♦ ❜ð✐ ri ∗ π fi (xj , yj ) = rbc xrji bi yj i i i+1 (yj + ξi ) + xj Ri (xj , yj ) , =1 ❈❤÷ì♥❣ ✹✳ ❑➻ ❞à ữớ ự ổ s ợ Ri (xj , yj ) ∈ C{xj , yj }✳ ❱➻ ✈➟② ri (yj + ξi ) + xj Ri (xj , yj ) = =1 ❝❤➼♥❤ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t tỹ sỹ tr ỗ (C2j ; xj , yj )✳ ❉♦ ✤â ❝â ri t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ❣✐❛♦ ✈ỵ✐ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Pi ) t↕✐ ❝→❝ ✤✐➸♠ ❣✐❛♦ (0, −ξi )✱ ≤ ri tr ỗ (C2j ; xj , yj )✳ ❚❛ ❦➼ ❤✐➺✉ ❝→❝ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ♥➔② ❧➔ E0i ✱ ✈ỵ✐ ≤ ≤ ri ✳ ❚ø ♣❤÷ì♥❣ tr➻♥❤ t❛ t❤➜② ✤÷đ❝ ❝→❝ t❤ỉ♥❣ sè tê ❤ñ♣ (N, ν) ❝õ❛ ❝→❝ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ❧➔ N (E0i ) = 1, ν(E0i ) = 1, ≤ i ≤ i, ≤ ≤ ri ◆➳✉ ≤ j ≤ m − ✈➔ ♥➳✉ Tj = Pi ✈ỵ✐ ♠å✐ ≤ i ≤ k ✱ t❤➻ E(Tj ) ❝❤➾ ❣✐❛♦ ✈ỵ✐ ✤ó♥❣ ❤❛✐ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ❦❤→❝ ✈➔ ❦❤ỉ♥❣ ❣✐❛♦ ✈ỵ✐ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ♥➔♦✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐ ❝õ❛ j ✱ ♥➳✉ T1 = P1 ✱ t❤➻ E(T1 ) ❝❤➾ ❣✐❛♦ ✈ỵ✐ ✤ó♥❣ ♠ët t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ❦❤→❝✱ ✈➔ ♥➳✉ Tm = Pk t❤➻ E(Tm ) ❝ơ♥❣ ❝❤➾ ❣✐❛♦ ✈ỵ✐ ✤ó♥❣ ♠ët t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ❦❤→❝✳ ❙❛✉ ✤➙② ❧➔ ữủ ỗ sỹ s t ♣❤➛♥ ❝→ ❜✐➺t ✈➔ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ❝õ❛ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à π ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ f (x, y) t↕✐ O✿ E(T2 ) E(Tj−1 ) (N (Tj−1 ), ν(Tj−1 )) ✳✳✳ ✳✳ ✳ E(T1 ) (N (T1 ), ν(T1 )) E0i1 ✲ (1, 1) E0i2 ✲ (1, 1) E0iri ✲ (1, 1) E(Tj+1 ) ✳✳✳ E(Tm ) E(Pi ) = E(Tj ) (N (Tj ), ν(Tj )) ✹✳✷ ❍➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❚r♦♥❣ ♠ö❝ ♥➔②✱ t❛ s➩ sû ❞ö♥❣ ❝→❝ ❞ú ❧✐➺✉ ❣✐↔✐ ❦➻ ❞à ♠ỉ t↔ tr♦♥❣ ▼ư❝ ✹✳✶ ✤➸ ự ỵ s ữỡ ữớ ự ổ s ỵ ✹✳✶✳ ❈❤♦ f (x, y) ❧➔ ♠ët ❦➻ ❞à t↕✐ O ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✤è✐ ✈ỵ✐ ✤❛ ❞✐➺♥ ◆❡✇t♦♥ ❝õ❛ ♥â✳ ●✐↔ sû ❜✐➸✉ t❤ù❝ ❝õ❛ f (x, y) ✤÷đ❝ ❝❤♦ tr♦♥❣ ✭✹✳✶✮ ✈➔ ✭✹✳✷✮✱ ✈➔ ❣✐↔ sû k i rt at − Di := rt bt = t=1 t=i+1 ✈ỵ✐ ♠å✐ ≤ i ≤ k✳ ❑❤✐ ✤â ❤➔♠ ③❡t❛ tæ♣æ ❝õ❛ f (x, y) t↕✐ O ❜➡♥❣ k top Zf,O (s) = i=1 −ri (N (Pi )ν(Pi ) + Di−1 Di )s − ri N (Pi )ν(Pi ) , Di−1 Di (s + 1)(N (Pi )s + ν(Pi )) tr♦♥❣ ✤â✱ ✈ỵ✐ ♠å✐ ≤ i ≤ k✱ ν(Pi) = + bi✱ i N (Pi ) = k rt bt + bi t=1 rt at t=i+1 ự ự ỵ ỹ sỹ ổ t tr trữợ ①✉②➳♥ π ❝õ❛ f (x, y) ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝ ✭✹✳✶✮✳ ❱➻ f (x, y) ❧➔ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ✤è✐ ✈ỵ✐ ✤❛ ❞✐➺♥ ◆❡✇t♦♥ ❝õ❛ ♥â ♥➯♥ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ π ❝ô♥❣ ❝❤➼♥❤ ❧➔ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à ❝õ❛ f (x, y) t↕✐ O✳ ❈→❝ ❞ú ❧✐➺✉ N (Tj ) ✈➔ ν(Tj ) ✈➔ sü s➢♣ ①➳♣ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ✈➔ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ❝õ❛ ♣❤➨♣ ❣✐↔✐ ✤➣ ✤÷đ❝ ①→❝ ✤à♥❤ tr♦♥❣ ▼ư❝ ✹✳✶✱ ❞♦ ✤â t❛ ❝❤➾ ❝á♥ ♣❤↔✐ t➼♥❤ ✤➦❝ tr÷♥❣ ❊✉❧❡r ❝õ❛ E(Tj )◦ ✈➔ E(Tj ) ∩ E(Tj )✳ ❚÷ì♥❣ tü ♥❤÷ tr♦♥❣ ▼ư❝ ✶✳✺✱ t❛ ❝â ❜↔♥❣ t➼♥❤ ✤➦❝ tr÷♥❣ ❊✉❧❡r s❛✉ ✤➙②✿ ●✐❛♦ ✣✳tr✳ ❊✉❧❡r ✣✐➲✉ ❦✐➺♥ E(T1 )◦ ✱ E(Tm )◦ E(Tj )◦ < j < m, Tj = Pi (∀ ≤ i ≤ k) E(Pi )◦ −ri 1≤i≤k ◦ E0i ≤ i ≤ k ✱ ≤ ≤ ri E0i ∩ E0i = ∅ E(Tj )∩E(Tj+1 ) = (1✤✐➸♠) 1≤j