dd - dãy, đặc trưng Euler - Poincaré và ứng dụng vào nghiên cứu cấu trúc một số lớp mở rộng của môđun Cohen - Macaulay : Luận án TS. Toán học: 62 46 05 01

▼ô❝ ▲ô❝ ▼ë ➤➬✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤➢➡♥❣ ✶✳ ❈❤✉➮♥ ❜Þ ✶✵ ✶✳✶✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ✳ ✳ ✳ ✶✳✷✳ ❑✐Ĩ✉ ➤❛ t❤ø❝ ✈➭ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ✶✳✸✳ ➜➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ❝❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ❈❤➢➡♥❣ ✷✳ ❞❞✲❉➲② ✈➭ ❝➳❝ ➤➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ❝❛♦ ✷✳✶✳ ❈➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ ❞❞✲❞➲② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳ ➜➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ❝❛♦ ❝đ❛ ♣❤ø❝ ❑♦s③✉❧ ✷✳✸✳ ▲✐➟♥ ❤Ư ✈í✐ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✶✻ ✳ ✳ ✳ ✳ ✳ ✶✼ ✳ ✳ ✳ ✳ ✳ ✷✺ ✳ ✳ ✳ ✳ ✳ ✸✸ ❈❤➢➡♥❣ ✸✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✹✺ ✸✳✶✳ ▲ä❝ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ✈➭ ❤Ư t❤❛♠ sè tèt ✸✳✷✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸✳ ➜➷❝ tr➢♥❣ ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ❈❤➢➡♥❣ ✹✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ✹✳✶✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ✳ ✳ ✹✳✷✳ ➜➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ❝❛♦ ✈➭ ❤➺♥❣ sè ✹✳✸✳ ➜➷❝ tr➢♥❣ t❤❛♠ sè ❑Õt ❧✉❐♥ ✳ ✳ ✳ ✳ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✶ ✼✵ ✳ ✳ ✳ ✳ IF (M ) ✳ ✳ ✳ ✳ ✼✶ ✳ ✳ ✳ ✳ ✼✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻ ▼ë ➤➬✉ ◆❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ♠➠➤✉♥ t❤➠♥❣ q✉❛ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ ➤é ❞➭✐ ❝ñ❛ ♠➠➤✉♥ ♠♦❞✉❧♦ ♠ét ❤Ö t❤❛♠ sè ❧➭ ♣❤➢➡♥❣ ♣❤➳♣ ➤➲ ①✉✃t ❤✐Ö♥ tõ ❧➞✉ tr♦♥❣ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥✳ ❚õ ♥❤÷♥❣ ♥➝♠ ✺✵ ❝đ❛ t❤Õ ❦û tr➢í❝✱ ❙❡rr❡ ➤➲ ❝❤Ø r❛ ❝ã t❤Ĩ ❞ï♥❣ ♣❤ø❝ ❑♦s③✉❧ ➤Ĩ tÝ♥❤ ❜é✐ ❝đ❛ ♠ét ♠➠➤✉♥ ➤è✐ ✈í✐ ♠ét ❤Ư t❤❛♠ sè✱ tõ ➤ã ➤➢❛ r❛ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❤➭♠ ➤é ❞➭✐✱ sè ❜é✐ ✈í✐ ➤é ❞➭✐ ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤å♥❣ ➤✐Ị✉ ❑♦s③✉❧✳ ❈➳❝ ♠è✐ ❧✐➟♥ ❤Ư ➤ã ➤➢ỵ❝ t✐Õ♣ tơ❝ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❝➳❝ ❝➠♥❣ tr×♥❤ ❝đ❛ ❆✉s❧❛♥❞❡r✲❇✉❝❤s❜❛✉♠ ✈➭ ❝➳❝ t➳❝ ❣✐➯ ❦❤➳❝✱ ❞➱♥ ➤Õ♥ ♥❤÷♥❣ ❦Õt q✉➯ ♠➭ ♥❣➭② ♥❛② trë t❤➭♥❤ ❝➡ ❜➯♥ tr♦♥❣ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥✳ ➜Ó ♣❤➳t ❜✐Ó✉ ❝❤Ý♥❤ ①➳❝✱ ✈í✐ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❑ý ❤✐Ư✉ m M ✱ R ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ ✱ tr♦♥❣ ➤ã x ✳ ( ) ❧➭ ❤➭♠ ➤é ❞➭✐✱ ❑❤✐ ❞✃✉ ➤➻♥❣ t❤ø❝ ①➯② r❛✱ (M/xM ) = e(x, M ) M ị ữ s ó ề e(x, M ) ➤è✐ ✈í✐ ❤Ư t❤❛♠ sè ❝❤♦ ❧➭ ♠ét ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ❧➭ ♠ét x = x1 , , x d ∈ m (M/xM ) M (R, m) t❛ ❧✉➠♥ ①Ðt M ✳ ◆♦❡t❤❡r dim M = d ✳ ❑❤✐ ➤ã t❛ ❧✉➠♥ ❝ã e(x, M ) ❧➭ sè ❜é✐ ❝ñ❛ ♥❣❤Ü❛ ❧➭ tå♥ t➵✐ x s❛♦ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❈ã t❤Ĩ ♥ã✐ ột tr ữ trú ợ ♥❣❤✐➟♥ ❝ø✉ ❦ü ✈➭ ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ ♥❤✃t tr♦♥❣ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥✳ ◆Õ✉ (M/xM ) = e(x, M ) ▼❛❝❛✉❧❛② t❛ ❝ị♥❣ ❝ã M ✈í✐ ♠ä✐ ❤Ư t❤❛♠ sè ❧➭ ❈♦❤❡♥✲ x M ❝ñ❛ ✳ ▼ë ré♥❣ ➤➬✉ t✐➟♥ t❤❡♦ ❤➢í♥❣ ♥➭② ❝đ❛ ❧í♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧➭ ❦❤➳✐ ♥✐Ö♠ ♠➠➤✉♥ ❇✉❝❤s❜❛✉♠ ❞♦ ❙t s❛♦ ❝❤♦ tå♥ t➵✐ ♠ét ❤➺♥❣ sè ✈í✐ ♠ä✐ ❤Ư t❤❛♠ sè I(M ) = ✳ ✱ ✳ ❝❦r❛❞ ✈➭ ❱♦❣❡❧ ➤➢❛ r❛✳ ➜ã ❧➭ ❝➳❝ ♠➠➤✉♥ I(M ) t❤á❛ ♠➲♥ M (M/xM ) ❝❤✃t (M/xM ) = e(x, M ) + I(M ) ◆❤➢ ✈❐② ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧➭ ❇✉❝❤s❜❛✉♠ ✈í✐ t➢➡♥❣ e(x, M ) + C tù C ♠➭ tå♥ t➵✐ ❤➺♥❣ sè ♥❤➢ ♠➠➤✉♥ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❤Ư t❤❛♠ sè ✳ ❍➺♥❣ sè ✈➭ ❝ã t➟♥ ❧➭ ❤➺♥❣ sè ❇✉❝❤s❜❛✉♠ ❝ñ❛ tÝ♥❤ M ◆➝♠ ✶✾✼✾✱ ❜❛ ♥❤➭ t♦➳♥ ❤ä❝ ◆✳ ❚✳ r ét M x u ă M ✳ C ♥❤á ♥❤✃t ➤➢ỵ❝ ❦ý ❤✐Ư✉ ❧➭ ❝đ❛ I(M ) ❈➳❝ ♠➠➤✉♥ ♥❤➢ ✈❐② ❝ã r✃t ♥❤✐Ò✉ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✶ x ✈➭ ➤➢ỵ❝ ❣ä✐ ❧➭ ❝➳❝ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❘â r➭♥❣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ ❇✉❝❤s❜❛✉♠ ❧➭ ❝➳❝ tr➢ê♥❣ ♠➠➤✉♥ ❤ỵ♣ r✐➟♥❣ ❝đ❛ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠➠➤✉♥ s✉② ré♥❣ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ➤➢ỵ❝ ♣❤➳t tr✐Ó♥ s✉② r✃t ré♥❣✳ ♥❤❛♥❤ ▲ý tr♦♥❣ t❤✉②Õt t❤❐♣ ❦û ✽✵ ✈➭ ♥❤÷♥❣ ♥➝♠ ➤➬✉ t❤❐♣ ❦û ✾✵ ❝đ❛ t❤Õ ❦û ✷✵ ❜ë✐ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ◆✳ ❚✳ ❈➢ê♥❣✱ ❙❝❤❡♥③❡❧✱ ◆✳ ❱✳ ❚r✉♥❣✱ ❙t ❨❛♠❛❣✐s❤✐✱ ❚❛❦❛②❛♠❛✱ ❍×♥❤ ọ số ý ệ u ă ó ề ø♥❣ ❞ô♥❣ tr♦♥❣ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ✈➭ x(n) = xn1 , , xnd d ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ t❤× ✈í✐ n1 , , n d ❝❦r❛❞✱ ❱♦❣❡❧✱ ▲✳ ❚✳ ❍♦❛✱ ❇r♦❞♠❛♥♥✱ ●♦t♦✱ ✱ ✈í✐ n1 , , n d > (M/x(n)M ) = n1 nd e(x, M ) + I(M ) ➤đ ❧í♥ ✭➤Ĩ ♥❣➽♥ ❣ä♥ t❛ sÏ ❞ï♥❣ ❦ý ❤✐Ö✉ (M/x(n)M ) M ✳ ◆Õ✉ ✮✱ ♥ã✐ M ❦❤➠♥❣ ♣❤➯✐ ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✱ ❙❤❛r♣ ➤➷t ❝➞✉ ❤á✐✿ ❤➭♠ (M/x(n)M ) r✐➟♥❣✱ ❝ã ❞➵♥❣ ➤❛ t❤ø❝ t❤❡♦ ❧➭ ♠ét ➤❛ t❤ø❝ t❤❡♦ n1 , , n d ❦❤✐ n1 , , n d n1 , , n d n1 , , n d ✳ ❑❤✐ ❦❤➠♥❣❄ ❑❤➠♥❣ ❦❤ã ó tể tì ợ í ụ ỉ r ❝➞✉ tr➯ ❧ê✐ ❧➭ ♣❤đ ➤Þ♥❤✱ ❞➱♥ ➤Õ♥ ❝➞✉ ❤á✐ t✐Õ♣ t❤❡♦ ❧➭ ✈í✐ ➤✐Ị✉ ❦✐Ư♥ ♥➭♦ t❤× ❤➭♠ (M/x(n)M ) ❝ã ❞➵♥❣ ➤❛ t❤ø❝✳ ▼ét ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ➤➢ỵ❝ ◆✳ ❚✳ ❈➢ê♥❣ ➤➢❛ r❛ tr♦♥❣ ❬✽❪ q✉❛ ❦❤➳✐ ♥✐Ư♠ ✉♣✲❞➲②✳ ❍➡♥ ♥÷❛✱ tr♦♥❣ ❜➭✐ ❜➳♦ ❬✾❪ ➠♥❣ ❝ị♥❣ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ tỉ♥❣ q✉➳t ❤➭♠ (M/x(n)M ) t❤➢➡♥❣ ❝đ❛ ♠ét ♠ét ❤Ư t❤❛♠ sè ❦ý ❤✐Ư✉ ❧✐♥❤ ✈➭♥❤ t❛ s✉② ✈➭♥❤ x ❝đ❛ ●♦r❡♥st❡✐♥✱ M s❛♦ ❝❤♦ ◆✳ ❚✳ ❈➢ê♥❣ (M/x(n)M ) a(M ) = a0 (M )a1 (M ) ad−1 (M ) ❤ã❛ ❧➭ ❧✉➠♥ ❜Þ ❝❤➷♥ tr➟♥ ❜ë✐ ♠ét ➤❛ t❤ø❝✳ tư ❝đ❛ t❤➢➡♥❣ r❛ ❧✉➠♥ ♠➠➤✉♥ ❝đ❛ tå♥ ➤è✐ ♠ét t➵✐ ➤å♥❣ ✈➭♥❤ ♠ét ❤Ư ➤✐Ị✉ ➤Þ❛ ●♦r❡♥st❡✐♥ t❤❛♠ ✈í✐ ➤➲ tõ xi ∈ a(M/(xi+1 , , xd )M ) i = 1, , d ✱ ✳ r➺♥❣ ❧✉➠♥ ❧➭ ✈➭♥❤ tå♥ t➵✐ (M ) = Ann(Hmi (M )) Hmi (M ) ♠ét ❦Õt x = x1 , , x d sè r❛ R ❧➭ ♠ét ➤❛ t❤ø❝✳ ❈ô t❤Ĩ ❤➡♥✱ ♣❤➢➡♥❣ t❤× ❝❤Ø ❑❤✐ ✈➭♥❤ ❝đ❛ q✉➯ t❤á❛ M ❝ñ❛ ♠➲♥ ✳ ❑❤✐ ❧➭ R ❙❝❤❡♥③❡❧✱ tÝ♥❤ ❝❤✃t ▼ét ❤Ư t❤❛♠ sè ♥❤➢ ✈❐② ➤➢ỵ❝ t➳❝ ❣✐➯ ◆✳ ❚✳ ❈➢ê♥❣ ❣ä✐ ❧➭ ♠ét ❤Ö t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝✱ ❤➡♥ ♥÷❛ ❦❤✐ ➤ã d (M/x(n)M ) = n1 ni e(x1 , , xi , (0 : xi+1 )M/(xi+2 , ,xd )M ), ✭ ✮ i=0 ❧➭ ♠ét ➤❛ t❤ø❝ ✈í✐ ♠ä✐ n1 , , n d > ✳ ❑❤➳✐ ♥✐Ư♠ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ s❛✉ ➤ã ➤➲ ➤➢ỵ❝ ❑❛✇❛s❛❦✐ sư ❞ơ♥❣ ♥❤➢ ♠ét ❝➠♥❣ ❝ơ t❤❡♥ ❝❤èt ➤Ĩ ❣✐➯✐ ❜➭✐ t♦➳♥ ✷ ▼❛❝❛✉❧❛② ❤ã❛ ♠ét ➤❛ t➵♣ ➤➵✐ sè ❞♦ ❋❛❧t✐♥❣s ➤➷t r❛✱ tõ ➤ã ➤➢❛ r❛ ❝➞✉ tr➯ ❧ê✐ ❦❤➻♥❣ ➤Þ♥❤ ❝❤♦ ❣✐➯ t❤✉②Õt ❝đ❛ ❙❤❛r♣ ✈Ị ➤✐Ị✉ ❦✐Ư♥ tå♥ t➵✐ ♣❤ø❝ ➤è✐ ♥❣➱✉✳ ❈➳❝ ❦Õt q✉➯ ➤ã t❤ó❝ ➤➮② ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ❦ü ❤➡♥ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❝➳❝ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ♥➭② ❝ị♥❣ ♥❤➢ ø♥❣ ❞ơ♥❣ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ✈➭♥❤ ✈➭ ♠➠➤✉♥✳ ❇➯♥ t❤➞♥ ❝➳❝ ❤Ö t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ❝ã r✃t ♥❤✐Ò✉ tÝ♥❤ ❝❤✃t tèt✳ ❍➬✉ ❤Õt ❝➳❝ tÝ♥❤ ❝❤✃t ♥➭② ➤Ị✉ ❞♦ ❝➳❝ ❤Ư t❤❛♠ sè ♥➭② t❤á❛ ♠➲♥ ❝➠♥❣ t❤ø❝ ✭ ✮ ë tr➟♥✳ ❱× ✈❐② tr♦♥❣ ❧✉❐♥ ➳♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ ➤➷t ✈✃♥ ➤Ị ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❝➳❝ ❤Ư t❤❛♠ sè t❤á❛ ♠➲♥ ✭ ✮ ❝ị♥❣ ♥❤➢ ❝➳❝ ø♥❣ ❞ơ♥❣ ❝đ❛ ❝❤ó♥❣✳ ❈➳❝ ❤Ư t❤❛♠ sè ♥❤➢ ✈❐② ❧➭ tr➢ê♥❣ ❤ỵ♣ r✐➟♥❣ ❝đ❛ ệ ợ ị ĩ tr ▼ét ♠ë ré♥❣ ❦❤➳❝ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t❤❡♦ ❤➢í♥❣ ❤♦➭♥ t♦➭♥ ❦❤➳❝ ❧➭ ❦❤➳✐ ♥✐Ö♠ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳ ❚❛ ❣ä✐ M ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✭t➢➡♥❣ ø♥❣✱ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✮ ♥Õ✉ tå♥ t➵✐ ♠ét ❧ä❝ ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ M M0 ⊂ M1 ⊂ ⊂ Mt = M, s❛♦ ❝❤♦ Mi /Mi−1 (M0 ) < ∞ dim M0 < dim M1 < < dim Mt = d t ứ s ỗ rộ ✈í✐ i = 1, 2, , t ✳ ❈➳❝ ❧ä❝ ♥❤➢ ✈❐② ➤➢ỵ❝ ❣ä✐ ❧➭ ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✭t➢➡♥❣ ø♥❣✱ ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ▼❛❝❛✉❧❛② ✐➤➟❛♥ ✈➭ ♥❣✉②➟♥ s✉② ré♥❣✮✳ ❈♦❤❡♥✲▼❛❝❛✉❧❛② tè ❧✐➟♥ ❦Õt ❝đ❛ ❈❤ó s✉② ý r➺♥❣ ré♥❣ ♠➠➤✉♥ ❧➭ t❤á❛ tr♦♥❣ ❦❤➠♥❣ ♠➲♥ ❦❤✐ ❝➳❝ tré♥ ♠➠➤✉♥ ❧➱♥✱ ❈♦❤❡♥✲ ♥❣❤Ü❛ ❧➭ dim R/p = dim M ❝➳❝ ❤♦➷❝ dim R/p = ✭tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✮✱ t❤× ❝➳❝ ✐➤➟❛♥ tè ♥❣✉②➟♥ ▼❛❝❛✉❧❛② s✉② ❧✐➟♥ ré♥❣ ❦Õt ❞➲② ❝đ❛ ❝ã ❝➳❝ ➤è✐ ♠➠➤✉♥ ❝❤✐Ị✉ ❦❤➳ ❈♦❤❡♥✲▼❛❝❛✉❧❛② tï② ý✱ t❤❛② ➤æ✐ tõ ❞➲② ✈➭ ➤Õ♥ ❈♦❤❡♥✲ dim M ✳ ➜➞② ❧➭ ♠ét ➤✐Ĩ♠ ❦❤➳❝ ❜✐Ưt ❝➡ ❜➯♥ ❣✐÷❛ ❝➳❝ ❧í♣ ♠➠➤✉♥ ♥➭②✳ ❈✃✉ tró❝ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ①✉✃t ❤✐Ư♥ tù ♥❤✐➟♥ tr♦♥❣ ❝➳❝ ø♥❣ ❞ơ♥❣ ❝đ❛ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ✈➭♦ ❝➳❝ ❜➭✐ t♦➳♥ tỉ ợ ợ t ị ĩ t tr ♠➠➤✉♥ ♣❤➞♥ ❜❐❝ tr♦♥❣ ❬✸✽❪✱ tr➢ê♥❣ ❤ỵ♣ ♠➠➤✉♥ tr➟♥ ✈➭♥❤ ị ợ ét ◆❤➭♥ ❬✶✽❪✱ ❙❝❤❡♥③❡❧ ❬✸✼❪✳ ❍✐Ö♥ ♥❛② ✈✐Ö❝ ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ♠➠➤✉♥ ♥➭② ➤❛♥❣ t❤✉ ❤ót sù q✉❛♥ t➞♠ ❝đ❛ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝✱ ➤➷❝ ❜✐Ưt ❧➭ ❝➳❝ ø♥❣ ❞ơ♥❣ tr tổ ợ ý tết tị ❬✷✵❪✱ ✳ ✳ ✳ ✮✳ ❇➟♥ ❝➵♥❤ ➤ã✱ ✈✐Ö❝ ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ❝➳❝ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② tõ ❦❤Ý❛ ❝➵♥❤ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ❝ị♥❣ ❧➭ ♠ét ✈✃♥ ➤Ị q✉❛♥ trä♥❣ ✈➭ t❤✉ ❤ót ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝✳ ❈➳❝ ❝➠♥❣ tr×♥❤ t✐➟✉ ❜✐Ĩ✉ t❤❡♦ ❤➢í♥❣ ♥➭② ❝ã t❤Ĩ ❦Ĩ ➤Õ♥ ❬✶✽❪✱ ❬✷✽❪✱ ❬✸✼❪✱ ❬✸✽❪✳ ❝❤✃t ▼ét tr♦♥❣ ♥❤÷♥❣ ❦Õt q✉➯ q✉❛♥ trä♥❣ ♥❤✃t ❧➭ ➤➷❝ tr➢♥❣ tÝ♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② q✉❛ tÝ♥❤ tr✐Ưt t✐➟✉ ✈➭ tÝ♥❤ ❝❤✃t ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝đ❛ ➤è✐ ♥❣➱✉ ▼❛t❧✐s ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳ ▼ét ♠ë ré♥❣ ❦❤➳❝ ❦❤➳ tù ♥❤✐➟♥ ❝đ❛ ❦❤➳✐ ♥✐Ư♠ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ▼➠➤✉♥ t➢ỵ♥❣ ❞➲② ➤➢ỵ❝ ◆✳ ❚✳ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t✐Õ♣ t❤❡♦ ❝đ❛ ❈➢ê♥❣ ❞➲② ❝❤ó♥❣ t➠✐ ✈➭ ✈➭ ▲✳ ❚✳ ◆❤➭♥ ➤➢❛ ❈♦❤❡♥✲▼❛❝❛✉❧❛② tr♦♥❣ ❧✉❐♥ ➳♥ ♥➭②✳ r❛ s✉② tr♦♥❣ ré♥❣ ❈❤ó♥❣ t➠✐ ❜➭✐ ❞➲② sÏ ❜➳♦ ❧➭ ❝❤Ø ❬✶✽❪✳ ❤❛✐ r❛ ➤è✐ r➺♥❣ ➤è✐ ✈í✐ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ❧✉➠♥ tå♥ t➵✐ ♠ét ❤Ö t❤❛♠ sè t❤á❛ ♠➲♥ ❝➠♥❣ t❤ø❝ ✭ ✮ ë tr➟♥✳ ❚õ ➤ã ❝❤ó♥❣ t➠✐ ø♥❣ ❞ơ♥❣ ➤Ĩ q✉❛② ❧➵✐ ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ❝➳❝ ♠➠➤✉♥ ♥➭②✳ ▼➷❝ ❞ï ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ❦❤➳ ❣✐è♥❣ ♥❤❛✉✱ t✉② ♥❤✐➟♥ ❝ò♥❣ t➢➡♥❣ tù ♥❤➢ ➤è✐ ✈í✐ ❝➳❝ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✱ ❝➳❝ ❦ü t❤✉❐t ❧➭♠ ✈✐Ư❝ ✈í✐ ❤❛✐ ❧í♣ ♠➠➤✉♥ ë tr➟♥ ❧➭ ❦❤➳❝ ♥❤❛✉✱ ♠❛♥❣ ➤➷❝ t❤ï ❝ñ❛ tõ♥❣ ❧í♣ ♠➠➤✉♥✳ ▲✉❐♥ ➳♥ ➤➢ỵ❝ ❝❤✐❛ t❤➭♥❤ ❜è♥ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✶ ❧➭ ❝❤➢➡♥❣ ❝❤✉➮♥ ❜Þ✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ ♥➟✉ ❧➵✐ ♥❣➽♥ ❣ä♥ ♠ét sè ❦Õt q✉➯ q✉❡♥ ❜✐Õt tr♦♥❣ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ➤Ĩ t✐Ư♥ ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ tr♦♥❣ ❝➳❝ ❝❤➢➡♥❣ s❛✉✳ ❈ơ t❤Ĩ ❧➭ tr♦♥❣ ❚✐Õt ✶✱ ❝❤ó♥❣ t➠✐ sÏ ♥➟✉ ❧➵✐ ❦❤➳✐ ♥✐Ö♠ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✱ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ❞✲❞➲②✱ ❞✲❞➲② ♠➵♥❤✱ ❤Ö t❤❛♠ sè ❝❤✉➮♥ t➽❝ ✈➭ ♠ét sè ❦Õt q✉➯ ❧✐➟♥ q✉❛♥✱ ❝❤ñ ②Õ✉ tõ ❝➳❝ ❜➭✐ ❜➳♦ ❬✹✻❪✱ ❬✹✸❪✳ ❚r♦♥❣ ❚✐Õt ✷✱ ❝❤ó♥❣ t➠✐ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ư♠ ❦✐Ĩ✉ ➤❛ t❤ø❝✱ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ✈➭ ♠ét số tí t ủ ú ợ trì tr ❬✾❪✱ ❬✶✵❪✱ ❬✸✵❪✱ ❬✸✶❪✳ ▼ét sè ❦Õt q✉➯ ✈Ò ➤➷❝ tr rPré ủ ứ s ợ trì ❜➭② tr♦♥❣ ❚✐Õt ✸✳ ❈➳❝ ❦Õt q✉➯ ♥➭② ❝❤ñ ②Õ✉ tõ ❬✶✼❪✱ ❬✹✺❪✳ ❈➳❝ ❚r♦♥❣ ❦Õt q✉➯ ❈❤➢➡♥❣ ❝ñ❛ ✷ ❧✉❐♥ ❝❤ó♥❣ ➳♥ t➠✐ ➜ã ❧➭ ❝➳❝ ❞➲② ❝➳❝ ♣❤➬♥ tö i = 1, 2, , s tì ợ trì tệ x1 , , x s ∈ m xn1 , , xni i ❧➭ ❞✲❞➲② tr♦♥❣ ♥✐Ö♠ ❝➳❝ ❈❤➢➡♥❣ ❞❞✲❞➲② tr➟♥ ✷✱ ♠ét ✸ tr➟♥ ✹✳ ♠➠➤✉♥✳ n1 , , n s ni+1 M/(xi+1 , , xns s )M ✳ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ✈➭ >0 ✱ ▼ét tr♦♥❣ ♥❤÷♥❣ ❦Õt q✉➯ ❝❤Ý♥❤ ❝ñ❛ ❈❤➢➡♥❣ ✷ ❧➭ ➤➷❝ tr➢♥❣ tÝ♥❤ ❝❤✃t ❞❞✲❞➲② ❝đ❛ ❤Ư t❤❛♠ sè t❤➠♥❣ q✉❛ ❤➭♠ ➤é ❞➭✐ ✈➭ sè ❜é✐✳ ❈ơ t❤Ĩ ❝❤ó♥❣ t➠✐ ❝ã ➤Þ♥❤ ❧ý ✭①❡♠ ➜Þ♥❤ ❧ý ✷✳✶✳✽✮✳ ➜Þ♥❤ ❧ý✳ ●✐➯ sư x = x1 , , x d ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ M✳ ❈➳❝ ➤✐Ị✉ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ x ❧➭ ♠ét ❞❞✲❞➲② tr➟♥ M ✳ ✭✐✐✮ ❱í✐ ♠ä✐ n1 , , nd > 0✱ d (M/x(n)M ) = n1 ni e(x1 , , xi , (0 : xi+1 )M/(xi+2 , ,xd )M ) i=0 ✭✐✐✐✮ ❚å♥ t➵✐ ❝➳❝ sè ♥❣✉②➟♥ a0 , a1 , , ad s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n1 , , nd > 0✱ d (M/(xn1 , , xnd d )M ) = n n i i=0 ▼ét ❤Ö q✉➯ ❝đ❛ ➤Þ♥❤ ❧ý ♥➭② ❧➭ ♠ä✐ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ➤Ị✉ ❧➭ ❞❞✲❞➲②✳ ◆❣➢ỵ❝ ❧➵✐✱ tõ ♠ét ❦Õt q✉➯ ❝đ❛ ◆✳ ❚✳ ❈➢ê♥❣ t❤× ♠ä✐ ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲② ✈í✐ sè ♠ị ➤đ ❧í♥ ❧✉➠♥ ❧➭ ♠ét ❤Ö t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝✳ t❤❛♠ sè t❤❛♠ sè k ❧➭ ❞❞✲❞➲② ✈➭ ❤Ö x = x1 , , x d ❝ñ❛ ♣❤ø❝ ❑♦s③✉❧ ❝ñ❛ t❤❛♠ ❝ñ❛ M M sè ♣✲❝❤✉➮♥ t➽❝ ❧➭ ❉♦ ➤ã sù tå♥ t ủ ệ t ỗ ệ t ➤Þ♥❤ ♥❣❤Ü❛ ➤➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ø♥❣ ✈í✐ x ❧➭ d (−1)i−k (Hi (x, M )), χk (x, M ) = i=k ✺ tr♦♥❣ ➤ã Hi (x, M ) ♥ã✐ r➺♥❣ χ0 (x, M ) = e(x, M ) ❞➱♥ ❧➭ ♠➠➤✉♥ ➤å♥❣ ➤✐Ò✉ ❑♦s③✉❧ t❤ø ✈➭ χk (x, M ) i ✳ ▼ét ❦Õt q✉➯ ❝đ❛ ❙❡rr❡ ✈í✐ ♠ä✐ k = 0, 1, , d ✱ χ1 (x, M ) = (H0 (x, M )) − χ0 (x, M ) = (M/xM ) − e(x, M ) ➤Õ♥ ✳ (M/x(n)M ) ◆❤➢ ✈❐② tÝ♥❤ ❝❤✃t ➤❛ t❤ø❝ ❝ñ❛ ❤➭♠ χ1 (x(n), M ) ✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ t❤ø❝ ❝đ❛ ❤➭♠ x t➢➡♥❣ ➤➢➡♥❣ ✈í✐ tÝ♥❤ ➤❛ ❧➭ ❞❞✲❞➲② t❤× d−1 χ1 (x(n), M ) = n1 ni e(x1 , , xi , (0 : xi+1 )M/(xi+2 , ,xd )M ) i=0 ➜✐Ò✉ ❣✐➯ ♥➭② ◆✳ ❚✳ ❞➱♥ ➤Õ♥ ❈➢ê♥❣ χk (x(n), M ) ❧➭ ♠ét ❬✶❪✿ ❝➞✉ ♣❤➯✐ ♠ét ➤❛ ❤á✐ ♠ë ❝❤➝♥❣ t❤ø❝ tr♦♥❣ x ♥Õ✉ t❤❡♦ ❧✉❐♥ ❧➭ ♠ét n1 , , n d ➳♥ t✐Õ♥ ❤Ư ✈í✐ sÜ t❤❛♠ ♠ä✐ ❦❤♦❛ sè ❤ä❝ ❝ñ❛ ♣✲❝❤✉➮♥ k > ❄ ❚r➯ t➳❝ t➽❝ ❧ê✐ t❤× ❝➞✉ ❤á✐ ♥➭② ❝❤ó♥❣ t➠✐ ❝ã ❦Õt q✉➯ q✉❛♥ trä♥❣ t❤ø ❤❛✐ ❝đ❛ ❈❤➢➡♥❣ ✷ ✭①❡♠ ➜Þ♥❤ ❧ý ✷✳✷✳✸✮✳ x = x1 , , x d ➜Þ♥❤ ❧ý✳ ❈❤♦ ❞❞✲❞➲② tr➟♥ ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ M✳ ●✐➯ sư x ❧➭ ♠ét M ✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ n1 , , nd > 0✱ d−k χk (x(n), M ) = n1 ni e(x1 , , xi , (0 : xi+1 )Hk−1 (xi+2 , ,xd ,M ) ) i=0 ❍Ö q✉➯ ❧❐♣ tø❝ ❧➭ tr➯ ❧ê✐ ❦❤➻♥❣ ị ò ỉ r tờ ủ ❤➭♠ ❝➞✉ ❤á✐ tr➟♥✳ χk (x(n), M ) ❍➡♥ ♥÷❛ ❝❤ó♥❣ t➠✐ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭②✳ P❤➬♥ ❝✉è✐ ❝đ❛ ❈❤➢➡♥❣ ✷ ➤➢ỵ❝ ❞➭♥❤ ➤Ĩ ♥❣❤✐➟♥ ❝ø✉ tÝ♥❤ ❝❤✃t ❝đ❛ ❝➳❝ ♠➠➤✉♥ ❝ã ♠ét ❤Ö t❤❛♠ ❞❞✲❞➲②✱ ✈➭♥❤ R sè ❧➭ M trì ó sử tờ ợ sư ❧➭ ❝ã ♣❤ø❝ ➤è✐ ♥❣➱✉✳ ➤➯♠ ♠ä✐ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ➤ã ❚r♦♥❣ M ❞ơ♥❣ ❦❤➳✐ ♥✐Ư♠ ●✐➯ t❤✐Õt ♥➭② ❜➯♦ ➤Ị✉ ❝ã ♠ét ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲②✱ ❜➟♥ ❝➵♥❤ ❝ã ♥❤✐Ị✉ tÝ♥❤ ❝❤✃t ❦❤➳❝ ➤ã♥❣ ✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ ❝➳❝ ♥❣❤✐➟♥ ❝ø✉ ❝ã sư ❞ơ♥❣ ❞❞✲❞➲② ♥❤➢ tÝ♥❤ ➤ã♥❣ ❝ñ❛ q✉Ü tÝ❝❤ ❦❤➠♥❣ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ tÝ♥❤ ❝❛t❡♥❛r②✱ ✳ ✳ ✳ ❚✉② ♥❤✐➟♥✱ tr♦♥❣ t❤ù❝ tÕ ❝ã ♥❤✐Ò✉ ✈Ý ❞ô ✈➭♥❤ R ❦❤➠♥❣ ❝ã ♣❤ø❝ ➤è✐ ♥❣➱✉ ♥❤➢♥❣ ✈➱♥ ❧➭ ❝❛t❡♥❛r②✱ ❝ã q✉Ü tÝ❝❤ ❦❤➠♥❣ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧➭ ➤ã♥❣ ✈➭ ❝ã ❤Ö t❤❛♠ sè ❧➭ ❞❞✲❞➲②✳ ❚r♦♥❣ t✐Õt ✻ ❝✉è✐ ❝đ❛ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ ❜á ❣✐➯ t❤✐Õt R ❝ã ♣❤ø❝ ➤è✐ ♥❣➱✉✱ ❝❤Ø ❣✐➯ sö M ❝ã ♠ét ❤Ö t❤❛♠ sè ❧➭ ❞❞✲❞➲② ✈➭ ♥❣❤✐➟♥ ❝ø✉ sù t❤❛② ➤ỉ✐ ❝đ❛ ❝➳❝ tÝ♥❤ ❝❤✃t ❦❤➳❝✳ ▼ét sè ❦Õt q✉➯ ❜❛♥ ➤➬✉ t❤❡♦ ❤➢í♥❣ ♥➭② ❧✐➟♥ q✉❛♥ ➤Õ♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛ ✈➭ ♠ét ➤ tr➢ê♥❣ ợ ệt ủ ị ý rệt ể ts ợ tr×♥❤ ❜➭② tr♦♥❣ t✐Õt ❝✉è✐ ❝đ❛ ❝❤➢➡♥❣ ♥➭②✳ ❈❤➢➡♥❣ ✸ ➤➢ỵ❝ ❞➭♥❤ ❝❤♦ ❝➳❝ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳ ❈❤ó♥❣ t➠✐ tr➢í❝ ❤Õt ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❧ä❝ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ✈➭ ❤Ư t❤❛♠ sè tèt✳ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ♥➭② ➤ã♥❣ ✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ ❝➳❝ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❝❤➢➡♥❣ ♥➭② ✈➭ ❝➯ ❝❤➢➡♥❣ s❛✉ ✈Ò ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳ ❚❛ ♥ã✐ ♠ét ❧ä❝ F : M0 ⊂ M1 ⊂ ⊂ Mt = M M ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❝đ❛ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ♥Õ✉ dim M0 < dim M1 < < dim Mt = dim M = d ✳ ✳ ❑ý ❤✐Ö✉ di = dim Mi sè tèt ➤è✐ ✈í✐ x1 , , x di F ♥Õ✉ x = x1 , , x d ▼ét ❤Ö t❤❛♠ sè (xdi +1 , , xd )M ∩ Mi = ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ Mi ❧➭ ♠ét ❤Ư t❤❛♠ i = 0, 1, , t ✈í✐ ✳ ❑❤✐ ➤ã ✈➭ t❛ ❝ã t❤Ĩ ①Ðt ❤✐Ư✉ t IF,M (x) = (M/xM ) − e(x1 , , xdi , Mi ) i=0 IF,M (x) ➤➲ ➤➢ỵ❝ ❝ã ♥❤✐Ị✉ tÝ♥❤ ❝❤✃t t➢➡♥❣ tù ♥❤➢ ❤✐Ư✉ ①Ðt ▼❛❝❛✉❧❛② IF,M (x) tr➢í❝ s✉② ❧✉➠♥ n1 , , n d IF,M (x) t❤× ré♥❣ ❧➭ ➤➞② ✈➭ ♠ét ❤➭♠ ♥❤✐Ò✉ sè ♥➭② t➢➡♥❣ ❦❤✐ ♥❣❤✐➟♥ ✈✃♥ ❝ø✉ ➤Ị ➞♠✱ ❦❤➠♥❣ ❣✐➯♠✱ ➤➢➡♥❣ ✈í✐ ♠➠➤✉♥ ❦❤➳❝ ❦❤➠♥❣ ❦❤✐ IM (x) = (M/xM )−e(x, M ) tr♦♥❣ ①Ðt ✳ ✳ ✳ ✳ ➤➵✐ sè IF,M (x(n)) ❈ò♥❣ (M/xM ) ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ❝❤ó ý ❣✐❛♦ ♥❤➢ r➺♥❣ ❤♦➳♥✳ ♠ét ❜✃t ❈♦❤❡♥✲ ❈ơ t❤Ĩ✱ ❤➭♠ t❤❡♦ ➤➻♥❣ t❤ø❝ t i=0 e(x1 , , xdi , Mi ) ❧➭ ♠ét ♠ë ré♥❣ ➤➳♥❣ ❝❤ó ý ❝đ❛ ❜✃t ➤➻♥❣ t❤ø❝ q✉❡♥ ❜✐Õt ❣✐÷❛ ➤é ❞➭✐ ✈➭ sè ❜é✐ (M/xM ) e(x, M ) ✳ ❑Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ➤Þ♥❤ ❧ý s❛✉ ✭①❡♠ ➜Þ♥❤ ❧ý ✸✳✸✳✷✮✳ ➜Þ♥❤ ❧ý✳ ❈➳❝ ♠Ư♥❤ ➤Ò s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ M ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳ ✼ ❈❤➢➡♥❣ ✸ ❝ã t❤Ó tã♠ t➽t tr♦♥❣ F ✭✐✐✮ ❚å♥ t➵✐ ♠ét ❧ä❝ x = x1 , , x d tèt x = x1 , , x d ◆❤➢ ✈❐②✱ ❦❤✐ F ➤è✐ ✈í✐ F ✭✐✐✐✮ ❚å♥ t➵✐ ♠ét ❧ä❝ M ⊂ Mt = M t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ư♥ ❝❤✐Ị✉ ✈➭ ♠ét ❤Ư t❤❛♠ sè tèt s❛♦ ❝❤♦ IF,M (x(n)) = 0✱ ✈í✐ ♠ä✐ n1 , , nd > 0✳ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❤Ư t❤❛♠ sè F ✱ IF,M (x) = 0✳ ➤è✐ ✈í✐ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✱ tå♥ t➵✐ ♠ét ❧ä❝ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❤Ư t❤❛♠ sè tèt F : M0 ⊂ M1 ⊂ x = x1 , , x d t❛ ❧✉➠♥ ❝ã t (M/x(n)M ) = n1 ndi e(x1 , , xdi , Mi ) i=0 ❧➭ ♠ét ➤➷❝ ➤❛ t❤ø❝ tr➢♥❣ ❝đ❛ ✈í✐ ♠ét ♠ä✐ ❤Ư n1 , , n d > ✱ t❤❛♠ sè ❧➭ tr♦♥❣ ❞❞✲❞➲② q✉❛ ➤ã ❤➭♠ di = dim Mi ➤é ❞➭✐ ë ✳ ❚õ ❈❤➢➡♥❣ ♠ét ✷✱ t❛ s✉② r❛ ♠ä✐ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ❧✉➠♥ ❝ã ♠ét ❤Ö t❤❛♠ sè ❧➭ ❞❞✲❞➲②✳ ❑❤✐ ①Ðt ❝➳❝ ❦Õt q✉➯ e(x1 , , xdi , Mi ) tr➟♥ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ✈➭♥❤ ❙t❛♥❧❡②✲❘❡✐s♥❡r✱ ❝➳❝ ❤Ư sè ➤➢ỵ❝ tÝ♥❤ t➢ê♥❣ ♠✐♥❤ t❤➠♥❣ q✉❛ sè ❝➳❝ ♠➷t ❝ù❝ ➤➵✐ ❝đ❛ ♣❤ø❝ ➤➡♥ ❤×♥❤ t➢➡♥❣ ø♥❣ ✈í✐ ✈➭♥❤ ➤ã✳ ❚r♦♥❣ ❝❤➢➡♥❣ ❝✉è✐ ❝ï♥❣ ▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳ ❝❤ó♥❣ t➠✐ ♥❣❤✐➟♥ ré♥❣ t❤× ❧✉➠♥ tå♥ t➵✐ ♠ét ❤➺♥❣ sè F IF,M (x) < C ✱ ❝➳❝ ❤Ö t❤❛♠ sè tèt ❝đ❛ ❝❤♦ ❧í♣ ❝➳❝ ♠➠➤✉♥ ❑Õt q✉➯ ➤➬✉ t✐➟♥ ❝❤ó♥❣ t➠✐ ❝❤Ø r❛ ❧➭ ♥Õ✉ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ✈➭ ➤è✐ ✈í✐ ❝ø✉ ➤ ✳ ➷t M C ❧➭ ♠ét ❧➭ ♠ét ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❤Ö t❤❛♠ sè tèt IF (M ) = supIF,M (x) tr♦♥❣ ➤ã x x ❝ñ❛ M ❝❤➵② tr➟♥ x ➤è✐ ✈í✐ ❧ä❝ IF,M (x(n)) = IF (M ) F M ❈♦❤❡♥✲ ✈í✐ ♠ä✐ F ✳ ▲✉➠♥ tå♥ t➵✐ ♠ét ❤Ư t❤❛♠ sè x s❛♦ n1 , , n d > ✳ ❉♦ ➤ã t (M/x(n)M ) = n1 ndi e(x1 , , xdi , Mi ) + IF (M ) i=0 ✈➭ x ❧➭ ♠ét ❞❞✲❞➲② tr➟♥ M ✳ ❍➺♥❣ sè IF (M ) ➤è✐ ✈í✐ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ➤ã♥❣ ✈❛✐ trß t➢➡♥❣ tù ♥❤➢ ❤➺♥❣ sè ❇✉❝❤s❜❛✉♠ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ✹ ❧➭ ✈✐Ö❝ tÝ♥❤ ❤➺♥❣ sè IF (M ) I(M ) ➤è✐ ✈í✐ ❑Õt q✉➯ q✉❛♥ trä♥❣ t❤ø ❤❛✐ ❝ñ❛ ❈❤➢➡♥❣ t❤➠♥❣ q✉❛ ➤é ❞➭✐ ❝ñ❛ ♠ét sè ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳ ❚❛ ❝ã ➤Þ♥❤ ❧ý ✭①❡♠ ➜Þ♥❤ ❧ý ✹✳✷✳✻✮✳ ✽ ➜Þ♥❤ ❧ý✳ ❈❤♦ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ▼❛❝❛✉❧❛② s✉② ré♥❣ M ✈í✐ ♠ét ❧ä❝ ❈♦❤❡♥✲ F : M0 ⊂ M1 ⊂ ⊂ Mt = M ✳ ➜➷t di = dim Mi ✱ i = 0, 1, , t − 1✳ ❑❤✐ ➤ã t di+1 −1 di+1 −1 IF (M ) = i=0 k=di ❑Õt q✉➯ q✉❛♥ trä♥❣ t❤ø ❜❛ j=1 ❝ñ❛ k−1 j−1 ❈❤➢➡♥❣ ✹ (Hmj (M/Mi )) ❧➭ ➤Þ♥❤ ❧ý s❛✉ ✭①❡♠ ➜Þ♥❤ ❧ý ✹✳✸✳✷✮✳ ➜Þ♥❤ ❧ý✳ ❈➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ M ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳ ✭✐✐✮ ❚å♥ t➵✐ ♠ét ❧ä❝ x = x1 , , xd ❝đ❛ M ✈í✐ ♠ä✐ F t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉✱ ♠ét ❤Ư t❤❛♠ sè tèt ➤è✐ ✈í✐ F ✈➭ ♠ét ❤➺♥❣ sè C s❛♦ ❝❤♦ IF,M (x(n)) C n1 , , nd > 0✳ ✭✐✐✐✮ ❚å♥ t➵✐ ♠ét ❧ä❝ F t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ s❛♦ ❝❤♦ IF (M ) < ∞✳ ❚õ ➤Þ♥❤ ❧ý ♥➭② ❝❤ó♥❣ t➠✐ ❝❤ø♥❣ ♠✐♥❤ ♠ét t✐➟✉ ❝❤✉➮♥ ❤÷✉ ❤➵♥ ➤Ó ❦✐Ó♠ tr❛ tÝ♥❤ ❝❤✃t ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✿ M s✉② ré♥❣ ❞➲② ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ tå♥ t➵✐ ♠ét ❧ä❝ ♠ét ❤Ư t❤❛♠ sè tèt x ❝đ❛ M ➤è✐ ✈í✐ F ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② F s❛♦ ❝❤♦ ✾ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ✈➭ IF,M (x) = IF,M (x21 , , x2d ) ✳ ➳ n1 , , n d > ✈í✐ ♠ä✐ ✳ ♣ ❞ơ♥❣ ❣✐➯ t❤✐Õt q✉✐ ♥➵♣ ➤è✐ ✈í✐ D t❛ ♥❤❐♥ ➤➢ỵ❝ (x1 , , xd−1 )[xd (M/D1 ) : xi ] ⊆ xd (M/D1 ) + (0 : xi )M/D1 1 ✳ ❙ư ❞ơ♥❣ ❣✐➯ t❤✐Õt q✉✐ ♥➵♣ t❛ s✉② r❛ (x2 , , xd )[xd (M/xn1 M ) : xi ] ⊆ xd (M/xn1 M ) + (0 : xi )M/xn1 M ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ❉ï♥❣ ➤ (x2 , , xd )[(xd , xn1 )M : xi ] + xn1 M ⊆ xd M + xn1 M : xi ✳ Þ♥❤ ❧ý ●✐❛♦ ❑r✉❧❧ t❛ ➤➢ỵ❝ (x2 , , xd )[(xd , xn1 )M : xi ] + xn1 M (x2 , , xd )(xd M : xi ) ⊆ n1 (xd M + xn1 M : xi ) = xd M + :M xi ⊆ n1 ❱× d1 > ♣❤➢➡♥❣ ♥➟♥ ♣❤➳♣ t❛ ❝ò♥❣ t➢➡♥❣ tù ❝ã ➤è✐ ID1 ,M/xn1 M (xn2 , xn1 , xn3 , , xnd d ) = c ✈í✐ ❞➲② x , x1 , x3 , , x d (x1 , x3 , , xd )(xd M : xi ) ⊆ xd M + :M xi x i ) ⊆ x d M + :M x i ✳ ❱❐② t❛ ❝ị♥❣ ✳ ♥❤❐♥ ❇➺♥❣ ➤➢ỵ❝ (x1 , , xd )(xd M : ✳ ➜Þ♥❤ ❧ý s❛✉ ❧➭ ❦Õt q✉➯ ❝❤Ý♥❤ ❝ñ❛ t✐Õt ♥➭② ❝❤♦ t❛ ♠ét ➤➷❝ tr➢♥❣ ❝ñ❛ tÝ♥❤ ❝❤✃t ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② t❤➠♥❣ q✉❛ ❤➭♠ ✽✼ IF,M (x(n)) ✳ M ➜Þ♥❤ ❧ý ✹✳✸✳✷✳ ❈❤♦ ❧➭ ♠ét R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝❤✐Ị✉ d✳ ❈➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ M ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳ ✭✐✐✮ ❚å♥ t➵✐ ♠ét ❧ä❝ F F ✭✐✐✐✮ ❚å♥ t➵✐ ♠ét ❧ä❝ x1 , , xd ➤è✐ ✈í✐ F t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ✈➭ (ii) ⇒ (iii) t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ✈➭ ♠ét ❤Ư t❤❛♠ sè tèt s❛♦ ❝❤♦ (i) ⇒ (ii) ❈❤ø♥❣ ♠✐♥❤✳ IF,M (x(n)) ❧➭ ❤➺♥❣ sè ✈í✐ ♠ä✐ n1 , , nd > 0✳ ➤➲ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ tr♦♥❣ ❇ỉ ➤Ị ✹✳✷✳✺✳ ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥ ✈× IF,M (x(n)) (iii) ⇒ (i) ✿ ●✐➯ sư ❧ä❝ ❝❤✐Ị✉ ❝đ❛ M ❧➭ ✳ q✉✐ ♥➵♣ d t❤❡♦ ✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ◆Õ✉ d = t❤× ❦❤➻♥❣ ➤Þ♥❤ ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥✳ ●✐➯ sư ❦❤➠♥❣ ❣✐➯♠✳ D : D0 ⊂ ⊂ Dt = M ❧✉❐♥ ♥❤➢ ♣❤➬♥ ➤➬✉ ❝❤ø♥❣ ♠✐♥❤ ❇ỉ ➤Ị ✹✳✸✳✶✱ n1 , , n d > IF (M ) < ∞✳ ID,M (x(n)) ✳ ▲❐♣ ❧➭ ❤➺♥❣ sè ✈í✐ ♠ä✐ M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ❜➺♥❣ M ❧✉➠♥ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ♥➟♥ d>1 ✳ ❳Ðt ❧ä❝ Dd : (xd M + D0 )/xd M ⊂ · · · ⊂ (xd M + Ds )/xd M ⊂ M/xd M tr♦♥❣ ➤ã s = t−1 ♥Õ✉ dt−1 < d − ✈➭ s = t−2 ♠ét ❞❞✲❞➲② ♥➟♥ ❧➭ ♠ét ❤Ư t❤❛♠ sè tèt ❝đ❛ ✈í✐ i = 0, 1, , t − ✱ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✺✱ r❛ ♥ã✐ x1 , , xd−1 nd−1 IDd ,M/xd M (xn1 , , xd−1 ) 0, nd = ✳ ✈í✐ ●ä✐ Dd t❤á❛ ✳ ❚❛ ❝ã ♠➲♥ ❧➭ ❞❞✲❞➲② tr➟♥ M/xd M ❦✐Ư♥ M/xd M ✈í✐ ❝❤✐Ị✉✳ ❍➡♥ Di ♥÷❛✱ ❚❛ ❞Ơ ❞➭♥❣ s✉② ♠ä✐ n1 , , nd−1 > ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧➭ ❧ä❝ ❝❤✐Ị✉ ❝đ❛ ✳ ❑❤✐ ➤ã✱ t❤❡♦ ❧❐♣ ❧✉❐♥ ❣✐è♥❣ ♥❤➢ ♣❤➬♥ ➤➬✉ ❝❤ø♥❣ ♠✐♥❤ ❇ỉ ➤Ị ✹✳✸✳✶ t❛ ❝ã n d−1 (M/(xn1 , ,xd−1 , xd )M ) t = ❧➭ ✳ D : D0 ⊂ D1 ⊂ ⊂ Dt = M/xd M dim Di = di x ✳ ❱× (Di + xd M )/xd M ➤✐Ò✉ = ID,M (x(n)) = c ❚❤❡♦ ❣✐➯ t❤✐Õt q✉✐ ♥➵♣ s✉② ré♥❣ ❞➲②✳ M/xd M r✐➟♥❣ M dt−1 = d − ♥Õ✉ n1 ndi e(x1 , , xdi , (Di + xd M )/xd M ) + c i=2 ✽✽ t = n1 ndi e(x1 , , xdi , Di ) + c i=2 ❉♦ ➤ã t = s+1 ✈➭ Di = :M/xd M xdi +1 xdi +1 )/(xd M + Di ) ✳ di = di ❉➱♥ Di /Di−1 M/xd M + Ds Dd s ✳ ❍➡♥ ♥÷❛✱ tõ ❇æ Di /((xd M + Di )/xd M ) ❝ã ➤é ❞➭✐ ❤÷✉ ❤➵♥✳ ré♥❣ ♥➟♥ t❤❡♦ ❇ỉ ➤Ị ✹✳✶✳✸✱ ❱× ✈❐② ➤Õ♥ i ✈í✐ ❱× D ➤Ị ✸✳✶✳✻✱ (xd M : ❧➭ ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ❝ò♥❣ ❧➭ ♠ét ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❧➭ ❝➳❝ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ✈í✐ ♠ä✐ ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ i s ✈➭ ❚❛ ①Ðt ❤❛✐ tr➢ê♥❣ ❤ỵ♣✳ ❚r➢ê♥❣ ❤ỵ♣ ✶✳ dt−1 < d − (s = t − 1) ✳ ò ứ s rộ ữ ❝❤♦ M ✈➭ N = Dt−1 ✈➭ ❞❞✲❞➲② ❚❤❛② xd x1 , , x d ❜ë✐ x3d ✱ M/Dt−1 ➳♣ ❞ơ♥❣ ▼Ư♥❤ ➤Ị ✷✳✸✳✸ t❛ ❝ã ❞➲② ❦❤í♣ −→ Hmi−1 (M/Dt−1 ) −→ Hmi−1 (M/x3d M +Dt−1 ) −→ Hmi (M/Dt−1 ) −→ 0, i 0✳ ➤Ò ✹✳✶✳✸✱ F ❱× dim Di /Mi = di − ♥➟♥ ♥Õ✉ t > t❤× t❤❡♦ ❇ỉ ❦❤➠♥❣ ❧➭ ♠ét ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ➤ Þ♥❤ ❧ý ✹✳✸✳✷ ➤➲ ➤➢❛ r❛ ♠ét ➤➷❝ tr➢♥❣ ❝❤♦ tÝ♥❤ ❝❤✃t ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② t❤➠♥❣ q✉❛ tÝ♥❤ ❦❤➠♥❣ ➤ỉ✐ ❝đ❛ ❤➭♠ ✳ IF,M (x(n)) ✈í✐ ♠ä✐ n1 , , n d > ❍Ư q✉➯ s❛✉ ❝đ❛ ❇ỉ ➤Ị ✸✳✸✳✸ ✈➭ ➜Þ♥❤ ❧ý ✹✳✸✳✷ ❝❤Ø r❛ r➺♥❣ t❛ ❝❤Ø ❝➬♥ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ➤ã ✈í✐ ❤÷✉ ❤➵♥ n1 , , n d ✳ ✾✶ ❍Ö q✉➯ ✹✳✸✳✻✳ M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❝ã ♠ét ❧ä❝ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ❝❤♦ ✈➭ ♠ét ❤Ö t❤❛♠ sè tèt x1 , , x d ➤è✐ ✈í✐ F s❛♦ IF,M (x) = IF,M (x21 , , x2d )✳ ❍Ö q✉➯ ✹✳✸✳✼✳ ❈❤♦ ❤Ö t❤❛♠ sè M x1 , , x d ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ✈➭ F F ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳ ▼ét ❝ñ❛ M ❧➭ ♠ét ❞❞✲❞➲② ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❝ã ♠ét ❧ä❝ t❤á❛ s❛♦ ❝❤♦ x1 , , x d IF,M (x1 , , xd ) = IF,M (x21 , , x2d )✳ ✾✷ ❧➭ ♠ét ❤Ö t❤❛♠ sè tèt ➤è✐ ✈í✐ F ❑Õt ❧✉❐♥ ◆❤➢ ✈❐②✱ tr♦♥❣ ❧✉❐♥ ➳♥ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ❝➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ s❛✉✳ ✶✳ ➤ ➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ❞❞✲❞➲② ✈➭ ➤➷❝ tr➢♥❣ ♠ét ❤Ö t❤❛♠ sè ❧➭ ❞❞✲❞➲② q✉❛ ❤➭♠ ➤é ❞➭✐✳ ❚õ ➤ã s✉② r❛ q✉❛♥ ❤Ư ❣✐÷❛ ❞❞✲❞➲② ✈➭ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝✳ ✷✳ ❈❤ø♥❣ ♠✐♥❤ tÝ♥❤ ➤❛ t❤ø❝ ❝ñ❛ ❝➳❝ ➤➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ❝❛♦ ➤è✐ ✈í✐ ❧ị② t❤õ❛ ❝đ❛ ♠ét ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲② ✈➭ ➤➢❛ r❛ ❞➵♥❣ t➢ê♥❣ ♠✐♥❤ ❝ñ❛ ❝➳❝ ➤❛ t❤ø❝ ➤ã✳ ✸✳ ❈❤Ø r❛ sù tå♥ t➵✐ ❝đ❛ ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲② ❦❤✐ ❝❤✉②Ĩ♥ q✉❛ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ q✉Ü tÝ❝❤ ❦❤➠♥❣ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛②✳ ✹✳ ●✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❧ä❝ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉✱ ❤Ư t❤❛♠ sè tèt✱ ❤➭♠ IF,M (x1 , , xd ) ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥✳ ✺✳ ❈❤Ø r❛ sù tå♥ t➵✐ ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲② ➤è✐ ✈í✐ ❝➳❝ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳ ✻✳ ◆❣❤✐➟♥ ❝ø✉ tÝ♥❤ ❝❤✃t ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ❧✐➟♥ q✉❛♥ ➤Õ♥ ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ❤Ö t❤❛♠ sè tèt✱ ❤➭♠ ➤é ❞➭✐✱ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳ ✼✳ ➤ ➷❝ tr➢♥❣ tÝ♥❤ ❝❤✃t ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② q✉❛ tÝ♥❤ tr✐Ưt t✐➟✉ ❝đ❛ ❝➳❝ ❤➭♠ IF,M (x1 , , xd ) ✳ ➳ ♣ ❞ơ♥❣ ❦Õt q✉➯ ➤ã tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♠➠➤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ①✃♣ ①Ø ✈➭ ✈➭♥❤ ❙t❛♥❧❡②✲❘❡✐s♥❡r✳ ✽✳ ◆❣❤✐➟♥ ❝ø✉ ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✱ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛✱ ✳ ✳ ✳ ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳ ❚Ý♥❤ t♦➳♥ ❝➳❝ ❤➭♠ ➤é ❞➭✐✱ ❤➭♠ IF,M (xn1 , , xnd d ) ✱ ❤➺♥❣ sè IF (M ) ✱ ❤➭♠ ❍✐❧❜❡rt✲ ❙❛♠✉❡❧ t❤➠♥❣ q✉❛ ❝➳❝ ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ✾✳ ➤ ❤➭♠ ➷❝ tr➢♥❣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② q✉❛ tÝ♥❤ ❝❤✃t ❜Þ ❝❤➷♥ ❝ñ❛ IF,M (xn1 , , xnd d ) ✳ ✾✸ ▼ét ✈➭✐ ❤➢í♥❣ ♣❤➳t tr✐Ĩ♥ ✶✳ ø ♥❣ ❞ơ♥❣ ❞❞✲❞➲② ✈➭♦ ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ❝➳❝ ❧í♣ ♠➠➤✉♥ ré♥❣ ❤➡♥ ❝➳❝ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳ ◆ã✐ r✐➟♥❣✱ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤➷❝ tr➢♥❣ ❝❤♦ ♠ét ♠➠➤✉♥ ❝ã ♠ét ❤Ö t❤❛♠ sè ❧➭ ❞❞✲❞➲②✳ ✷✳ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ q✉❛♥ ❤Ư ❣✐÷❛ tÝ♥❤ ❝❤✃t ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈➭ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ❝ñ❛ ♠➠➤✉♥ ✈➭ ❝➳❝ ♠➠➤✉♥ ♣❤➞♥ ❜❐❝ ❧✐➟♥ ❦Õt✱ ♠➠➤✉♥ ❘❡❡s ➤è✐ ✈í✐ ♠ét ♣❤➬♥ ❝đ❛ ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲②✳ ✸✳ ➳ ♣ ❞ơ♥❣ ❝➳❝ ❦Õt q✉➯ ➤➲ ➤➵t ➤➢ỵ❝ ➤è✐ ✈í✐ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈➭♦ tr➢ê♥❣ ❤ỵ♣ ✈➭♥❤ ❙t❛♥❧❡②✲❘❡✐s♥❡r ✈➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤➷❝ tr➢♥❣ ✈Ị tỉ ❤ỵ♣ ❝đ❛ ♣❤ø❝ ➤➡♥ ❤×♥❤ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳ ❈➳❝ ❦Õt q✉➯ tr♦♥❣ ❧✉❐♥ ➳♥ ➤➲ ➤➢ỵ❝ ❜➳♦ ❝➳♦ ✈➭ t❤➯♦ ❧✉❐♥ t➵✐✿ ✲ ❙❡♠✐♥❛r ♣❤ß♥❣ ➤ ➵✐ sè✱ ✈✐Ư♥ ❚♦➳♥