Chỉ số chính quy của một số tập điểm béo

✣❸❍➴❈ ■❍❯➌ ❚❘×❮◆● ✣❸❍➴❈ ■❙× P❍❸▼ ∗∗∗∗∗∗ ❚❘❺◆ ❚❍➚ ❱■➏❚ ❚❘■◆❍ ❈❍➓ ❙➮ ❈❍➑◆❍ ◗❯❨ ❈Õ❆ ▼❐❚ ❙➮ ❚❾✣■➎▼ P❇➆❖ ▲❯❾◆ ❿❱❚❍❸ ❈◆❙➒❚⑩◆ ❍➴❈ ❖ ❚❍❊❖ ✣➚◆❍ ❍×❰◆● ◆●❍■➊◆ ❈Ù❯ ❚❤ø❛ ❚❤✐➯♥ ❍✉➳✱ ♥➠♠ ✷✵✶✽ ∗∗∗∗∗∗ ✣❸❍➴❈ ■❍❯➌ ❚❘×❮◆● ✣❸❍➴❈ ■❙× P▼❍❸ ∗∗∗∗∗∗ ∗∗∗∗∗∗ ❚❘❺◆ ❚❍➚ ❱■➏❚ ❚❘■◆❍ ❈❍➓ ❙➮ ❈❍➑◆❍ ◗❯❨ ❈Õ❆ ▼❐❚ ❙➮ ❚❾✣■➎▼ P❇➆❖ ❈❤✉②➯♥ ♥❣➔♥❤✿ ✣ ❸❙➮ ■⑨ ❱Þ ▲❚❍❯❨➌❚ ❙➮ ▼➣ sè✿ ✻✵ ✹✵✻✵✹✶ ▲❯❾◆ ❿❱❚❍❸ ❈◆❙➒❚⑩◆ ❍➴❈ ❖ ❚❍❊❖ ✣➚◆❍ ❍×❰◆● ◆●❍■➊◆ ❈Ù❯ ❈→♥ ởữợ P P ứ ❚❤✐➯♥ ❍✉➳✱ ♥➠♠ ✷✵✶✽ ✐ ▲❮■ ❈❆▼ ✣❖❆◆ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐✱ ❝→❝ sè ❧✐➺✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❣❤✐ tr tr tỹ ữủ ỗ t ❣✐↔ ❝❤♦ ♣❤➨♣ sû ❞ư♥❣ ✈➔ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❝ỉ♥❣ ❜è tr♦♥❣ ❜➜t ❦➻ ♠ët ❝æ♥❣ tr➻♥❤ ♥➔♦ ❦❤→❝✳ ❍✉➳✱ ♥❣➔② ✶✺ t❤→♥❣ ✾ ♥➠♠ ✷✵✶✽ ❍å❝ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ❚r➛♥ ❚❤à ❱✐➺t ❚r✐♥❤ ✐✐ ▲❮■ ❈❷▼ ❒◆ ▲✉➟♥ ✈➠♥ ữủ t ữợ sỹ ữợ ❣✐→♦✱ P●❙✳❚❙ P❤❛♥ ❱➠♥ ❚❤✐➺♥✳ ❚æ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✈➔ sü ❦➼♥❤ trå♥❣ ✤è✐ ✈ỵ✐ t t ữợ ú ù tổ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ❝ơ♥❣ ♥❤÷ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ổ ỷ ỡ qỵ ❚❤➛② ❝æ ❑❤♦❛ ❚♦→♥✱ ❝→❝ ❚❤➛② ð ✣↕✐ ❤å❝ ❍✉➳ ✈➔ ❱✐➺♥ ❚♦→♥ ❤å❝ ✤➣ ❞↕② ❞é ✈➔ tr✉②➲♥ ✤↕t ❦✐➳♥ t❤ù❝ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉ tr÷í♥❣ ✣❍❙P ❍✉➳✱ ♣❤á♥❣ ✣➔♦ t↕♦ s❛✉ ✣↕✐ ❤å❝✱ ❦❤♦❛ ❚♦→♥ tr÷í♥❣ ✣❍❙P ❍✉➳ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tỉ✐ tr♦♥❣ s✉èt ❦❤â❛ ❤å❝✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ❝→❝ ❛♥❤ ❝❤à ❈❛♦ ❤å❝ ❚♦→♥ ❦❤â❛ trữớ P số ỵ t❤✉②➳t sè ✈➻ sü ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈ø❛ q✉❛✳ ❉♦ ✤➙② ❧➔ ❧➛♥ ✤➛✉ t✐➯♥ t❤ü❝ ❤✐➺♥ ❝æ♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ♥➯♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ tæ✐ r➜t ♠♦♥❣ ữủ sỹ õ õ ỵ t ❝ỉ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❜➔✐ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥✳❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍✉➳✱ ♥❣➔② ✶✺ t❤→♥❣ ✾ ♥➠♠ ✷✵✶✽ ❍å❝ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ❚r➛♥ ❚❤à ❱✐➺t ❚r✐♥❤ ✐✐✐ ▼Ư❈ ▲Ư❈ ❚r❛♥❣ ♣❤ư ❜➻❛ ✐ ▲í✐ ❝❛♠ ✤♦❛♥ ✐✐ ▲í✐ ❝↔♠ ì♥ ✐✐✐ ▼ư❝ ❧✉❝ ✶ ▼❐❚ ị ì ề ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✼ ✶✳✶ ❱➔♥❤ ♣❤➙♥ ❜➟❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ ❝❤✐➲✉ ❑r✉❧❧ ❝õ❛ ✈➔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✶✳✶ ❱➔♥❤ ♣❤➙♥ ❜➟❝ ✈➔ ♠æ✤✉♥ ♣❤➙♥ ❜➟❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✶✳✷ ✣❛ t↕♣ ①↕ ↔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✳✸ ❈❤✐➲✉ ❑r✉❧❧ ❝õ❛ ✈➔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✳✹ ❱➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷✳✶ ❍➔♠ ❍✐❧❜❡rt ✈➔ ✤❛ t❤ù❝ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷✳✷ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷✳✸ ▼è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ ✈ỵ✐ ❝❤➾ sè ❝❤➼♥❤ q✉② ❈❛st❡❧♥✉♦✈♦✲▼✉♠❢♦r❞ ❝õ❛ ✈➔♥❤ t♦↕ ✤ë✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ❈❤÷ì♥❣ ✷ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët sè t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤ P ✷✵ n ✷✳✶ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ ♥➡♠ tr➯♥ ✷ ✤÷í♥❣ t❤➥♥❣ ♣❤➙♥ ❜✐➺t ✷✵ ✷✳✷ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ s ✤✐➸♠ ❜➨♦ ♣❤➙♥ ❜✐➺t tr♦♥❣ Pn ✱ s ≤ ✺ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸ ✷✻ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ♥✰✸ ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ✳ ✳ ❑➳t ❧✉➟♥ ✸✸ ✶ ❚➔✐ t ị ì ❉Ị◆● ❑➼ ❤✐➺✉ Þ ♥❣❤➽❛ ❚➟♣ sè ♥❣✉②➯♥ ❚➟♣ sè ♥❣✉②➯♥ ❞÷ì♥❣ ❑❤ỉ♥❣ ❣✐❛♥ ①↕ ↔♥❤ ♥✲❝❤✐➲✉ tr➯♥ tr÷í♥❣ ✤â♥❣ ✤↕✐ sè ❦ R := k[x0 , , xn ] ❱➔♥❤ ✤❛ t❤ù❝ t❤❡♦ ❝→❝ ❜✐➳♥ x0 , , xn tr➯♥ tr÷í♥❣ k ❆♥♥(M ) ❆♥♥✐❤✐t♦r ❝õ❛ R✲♠ỉ✤✉♥ M e(A) ❙è ❜ë✐ ❝õ❛ ✈➔♥❤ t♦↕ ✤ë t❤✉➛♥ ♥❤➜t A HM (t) ❍➔♠ ❍✐❧❜❡rt ❝õ❛ ♠æ✤✉♥ ♣❤➙♥ ❜➟❝ M r❡❣(Z) ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ Z (S) ✭❤❛② S ✮ ■❞❡❛❧ ♥❣✉②➯♥ tè t❤✉➛♥ ♥❤➜t ①→❝ ✤à♥❤ ❜ð✐ t➟♣ S M ❚ê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ♥❤â♠ ❝♦♥ Md d d ❞✐♠B ❈❤✐➲✉ ✭❑r✉❧❧✮ ❝õ❛ ✈➔♥❤ B Z(T ) ❚➟♣ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ t➟♣ T ❝→❝ ♣❤➛♥ tû t❤✉➛♥ ♥❤➜t ❝õ❛ ✈➔♥❤ R = k[x0 , , xn ] rf t ỗ f [a] ❙è ♥❣✉②➯♥ ❧ỵ♥ ♥❤➜t b s❛♦ ❝❤♦ b ≤ a✱ a ∈ Q Z Z+ Pn := Pnk ✸ ▲❮■ ◆➶■ ✣❺❯ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ ❝â t❤➸ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ t❤ỉ♥❣ q✉❛ ❤➔♠ ❍✐❧❜❡rt✱ ❝ư t❤➸ ♥❤÷ s❛✉✿ ❈❤♦ X = {P1 , , Ps } ❧➔ t➟♣ ❝→❝ ✤✐➸♠ ♣❤➙♥ ❜✐➺t tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ①↕ ↔♥❤ Pn := Pnk ✱ ✈ỵ✐ k ❧➔ ♠ët tr÷í♥❣ ✤â♥❣ ✤↕✐ sè✳ ●å✐ ℘1 , , ℘s ❧➔ ❝→❝ ✐❞❡❛❧ ♥❣✉②➯♥ tè t❤✉➛♥ ♥❤➜t ❝õ❛ ✈➔♥❤ ✤❛ t❤ù❝ R := k [x0 , , xn ] tữỡ ự ợ P1 , , Ps ❈❤♦ ms m1 , , ms ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ ✤➦t I := ℘m ∩ ∩ ℘s ✳ ❚❛ ❣å✐ (X, I) ❧➔ t➟♣ ✤✐➸♠ ❜➨♦ tr Pn ỵ Z := m1 P1 + · · · + ms Ps ❱➔♥❤ t♦↕ ✤ë t❤✉➛♥ ♥❤➜t ❝õ❛ Z ❧➔ A := R/I ✳ ❱➔♥❤ A = ♣❤➙♥ ❜➟❝ ✈ỵ✐ ❜ë✐ ❝õ❛ ♥â ❧➔ s e(A) := i=1 t≥0 At ❧➔ ♠ët ✈➔♥❤ mi + n − n ❍➔♠ ❍✐❧❜❡rt ❝õ❛ Z ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ HA (t) := dimk At ✱ t➠♥❣ ❝❤➦t ❝❤♦ ✤➳♥ ❦❤✐ ✤↕t ✤÷đ❝ sè ❜ë✐ e(A)✱ t↕✐ ✤â ♥â ❞ø♥❣✳ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ Z ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ sè ♥❣✉②➯♥ ❜➨ ♥❤➜t t s❛♦ ❝❤♦ HA (t) = e(A) ✈➔ ♥â ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ r❡❣(Z)✳ ❱➜♥ ✤➲ t➻♠ ❝❤➦♥ tr➯♥ ❝õ❛ ❝❤➾ sè ❝❤➼♥❤ q✉② r❡❣(Z) ✤➣ ✤÷đ❝ r➜t ♥❤✐➲✉ ♥❣÷í✐ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✳ ◆➠♠ ✶✾✻✶✱ ❙❡❣r❡ ✭①❡♠ ❬✶✼❪✮ ✤➣ ❝❤➾ r❛ ✤÷đ❝ ❝❤➦♥ tr➯♥ ❝õ❛ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝❤♦ ❝→❝ t➟♣ ✤✐➸♠ ❜➨♦ tê♥❣ q✉→t Z = m1 P1 + · · · + ms Ps tr♦♥❣ P2 ✿ reg(Z) ≤ max m1 + m2 − 1, m1 + · · · + ms ✈ỵ✐ m1 ≥ · · · ≥ ms ✳ ✣➳♥ ♥➠♠ ✶✾✾✶✱ ❈❛t❛❧✐s❛♥♦ ✭①❡♠ ❬✻❪✮ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ tr➯♥ ❝❤♦ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ ð ✈à tr➼ tê♥❣ q✉→t tr♦♥❣ P2 ✳ ❱➔♦ ♥➠♠ ✶✾✾✸✱ ❈❛t❛❧✐s❛♥♦✱ ❚r✉♥❣ ✈➔ ❱❛❧❧❛ ✭①❡♠ ❬✼❪✮ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ ♥➔② ❝❤♦ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ ð ✈à tr➼ tê♥❣ q✉→t tr♦♥❣ Pn ✿ m1 + · · · + ms + n − n ◆➠♠ ✶✾✾✻✱ ◆✳❱✳❚r✉♥❣ ✭①❡♠ ❬✷✵❪✮ ✤➣ ❞ü ✤♦→♥ r➡♥❣ ❝❤➦♥ tr ởt t tý ỵ Z = m1 P1 + · · · + ms Ps tr♦♥❣ Pn ❧➔ reg(Z) ≤ max m1 + m2 − 1, r❡❣(Z) ≤ max {Tj |j = 1, , n} , tr♦♥❣ ✤â ✹ Tj = max q l=1 mi l + j − |Pi1 , , Piq ♥➡♠ tr➯♥ ♠ët ❥✲♣❤➥♥❣ ✳ j ❍✐➺♥ ♥❛②✱ ❝❤➦♥ tr♦♥❣ ❞ü ✤♦→♥ ❝õ❛ ◆✳❱✳❚r✉♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❝❤➦♥ tr➯♥ ❝õ❛ ❙❡❣r❡✳ ❈❤➦♥ tr➯♥ ❙❡❣r❡ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ✤ó♥❣ tr♦♥❣ ♥❤✐➲✉ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✿ tr÷í♥❣ ❤đ♣ n = 2, ✭①❡♠ ❬✾❪✱ ❬✶✵❪✮✱ ❬✶✽❪✱ ❬✶✾❪✮✱ ❝❤♦ t➟♣ ✤✐➸♠ ❦➨♣ Z = 2P1 + · · · + 2Ps tr♦♥❣ P4 ✭①❡♠ ❬✷✵❪✮✱ ❝❤♦ t➟♣ n + ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ✭①❡♠ ❬✸❪✮✱ ❝❤♦ n + ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ✭①❡♠ ❬✷❪✮✱✳✳✳✳ ●➛♥ ✤➙②✱ ❯✳ ◆❛❣❡❧ ✈➔ ❇✳ ❚r♦❦ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❝❤➦♥ tr➯♥ ❝õ❛ ❙❡❣r❡ ✤ó♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t ✭①❡♠ ❬✷✹❪✮✳ ❇➔✐ t♦→♥ t➼♥❤ ✤÷đ❝ ❝❤➾ sè ❝❤➼♥❤ q✉② r❡❣(Z) ❧➔ ❦❤â ❤ì♥ ❜➔✐ t♦→♥ t➻♠ ❝❤➦♥ tr➯♥ ❝❤♦ r❡❣(Z)✳ ◆➠♠ ✶✾✽✹✱ ❉❛✈✐s ✈➔ ●❡r❛♠✐t❛ ✭①❡♠ ❬✽❪✮ ✤➣ t➼♥❤ ✤÷đ❝ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝❤♦ t➟♣ ✤✐➸♠ ❜➨♦ Z = m1 P1 + · · · + ms Ps ♥➡♠ tr➯♥ ♠ët ✤÷í♥❣ t❤➥♥❣ ❝õ❛ Pn ✿ reg(Z) = m1 + · · · + ms − ❱➔♦ ♥➠♠ ✶✾✾✸✱ ❈❛t❛❧✐s❛♥♦✱ ❚r✉♥❣ ✈➔ ❱❛❧❧❛ ✭①❡♠ ❬✼❪✮ ✤➣ t➼♥❤ ✤÷đ❝ ❝❤➾ sè ❝❤➼♥❤ q✉② r❡❣(Z) ❝❤♦ t➟♣ ✤✐➸♠ ❜➨♦ ♥➡♠ tr➯♥ ♠ët ✤÷í♥❣ ❝♦♥❣ ❤ú✉ t✛ ❝❤✉➞♥ tr♦♥❣ Pn ✿ mi + n − n ◆➠♠ ✷✵✶✷✱ ❚❤✐➺♥ ✭①❡♠ ❬✷✶❪✮ ❝ơ♥❣ ✤➣ t➼♥❤ ✤÷đ❝ ❝❤➾ sè ❝❤➼♥❤ q✉② r❡❣(Z) ❝❤♦ s✰✷ ✤✐➸♠ ❜➨♦ s❛♦ ❝❤♦ ❝❤ó♥❣ ❦❤ỉ♥❣ ♥➡♠ tr➯♥ ✭s−✶✮✲♣❤➥♥❣ tr♦♥❣ Pn ✿ r❡❣(Z) = max m1 + m2 − 1, r❡❣(Z) = max {Tj |j = 1, , n} , tr♦♥❣ ✤â Tj = max q l=1 mil + j − |Pi1 , , Piq ♥➡♠ tr➯♥ ♠ët ❥✲♣❤➥♥❣ j ◆➠♠ ✷✵✶✼✱ ❚❤✐➺♥ ✈➔ ❙✐♥❤ ✭①❡♠ ❬✷✸❪✮ ✤➣ t➼♥❤ ✤÷đ❝ ❝❤➾ sè ❝❤➼♥❤ q✉② r❡❣✭Z ✮ t s ỗ s ú ❦❤æ♥❣ ♥➡♠ tr➯♥ ✭r − 1✮✲♣❤➥♥❣✱ s ≤ r + tr♦♥❣ Pn ✿ r❡❣(Z) = max {Tj |j = 1, , n} , tr♦♥❣ ✤â Tj = max mq + j − |Pi1 , , Piq ♥➡♠ tr➯♥ ởt j ợ ố ữủ t ✈➔ ♥❣❤✐➯♥ ❝ù✉ t❤➯♠ ✈➲ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤ Pn ✈➔ ữủ sỹ ữợ t P P ❱➠♥ ❚❤✐➺♥✱ tæ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐✿ ✧❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët sè t➟♣ ✤✐➸♠ ❜➨♦✧ ✤➸ t✐➳♥ ❤➔♥❤ ♥❣❤✐➯♥ ❝ù✉✳ ❈❤ó♥❣ tỉ✐ ✤➣ t➼♥❤ ✤÷đ❝ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët sè t➟♣ ✤✐➸♠ ❜➨♦ ❝❤÷❛ ♥➡♠ tr♦♥❣ trữớ ủ tr ỗ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ t➟♣ ✤✐➸♠ ❜➨♦ ✈➔ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ①↕ ↔♥❤✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ sü t➼♥❤ t♦→♥ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët sè t➟♣ ✤✐➸♠ ❜➨♦✱ ❝ö t❤➸ ❧➔✿ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ s ✤✐➸♠ ❜➨♦ ♣❤➙♥ ❜✐➺t tr♦♥❣ Pn ✱ s ≤ 5❀ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ ♥➡♠ tr➯♥ ✷ ✤÷í♥❣ t❤➥♥❣ ✈➔ ♠ët sè tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ n + ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ✳ ✻ ❍➺ q✉↔ ✷✳✶✳✶✳ ✭❬✸❪✱ ▲❡♠♠❛ ✹✳✹✮ ❈❤♦ Z = m1 P1 + · · · + ms Ps ❧➔ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ P ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ♠ët r✲❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ α ∼ = Pr ✳ ❚❛ ❝â t❤➸ ①❡♠ r✲❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ α ♥❤÷ ❦❤ỉ♥❣ ❣✐❛♥ ①↕ ↔♥❤ r✲❝❤✐➲✉ Pr ❝❤ù❛ ❝→❝ ✤✐➸♠ P1 := P1 , , Ps := Ps ✈➔ Zα = m1 P1 + · · · + ms Ps ♥❤÷ ❧➔ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ Pr ✳ ◆➳✉ ❝â ♠ët sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ t s❛♦ ❝❤♦ r❡❣✭Zα ✮≤ t tr♦♥❣ Pr ✱ t❤➻ n r❡❣(Z) ≤ t tr Pn ỵ r Z = m P 1 + · · · + ms Ps ởt t tý ỵ tr♦♥❣ P ✳ ❑❤✐ ✤â reg(Z) ≤ max {T1 (Z), T2 (Z), T3 (Z)} ❉ü❛ ✈➔♦ ❝→❝ ❦➳t q✉↔ tr➯♥ ❝❤ó♥❣ tỉ✐ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦➳t q✉↔ s❛✉✿ ỵ Z = m P + à · · + ms Ps ❧➔ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ Pn ✳ ◆➳✉ P1 , , Ps ♥➡♠ tr➯♥ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ ♣❤➙♥ ❜✐➺t l1 ✱ l2 t❤➻ 1 T (Z) − ≤ reg(Z) ≤ T (Z) ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû Pi1 , , Pir ❧➔ ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ s❛♦ ❝❤♦ T1 (Z) = mi1 + · · · + mir − ●å✐ Y = mi1 Pi1 + · · · + mir Pir ✱ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✶ t❛ ❝â reg(Z) ≥ reg(Y ) = T1 (Z) ✭✷✳✶✮ ▼➦t ❦❤→❝✱ ❞♦ ❝→❝ ✤✐➸♠ P1 , , Ps ♥➡♠ tr➯♥ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ l1 , l2 tỗ t ởt ự q ỵ t ❝â reg(Z) ≤ max{T1 (Z), T2 (Z), T3 (Z)} ✭✷✳✷✮ ❚❛ ❝â T3 (Z) = max q l=1 mil + |Pi1 , , Piq ♥➡♠ tr➯♥ ♠ët ✸✲♣❤➥♥❣ t✉②➳♥ t➼♥❤ ✷✶ ❙✉② r❛ s i=1 mi + 2T1 (Z) + ≤ ✭ ❞♦ P1 , , Ps ♥➡♠ tr➯♥ ✷ ✤÷í♥❣ t❤➥♥❣ ✮ 3T1 (Z) + ≤ ≤ T1 (Z) + = T1 (Z) T3 (Z) ≤ ✭✷✳✸✮ ❚❛ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉✿ ❚r÷í♥❣ ❤đ♣ ✶✿ ◆➳✉ T1 (Z) ≥ T2 (Z) t❤➻ t❛ ❝â r❡❣(Z) ≤ T1 (Z) ✭t❤❡♦ ✭✷✳✷✮ ✈➔ ✭✷✳✸✮✮ ❑➳t ❤đ♣ ✈ỵ✐ ✭✷✳✶✮ t❛ ❝â r❡❣(Z) = T1 (Z) = T (Z) ❚r÷í♥❣ ❤đ♣ ✷✿ ◆➳✉ T1 (Z) < T2 (Z) t❤➻ t❤❡♦ ✭✷✳✷✮ t❛ ❝â r❡❣(Z) ≤ max{T1 (Z), T2 (Z), T3 (Z)} ❙✉② r❛ r❡❣(Z) ≤ T2 (Z) = T (Z) ❑➳t ❤đ♣ ✈ỵ✐ ✭✷✳✶✮ t❛ ❝â T1 (Z) ≤ r❡❣(Z) ≤ T2 (Z) ❚❛ ❝â T1 (Z) + = mi1 + · · · + mir ≥ s mk k=1 ≥ T2 (Z) ✭❞♦ P1 , , Ps ♥➡♠ tr➯♥ ✷ ✤÷í♥❣ t❤➥♥❣✮ ▼➔ T1 (Z) < T2 (Z) ♥➯♥ T1 (Z) + = T2 (Z) ✷✷ ✭✷✳✹✮ ❑➳t ❤đ♣ ✈ỵ✐ ✭✷✳✹✮ t❛ ❝â T (Z) − reg(Z) T (Z) ự ỵ ✤➣ ❤♦➔♥ t❤➔♥❤✳ ✷✳✷ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ s ✤✐➸♠ ❜➨♦ ♣❤➙♥ ❜✐➺t tr♦♥❣ Pn✱ s ≤ ✺ r ú tổ s ữợ ữủ số ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ s ✤✐➸♠ ❜➨♦ ♣❤➙♥ ❜✐➺t tr♦♥❣ Pn ✱ s ≤ 5✳ ◆❣♦➔✐ ❝→❝ ❦➳t q✉↔ ✤➣ tr➻♥❤ ❜➔② tr♦♥❣ ♣❤➛♥ ✷✳✶✱ ❝❤ó♥❣ tỉ✐ ❝➛♥ ❞ị♥❣ t❤➯♠ ♠ët sè ❦➳t q✉↔ s❛✉ tr♦♥❣ ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ♠➻♥❤✳ ❇ê ✤➲ ✷✳✷✳✶✳ ✭❬✷✶❪✱ ❚❤❡♦r❡♠ ✸✳✹✮ ❈❤♦ P , , P ❧➔ ❝→❝ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❦❤æ♥❣ ♥➡♠ tr➯♥ ♠ët (s − 1)✲♣❤➥♥❣ t✉②➳♥ t➼♥❤ tr♦♥❣ P ✱ s ≤ n✱ ✈➔ m1 , , ms+2 ❧➔ ❝→❝ sè ♥❣✉②➯♥ ms+2 ❞÷ì♥❣✳ ✣➦t I = ℘m ∩ ∩ ℘s+2 ✱ A = R/I ✳ ❑❤✐ ✤â✱ s+2 n r(A) = T (Z) ỵ ✭❬✷✸❪✱ ❚❤❡♦r❡♠ ✸✳✶✮ ❈❤♦ P , , P s ❧➔ ❝→❝ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ð ✈à tr➼ tê♥❣ q✉→t tr➯♥ ♠ët r✲♣❤➥♥❣ t✉②➳♥ t➼♥❤ α tr♦♥❣ P ✱ s ≤ r + 3✳ ❈❤♦ m1 , , ms ❧➔ ❝→❝ sè n ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ Z = m1 P1 + · · · + ms Ps ✳ ❑❤✐ ✤â reg(Z) = max {T1 (Z), Tr (Z)} ▼➺♥❤ ✤➲ ✷✳✷✳✶✳ ✭❬✷✷❪✱ Pr♦♣♦s✐t✐♦♥ ✽✮ ❈❤♦ Z = m P + · · · + ms Ps ❧➔ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ P ✳ ◆➳✉ P1 , , Ps ♥➡♠ tr➯♥ ♠ët ✤÷í♥❣ t❤➥♥❣ t❤➻ 1 n reg(Z) = m1 + · · · + ms − ✣à♥❤ ỵ r P , , P s ❧➔ ❝→❝ ✤✐➸♠ ♣❤➙♥ ❜✐➺t tr♦♥❣ P2 ✈➔ Z = m1 P1 + · · · + ms Ps ❧➔ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ P2 ✳ ❑❤✐ ✤â reg(Z) ≤ max h − 1, s mi , i=1 tr♦♥❣ ✤â h := max{ k j=1 mij |Pi1 , , Pik t❤➥♥❣ ❤➔♥❣ }✳ ❉ü❛ ✈➔♦ ❝→❝ ❦➳t q✉↔ tr➯♥✱ ❝❤ó♥❣ tỉ✐ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦➳t q s ỵ Z = m P 1 + · · · + ms Ps ❧➔ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ Pn ✱ s ≤ 5✳ ▲ó❝ ✤â T (Z) − ≤ reg(Z) ≤ T (Z) ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ①➨t ✷ tr÷í♥❣ ❤đ♣ s❛✉✿ ❚r÷í♥❣ ❤đ♣ ✶✿ ◆➳✉ P , , P s ♥➡♠ tr➯♥ ♠ët ✤÷í♥❣ t❤➥♥❣ t❤➻ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✷✳✶ t❛ ❝â reg(Z) = T1 (Z) = m1 + · · · + ms − ❚r÷í♥❣ ❤đ♣ ✷✿ ◆➳✉ P , , P s ❦❤æ♥❣ ♥➡♠ tr➯♥ ♠ët ✤÷í♥❣ t❤➥♥❣ t❤➻ s ≥ 3✳ ❚❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ ❝♦♥ s❛✉✿ ❚r÷í♥❣ ❤đ♣ ✷✳✶✿ ❚r÷í♥❣ ❤đ♣ s = ❤♦➦❝ s = t❤➻ ❞♦ P , , P s ❦❤æ♥❣ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ ♥➯♥ t❤❡♦ ❇ê ✤➲ ✷✳✷✳✶ t❛ ❝â reg(Z) = T (Z) ❚r÷í♥❣ ❤đ♣ ✷✳✷✿ ❚r÷í♥❣ ❤đ♣ s = 5✳ ❚❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ ❝♦♥ s❛✉✿ ❚r÷í♥❣ ❤đ♣ ✷✳✷✳✶✿ ◆➳✉ P1 , , Ps ❦❤æ♥❣ ♥➡♠ tr➯♥ ✷✲♣❤➥♥❣ t✉②➳♥ t➼♥❤ t❤➻ t❤❡♦ ❇ê ✤➲ ✷✳✷✳✶ t❛ ❝â reg(Z) = T (Z) ❚r÷í♥❣ ❤đ♣ ✷✳✷✳✷✿ ◆➳✉ P1 , , Ps ♥➡♠ tr➯♥ ✷✲♣❤➥♥❣ t✉②➳♥ t➼♥❤ α ✈➔ ❝❤ó♥❣ ð ✈à tr➼ tê♥❣ q✉→t tr➯♥ t t ỵ t õ reg(Z) = max{T1 (Z), T2 (Z)} = T (Z) ❚r÷í♥❣ ❤đ♣ ✷✳✷✳✸✿ ◆➳✉ P1 , , Ps ♥➡♠ tr➯♥ ✷✲♣❤➥♥❣ t✉②➳♥ t➼♥❤ α ✈➔ ❝❤ó♥❣ ❦❤ỉ♥❣ ð ✈à tr➼ tê♥❣ q✉→t tr➯♥ α t❤➻ ❝â ✷ ✤÷í♥❣ t❤➥♥❣ l1 , l2 ❝❤ù❛ P1 , , Ps ✳ ●✐↔ sû Pi1 , , Pir ❧➔ ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ s❛♦ ❝❤♦ T1 (Z) = mi1 + · · · + mir − ●å✐ Y = mi1 Pi1 + · · · + mir Pir ✱ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✶ t❛ ❝â reg(Z) ≥ reg(Y ) = T1 (Z) ✷✹ ✭✷✳✺✮ ❚❤❡♦ q ỵ t õ reg(Z) ≤ max h − 1, mi i=1 = max {h − 1, T2 (Z)} , tr♦♥❣ ✤â✱ k mij |Pi1 , , Pik t❤➥♥❣ ❤➔♥❣ h := max j=1 ❙✉② r❛ T1 (Z) = h − 1✳ ❉♦ ✤â reg(Z) ≤ max{T1 (Z), T2 (Z)} ✭✷✳✻✮ t trữớ ủ s ã T2 (Z) ≤ T1 (Z) t❤➻ t❤❡♦ ✭✷✳✻✮ t❛ ❝â reg(Z) ≤ T1 (Z) ✭✷✳✼✮ ❚ø ✭✷✳✺✮ ✈➔ ✭✷✳✼✮ t❛ ❝â reg(Z) = T1 (Z)✳ • ◆➳✉ T2 (Z) > T1 (Z) t❤➻ t❤❡♦ ✭✷✳✺✮ ✈➔ ✭✷✳✻✮ t❛ ❝â T1 (Z) ≤ reg(Z) ≤ max{T1 (Z), T2 (Z)} = T2 (Z) ❚❛ ❝â T1 (Z) + = mi1 + · · · + mir ≥ mk k=1 = T2 (Z)✭ ❞♦ P1 , , Ps ♥➡♠ tr➯♥ ✷✲♣❤➥♥❣ ✮ ✷✺ ✭✷✳✽✮ ❙✉② r❛ T1 (Z) + ≥ T2 (Z)✳ ❉♦ T1 (Z) < T2 (Z) ♥➯♥ T1 (Z) + = T2 (Z) ❑➳t ❤đ♣ ✈ỵ✐ ✭✷✳✽✮ t❛ ❝â T (Z) − ≤ reg(Z) ≤ T (Z) ự ỵ t ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ♥✰✸ ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ❚➟♣ n + ✤✐➸♠ ❜➨♦ Z = m1 P1 + + mn+3 Pn+3 tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ①↕ ↔♥❤ Pn ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ♥➳✉ t➜t ❝↔ ❝→❝ ✤✐➸♠ P1 , , Pn+3 ❦❤ỉ♥❣ ❝ị♥❣ ♥➡♠ tr➯♥ ♠ët s✐➯✉ ♣❤➥♥❣✳ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ú tổ s ữợ ữủ số q n + ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ✳ ❈❤ó♥❣ tỉ✐ ❝➛♥ ❞ị♥❣ t❤➯♠ ❝→❝ ❦➳t q✉↔ s❛✉ tr♦♥❣ ự t q ỵ ✭❬✷❪✱ ❚❤❡♦r❡♠ ✷✳✶✮ ❈❤♦ Z := n+3 i=1 mi Pi ❧➔ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ✳ ❑❤✐ ✤â✱ Z t❤♦↔ ♠➣♥ ❝➟♥ tr➯♥ ❙❡❣r❡ tê♥❣ qt tự reg(Z) T (Z) ỵ ✭❬✷✸❪✱ ❚❤❡♦r❡♠ ✹✳✻✮ ❈❤♦ P , , P s ❧➔ ❝→❝ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❦❤æ♥❣ ♥➡♠ tr➯♥ (r − 1)✲♣❤➥♥❣ tr♦♥❣ P ✱ s ≤ r + ✈➔ ♠ ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✱ m = 2✳ ❈❤♦ n Z = mP1 + · · · + mPs t ỗ õ reg(Z) = T (Z) ỹ ỵ tr t q trữợ ú tổ ự ữủ t q s ỵ Z := n+3 i=1 mi Pi ❧➔ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ✳ ●✐↔ sû r➡♥❣ m1 > m2 > m3 ≥ m4 ≥ · · · ≥ mn+3 ✳ ❑❤✐ ✤â T (Z) − ≤ reg(Z) ≤ T (Z) ự ỵ t õ reg(Z) ≤ T (Z) ✭✷✳✾✮ ❚❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉ t❤❡♦ ♥✿ ❚r÷í♥❣ ❤đ♣ ✶✿ n ≤ t❤➻ t❤❡♦ ỵ t õ T (Z) reg(Z) ≤ T (Z) ❚r÷í♥❣ ❤đ♣ ✷✿ n ≥ 3✿ ❞♦ Z = m P + · · · + mn+3 Pn+3 ❧➔ ❝→❝ ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ ♥➯♥ ❝â tè✐ ✤❛ ✹ ✤✐➸♠ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣✳ ●✐↔ sû Pi1 , , Pir ❧➔ ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ s❛♦ ❝❤♦ T1 (Z) = mi1 + · · · + mir − 1 ❑❤✐ ✤â✱ ≤ r ≤ 4✳ ●å✐ Y = mi1 Pi1 + · · · + mir Pir ✱ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✶ t❛ ❝â reg(Z) ≥ reg(Y ) = T1 (Z) ✭✷✳✶✵✮ ❉♦ Z = m1 P1 + · · · + mn+3 Pn+3 ❧➔ ❝→❝ ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ ♥➯♥ ❝â tè✐ ✤❛ j + ✤✐➸♠ ♥➡♠ tr➯♥ ❥✲♣❤➥♥❣✱ ≤ j ≤ n✳ ❚❛ s♦ s→♥❤ ❝→❝ Ts (Z) ợ T1 (Z) s n rữớ ❤đ♣ ✷✳✶✿ ❱ỵ✐ ≤ s ≤ n✱ ❧ó❝ ✤â ❝â tè✐ ✤❛ s + ✤✐➸♠ ♥➡♠ tr➯♥ s✲♣❤➥♥❣✳ ●✐↔ sû Pq1 , , Pqk ❧➔ ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ s✲♣❤➥♥❣ s❛♦ ❝❤♦ Ts (Z) = k l=1 mql + s − , mq1 ≥ mq2 ≥ · · · ≥ mqk s ❑❤✐ ✤â✱ s + ≤ k ≤ s + 3✳ ❚❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ ❝♦♥ s❛✉✿ ❚r÷í♥❣ ❤đ♣ ✷✳✶✳✶✿ ❦ ❂ s✰✸✱ ❧ó❝ ✤â Ts (Z) = s+3 l=1 mql + s − s ❚❛ ❝â mq1 ≤ m1 ✱ mq2 ≤ m1 − 1✱✳✳✳✱ mqs ≤ m1 − 1✱ mqs+1 ≤ m2 − 1✱ mqs+2 ≤ m2 − 1✱ mqs+3 ≤ m2 − 1✳ ❙✉② r❛ Ts (Z) ≤ sm1 + 3m2 − ≤ m1 + m2 − s ❉♦ ✤â Ts (Z) ≤ T1 (Z) ✷✼ ❚r÷í♥❣ ❤đ♣ ✷✳✶✳✷✿ k = s + 2✱ ❧ó❝ ✤â Ts (Z) = s+2 l=1 mql + s − s ❚❛ ❝â mq1 ≤ m1 ✱ mq2 ≤ m1 − 1✱✳✳✳✱ mqs ≤ m1 − 1✱ mqs+1 ≤ m2 − 1✱ mqs+2 ≤ m2 − 1✳ ❙✉② r❛ Ts (Z) ≤ sm1 + 2m2 − ≤ m1 + m2 − s ❉♦ ✤â Ts (Z) ≤ T1 (Z) ❚r÷í♥❣ ❤đ♣ ✷✳✶✳✸✿ k = s + 1✱ ❧ó❝ ✤â Ts (Z) = s+1 l=1 mql + s − s ❚❛ ❝â mq1 ≤ m1 ✱ mq2 ≤ m1 − 1✱✳✳✳✱ mqs ≤ m1 − 1✱ mqs+1 ≤ m2 − 1✳ ❙✉② r❛ Ts (Z) ≤ sm1 + m2 − ≤ m1 + m2 − s ❉♦ ✤â Ts (Z) ≤ T1 (Z) ❱➟②✱ tr♦♥❣ ❚r÷í♥❣ ❤đ♣ ✷✳✶ t❛ ❝â Ts (Z) ≤ T1 (Z) rữớ ủ ợ s = t t trữớ ủ s ã T1 (Z) T2 (Z) t❤➻ ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✶✶✮ t❛ ❝â T (Z) = T1 (Z) ❚❤❡♦ ✭✷✳✾✮ t❛ ❝â reg(Z) ≤ T (Z) = T1 (Z) ▼➔ reg(Z) ≥ T1 (Z) = T (Z) ✭t❤❡♦ ✭✷✳✶✵✮✮ ♥➯♥ reg(Z) = T1 (Z) = T (Z) ✷✽ ✭✷✳✶✶✮ • ◆➳✉ T1 (Z) < T2 (Z) t❤➻ ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✶✶✮ t❛ ❝â T (Z) = T2 (Z) ❚❤❡♦ ✭✷✳✾✮ ✈➔ ✭✷✳✶✵✮ t❛ ❝â T1 (Z) ≤ reg(Z) ≤ T (Z) = T2 (Z) ●✐↔ sû Pj1 , , Pjk ❧➔ ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ ✷✲♣❤➥♥❣ s❛♦ ❝❤♦ k l=1 T2 (Z) = mj l ❑❤✐ ✤â ≤ k ≤ 5✳ ❚❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ ❝♦♥ s❛✉✿ ❚r÷í♥❣ ❤đ♣ ✷✳✷✳✶✿ k = 5✱ ❧ó❝ ✤â Pj1 , , Pj5 ❧➔ ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ ✷✲♣❤➥♥❣ s❛♦ ❝❤♦ l=1 T2 (Z) = mjl ❉♦ Pj1 , , Pj5 ♥➡♠ tr➯♥ ✷✲♣❤➥♥❣ ♥➯♥ ❝â tè✐ ✤❛ ✹ ✤✐➸♠ tr♦♥❣ ❝→❝ ✤✐➸♠ tr➯♥ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣✳ ∗ ◆➳✉ Pjt , Pjq , Pjk , Pjh ❧➔ ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ s❛♦ ❝❤♦ mjt + mjq + mjk + mjh ❧ỵ♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ tê♥❣ ❝→❝ sè ❜ë✐ ❝õ❛ ❝→❝ ✤✐➸♠ ❜➜t ❦➻ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ tr♦♥❣ ❝→❝ ✤✐➸♠ Pj1 , , Pj5 ✳ ▲ó❝ ✤â r mil ≥ mjt + mjq + mjk + mjh T1 (Z) + = l=1 ≥ l=1 mj l = T2 (Z) ❙✉② r❛ T1 (Z) + ≥ T2 (Z)✳ ❉♦ T1 (Z) < T2 (Z) ♥➯♥ T1 (Z) + = T2 (Z) ❉♦ ✤â T (Z) − ≤ reg(Z) ≤ T (Z) ∗ ◆➳✉ Pjt , Pjq , Pjk , ❧➔ ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ s❛♦ ❝❤♦ mjt + mjq + mjk ❧ỵ♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ tê♥❣ ❝→❝ sè ❜ë✐ ❝õ❛ ❝→❝ ✤✐➸♠ ❜➜t ❦➻ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ tr♦♥❣ ❝→❝ ✤✐➸♠ Pj1 , , Pj5 ✳ ▲ó❝ ✤â ✷✾ r mil ≥ mjt + mjq + mjk T1 (Z) + = l=1 l=1 ≥ mjl = T2 (Z) ❙✉② r❛ T1 (Z) + ≥ T2 (Z)✳ ❉♦ T1 (Z) < T2 (Z) ♥➯♥ T1 (Z) + = T2 (Z) ❉♦ ✤â T (Z) − ≤ reg(Z) ≤ T (Z) ∗ ◆➳✉ Pjt , Pjq ❧➔ ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ s❛♦ ❝❤♦ mjt +mjq ❧ỵ♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ tê♥❣ ❝→❝ sè ❜ë✐ ❝õ❛ ❝→❝ ✤✐➸♠ ❜➜t ❦➻ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ tr♦♥❣ ❝→❝ ✤✐➸♠ Pj1 , , Pj5 ✳ ▲ó❝ ✤â r mi l ≥ mj t + mj q T1 (Z) + = l=1 ≥ l=1 mjl = T2 (Z) ❙✉② r❛ T1 (Z) + ≥ T2 (Z)✳ ❉♦ T1 (Z) < T2 (Z) ♥➯♥ T1 (Z) + = T2 (Z) ❉♦ ✤â T (Z) − ≤ reg(Z) ≤ T (Z) ❱➟②✱ tr♦♥❣ ❚r÷í♥❣ ❤đ♣ ✷✳✷✳✶ t❛ ❝â T (Z) − ≤ reg(Z) ≤ T (Z) ❚r÷í♥❣ ❤đ♣ ✷✳✷✳✷✿ k = 4✱ ❧ó❝ ✤â Pj1 , , Pj4 ♥➡♠ tr➯♥ ✷✲♣❤➥♥❣ s❛♦ ❝❤♦ l=1 T2 (Z) = mj l ❉♦ Pj1 , , Pj4 ♥➡♠ tr➯♥ ✷✲♣❤➥♥❣ ♥➯♥ ❝â tè✐ ✤❛ ✸ ✤✐➸♠ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣✳ ✸✵ ∗ ◆➳✉ Pjt , Pjq , Pjk , ❧➔ ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ s❛♦ ❝❤♦ mjt + mjq + mjk ❧ỵ♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ tê♥❣ ❝→❝ sè ❜ë✐ ❝õ❛ ❝→❝ ✤✐➸♠ ❜➜t ❦➻ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ tr♦♥❣ ❝→❝ ✤✐➸♠ Pj1 , , Pj4 ✳ ▲ó❝ ✤â r mil ≥ mjt + mjq + mjk T1 (Z) + = l=1 l=1 ≥ mjl = T2 (Z) ❙✉② r❛ T1 (Z) + ≥ T2 (Z)✳ ❉♦ T1 (Z) < T2 (Z) ♥➯♥ T1 (Z) + = T2 (Z) ❉♦ ✤â T (Z) − ≤ reg(Z) ≤ T (Z) ∗ ◆➳✉ Pjt , Pjq ❧➔ ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ s❛♦ ❝❤♦ mjt +mjq ❧ỵ♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ tê♥❣ ❝→❝ sè ❜ë✐ ❝õ❛ ❝→❝ ✤✐➸♠ ❜➜t ❦➻ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ tr♦♥❣ ❝→❝ ✤✐➸♠ Pj1 , , Pj4 ✳ ▲ó❝ ✤â r mi l ≥ mj t + mj q T1 (Z) + = l=1 ≥ l=1 mjl = T2 (Z) ❙✉② r❛ T1 (Z) + ≥ T2 (Z)✳ ❉♦ T1 (Z) < T2 (Z) ♥➯♥ T1 (Z) + = T2 (Z) ❉♦ ✤â T (Z) − ≤ reg(Z) ≤ T (Z) ❱➟②✱ tr♦♥❣ ❚r÷í♥❣ ❤ñ♣ ✷✳✷✳✷ t❛ ❝â T (Z) − ≤ reg(Z) ≤ T (Z) ❚r÷í♥❣ ❤đ♣ ✷✳✷✳✸✿ k = 3✱ ❧ó❝ ✤â Pj1 , Pj2 , Pj3 ♥➡♠ tr➯♥ ✷✲♣❤➥♥❣ s❛♦ ❝❤♦ l=1 T2 (Z) = ✸✶ mj l ●å✐ Pjt , Pjq ❧➔ ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ s❛♦ ❝❤♦ mjt + mjq ❧ỵ♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ tê♥❣ ❝→❝ sè ❜ë✐ ❝õ❛ ❝→❝ ✤✐➸♠ ❜➜t ❦➻ ♥➡♠ tr➯♥ ✶✲♣❤➥♥❣ tr♦♥❣ ❝→❝ ✤✐➸♠ Pj1 , Pj2 , Pj3 ✳ ▲ó❝ ✤â r mil ≥ mjt + mjq T1 (Z) + = l=1 l=1 ≥ mjl = T2 (Z) ❙✉② r❛ T1 (Z) + ≥ T2 (Z)✳ ❉♦ T1 (Z) < T2 (Z) ♥➯♥ T1 (Z) + = T2 (Z) ❉♦ ✤â T (Z) − ≤ reg(Z) ≤ T (Z) ❱➟②✱ tø ❝→❝ ❦➳t q✉↔ tr➯♥ t❛ t❤➜② tr♦♥❣ ❚r÷í♥❣ ❤đ♣ ✷✳✷ t❛ ❝â T (Z) − ≤ reg(Z) ≤ T (Z) ◆❤➟♥ ①➨t ✷✳✸✳✶✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ m = m2 = · · · = mn+3 = 2✱ t❛ t❤➜② ❞♦ P1 , , Pn+3 ❧➔ ❝→❝ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❦❤æ♥❣ s✉② ❜✐➳♥ ♥➯♥ P1 , , Pn+3 ❦❤æ♥❣ ♥➡♠ tr➯♥ (n 1) ỵ t õ reg(Z) = T (Z) ✸✷ ❑➌❚ ▲❯❾◆ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët sè t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Pn ữủ ởt số s ã ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤✱ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ởt t ã ữ r ổ tự ữợ ❧÷đ♥❣ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ ♥➡♠ tr ữớ t t tr Pn ỵ ã ữ r ổ tự ữợ ữủ số ❝❤➼♥❤ q✉② ❝õ❛ s ✤✐➸♠ ❜➨♦ ♣❤➙♥ ❜✐➺t tr♦♥❣ P s ỵ n ã ữ r ổ tự ữợ ữủ số q n + ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ỵ t q tr t ♠ỵ✐✳ ❚r♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✱ tỉ✐ ✤➣ ❝è ❣➢♥❣ ❧➔♠ ✈✐➺❝ ♥❣❤✐➯♠ tó❝✱ t✉② ♥❤✐➯♥ ❞♦ ❤↕♥ ❝❤➳ ✈➲ ♠➦t t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝ ❝â ❤↕♥ ♥➯♥ ❦➳t q✉↔ ❧✉➟♥ ✈➠♥ ❝á♥ ❦❤→ ❦❤✐➯♠ tè♥✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐ ❦❤✐ ❝â ✤✐➲✉ ❦✐➺♥✱ tỉ✐ ♠♦♥❣ ♠✉è♥ t✐➳♣ tö❝ t➻♠ ❤✐➸✉ ✈➲ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët sè t➟♣ ✤✐➸♠ ❜➨♦ ❦❤→❝✳ ▼➦❝ ❞ò ❜↔♥ t❤➙♥ ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ s♦♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ✈➲ ♠➦t ♥ë✐ ❞✉♥❣ ✈➔ tự ữủ sỹ õ ỵ ❣✐ó♣ ✤ï ❝õ❛ t❤➛② ❝ỉ ✈➔ ❜↕♥ ✤å❝✳ ✸✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❆t✐②❛❤ ▼✳❋✳ ❛♥❞ ▼❛❝❞♦♥❛❧❞ ■✳●✳ ✭✶✾✻✾✮✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛ ✱ ❯♥✐✈❡rs✐t② ♦❢ ❖①❢♦r❞✳ ❬✷❪ ❇❛❧❧✐❝♦ ❊✳✱ ❉✉♠✐tr❡s❝✉ ❖✳ ❛♥❞ P♦st✐♥❣❤❡❧ ❊✳ ✭✷✵✶✻✮✱ ❖♥ ❙❡❣r❡✬s ❜♦✉♥❞ ❢♦r ❢❛t ♣♦✐♥ts ✐♥ Pn ✱ ❏✳ P✉r❡ ❆♣♣❧✳ ❆❧❣❡❜r❛ ✷✷✵✱ ■ss✉❡ ✷✸✵✼✲✷✸✷✸✳ ❬✸❪ ❇❡♥❡❞❡tt✐ ❇✳✱ ❋❛t❛❜❜✐✳●✳ ❛♥❞ ▲♦r❡♥③✐♥✐✳❆✳ ✭✷✵✶✷✮✱ ❙❡❣r❡✬s ❜♦✉♥❞ ❛♥❞ t❤❡ ❝❛s❡ ♦❢ ♥✰✷ ❢❛t ♣♦✐♥ts ♦❢ Pn ✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✹✵✱ ✸✾✺✲✺✹✼✸✳ ❬✹❪ ❇♦✉r❜❛❦✐✳ ◆ ✭✶✾✼✹✮✱ ❆❧❣❡❜r❛ ■ ✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✳ ❬✺❪ ❇r♦❞♠❛♥♥ ▼✳P ❛♥❞ ❙❤❛r♣ ✭✶✾✾✽✮✱ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣②✿ ❛♥ ❛❧❣❡❜r❛✐❝ ❛♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ❣❡♦♠❡tr✐❝ ❛♣♣❧✐❝❛t✐♦♥s ✱ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✻❪ ❈❛t❛❧✐s❛♥♦ ▼✳❱✳ ✭✶✾✾✶✮✱ ❋❛t ♣♦✐♥ts ♦♥ ❛ ❝♦♥✐❝✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✶✾✱ ✷✶✺✸✲✷✶✻✽✳ ❬✼❪ ❈❛t❛❧✐s❛♥♦ ▼✳❱✳✱ ❚r✉♥❣✳◆✳❱✳ ❛♥❞ ❱❛❧❧❛ ●✳ ✭✶✾✾✸✮✱ ❆ s❤❛r♣ ❜♦✉♥❞ ❢♦r t❤❡ r❡❣✉❧❛r✐t② ✐♥❞❡① ♦❢ ❢❛t ♣♦✐♥ts ✐♥ ❣❡♥❡r❛❧ ♣♦s✐t✐♦♥✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✶✶✽✱ ✼✶✼✲✼✷✹✳ ❬✽❪ ❉❛✈✐s ❊✳❉✳ ❛♥❞ ●❡r❛♠✐t❛ ❆✳❱✳ ✭✶✾✽✹✮✱ ❚❤❡ ❍✐❧❜❡rt ❢✉♥t✐♦♥ ♦❢ ❛ s♣❡❝✐❛❧ ❝❧❛ss ♦❢ ✶✲❞✐♠❡♥s✐♦♥ ❈♦❤❡♥ ✲ ▼❛❝❛✉❧❛② ❣r❛❞❡ ❛❧❣❡❜r❛s✱ ❚❤❡ ❈✉r✈❡s ❙❡♠✐♥❛r ❛t ◗✉❡❡♥✬s✱ ◗✉❡❡♥✬s P❛♣❡r ✐♥ P✉r❡ ❛♥❞ ❆♣♣❧✳ ▼❛t❤✳ ✻✼✱✶✲✷✾✳ ❬✾❪ ❋❛t❛❜❜✐ ●✳ ✭✶✾✾✹✮✱ r❡❣✉❧❛r✐t② ✐♥❞❡① ♦❢ ❢❛t ♣♦✐♥ts ✐♥ t❤❡ ♣r♦❥❡❝t✐✈❡ ♣❧❛♥❡✱ ❏✳ ❆❧❣❡❜r❛ ✶✼✵✱ ✾✶✻✲✾✷✽✳ ❬✶✵❪ ❋❛t❛❜❜✐ ●✳ ❛♥❞ ▲♦r❡♥③✐♥✐ ❆✳ ✭✷✵✵✶✮✱ ❖♥ t❤❡ s❤❛r♣ ❜♦✉♥❞ ❢♦r t❤❡ r❡❣✉❧❛r✐t② ✐♥❞❡① ♦❢ ❛♥② s❡t ♦❢ ❢❛t ♣♦✐♥ts✱ ❏✳ P✉r❡ ❆♣♣❧✳ ❆❧❣❡❜r❛ ✶✻✶✱ ✾✶✲✶✶✶✳ ❬✶✶❪ ❋✉❧t♦♥ ❲✳ ✭✶✾✻✾✮✱ ❆❧❣❡❜r❛✐❝ ❈✉r✈❡s ✱ ▼❛t❤✳ ▲❡❝t✳ ◆♦t❡ ❙❡r✐❡s✱ ❇❡♥❥❛♠✐♥✱ ◆❡✇ ❨♦r❦✳ ❬✶✷❪ ❏♦s❡♣❤ ❏✳ ❘♦t♠❛♥ ✭✶✾✼✾✮✱ ❆♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ ❤♦♠♦❧♦❣✐❝❛❧ ❛❧❣❡❜r❛ ✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ◆❡✇ ❨♦r❦✳ ✸✹ ❬✶✸❪ ❍❛rts❝❤♦r♥❡ ❘✳ ✭✶✾✼✼✮✱ ❆❧❣❡❜r❛ ●❡♦♠❡♦tr② ✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✳ ❬✶✹❪ ❑✉♥③ ❊✳ ✭✶✾✽✺✮✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛ ❛♥❞ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡♦tr② ✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✳ ❬✶✺❪ ▼❛ts✉♠✉r❛ ❍✳ ✭✶✾✼✵✮✱ ❈♦♠♠✉t❛t✐♦♥ ❆❧❣❡❜r❛ ✱ ❲✳ ❆✳ ❇❡♥❥❛♠✐♥✱ ■♥❝✳✱ ◆❡✇ ❨♦r❦✳ ❬✶✻❪ ❙❝❤❡♥❝❦ ❍✳ ✭✷✵✵✸✮✱ ❈♦♠♠✉t❛t✐♦♥❛❧ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡♦tr② ✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✶✼❪ ❙❡❣r❡✳❇✳ ✭✶✾✻✶✮✱ ❆❧❝✉♥❡ q✉❡st✐♦♥ s✉ ✐♥s✐❡♠✐ ❢✐♥✐t✐ ❞✐ ♣✉♥t✐ ✐♥ ❣❡♦♠❡tr✐❛ ❛❧❣❡❜r✐❝❛✱ ❆tt✐✳ ❈♦♥✈❡r❣♥♦✳ ■♥t❡r♥✳ ❞✐ ❚♦r✐♥♦✱ ✶✺✲✸✸✳ ❬✶✽❪ ❚❤✐❡♥ P✳❱✳ ✭✶✾✾✾✮✱ ❖♥ ❙❡❣r❡ ❜♦✉♥❞ ❢♦r t❤❡ r❡❣✉❧❛r✐t② ✐♥❞❡① ♦❢ ❢❛t ♣♦✐♥ts ✐♥ P2 ✱ ❆❝t❛ ▼❛t❤✳ ❱✐❡t♥❛♠✐❝❛ ✷✹✱ ✼✺✲✽✶✳ ❬✶✾❪ ❚❤✐❡♥ P✳❱✳ ✭✷✵✵✵✮✱ ❙❡❣r❡ ❜♦✉♥❞ ❢♦r t❤❡ r❡❣✉❧❛r✐t② ✐♥❞❡① ♦❢ ❢❛t ♣♦✐♥ts ✐♥ P3 ✱ ❏✳ P✉r❡ ❛♥❞ ❆♣♣❧✳ ❆❧❣❡❜r❛ ✶✺✶✱ ✶✾✼✲✷✶✹✳ ❬✷✵❪ ❚❤✐❡♥ P✳❱✳ ✭✷✵✵✷✮✱ ❙❤❛r♣ ✉♣♣❡r ❜♦✉♥❞ ❢♦r t❤❡ r❡❣✉❧❛r✐t② ✐♥❞❡① ♦❢ ③❡r♦✲s❝❤❡♠❡s ♦❢ ❞♦✉❜❧❡ ♣♦✐♥ts ✐♥ P4 ✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✸✵✱ ✺✽✷✺✲✺✽✹✼✳ ❬✷✶❪ ❚❤✐❡♥ P✳❱✳ ✭✷✵✶✷✮✱ ❘❡❣✉❧❛r✐t② ✐♥❞❡① ♦❢ s✰✷ ❢❛t ♣♦✐♥ts ♥♦t ♦♥ ❛ ✭s✲✶✮✲s♣❛❝❡✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✹✵✱ ✸✼✵✹✲✸✼✶✺✳ ❬✷✷❪ ❚❤✐❡♥ P✳❱✳ ✭✷✵✶✻✮✱ ▲♦✇❡r ❜♦✉♥❞ ❢♦r t❤❡ r❡❣✉❧❛r✐t② ✐♥❞❡① ♦❢ ❢❛t ♣♦✐♥ts✱ ■♥t❡r✲ ♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡r♠❛t✐❝s✱ ❱♦❧✉♠❡ ✶✵✾✱ ◆♦✳ ✸✱ ✭✷✵✶✻✮✱ ✼✹✺✲✼✺✺✳ ❬✷✸❪ ❚❤✐❡♥ P✳❱✳ ❛♥❞ ❙✐♥❤✳❚✳◆✳✭✷✵✶✼✮✱ ❖♥ t❤❡ r❡❣✉❧❛r✐t② ✐♥❞❡① ♦❢ s ❢❛t ♣♦✐♥ts ♥♦t ♦♥ ❛ ❧✐♥❡❛r ✭r✲✶✮✲s♣❛❝❡✱ s ≤ r + 3✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✹✺✱ ✹✶✷✸✲✹✶✸✽✳ ❬✷✹❪ ❯✇❡ ◆❛❣❡❧ ❛♥❞ ❇✐❧❧ ❚r♦❦ ✭✷✵✶✻✮✱ ❙❡❣r❡✬s ❘❡❣✉❧❛r✐t② ❇♦✉♥❞ ❢♦r ❋❛t P♦✐♥t ❙❝❤❡♠❡s✱ ❛r❳✐✈✿✶✻✶✶✳✵✻✷✼✾✳ ✸✺ ... ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ❝→❝ ❛♥❤ ❝❤à ❈❛♦ ❤å❝ ❚♦→♥ ❦❤â❛ ❳❳❱ tr÷í♥❣ ✣❍❙P ❍✉➳ ❝❤✉②➯♥ ♥❣➔♥❤ số ỵ tt số sỹ ❣✐ó♣ ✤ï tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈ø❛ q✉❛✳ ❉♦ ✤➙② ❧➔ ❧➛♥ ✤➛✉ t✐➯♥ t❤ü❝ ❤✐➺♥ ❝æ♥❣ ✈✐➺❝... ▼ët t➟♣ ❝♦♥ Y ❝õ❛ Pn ữủ t số tỗ t ởt t➟♣ T ❝→❝ ♣❤➛♥ tû t❤✉➛♥ ♥❤➜t ❝õ❛ R s❛♦ ❝❤♦ Y = Z(T )✳ ▼➺♥❤ ✤➲ ✶✳✶✳✷✳✶✳ ❍ñ♣ ❝õ❛ ❤❛✐ t➟♣ ✤↕✐ sè ❧➔ t➟♣ ✤↕✐ sè✳ ●✐❛♦ ♠ët tý ỵ t số t sè✳ Pn ✈➔ ∅ ❧➔ ❝→❝... ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ ♥➡♠ tr➯♥ ữớ t t tr Pn ỵ ã ữ r ổ tự ữợ ữủ số q✉② ❝õ❛ s ✤✐➸♠ ❜➨♦ ♣❤➙♥ ❜✐➺t tr♦♥❣ P ✱ s ỵ n ã ữ r ổ tự ữợ ữủ số q n + ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ✭✣à♥❤ ỵ t q tr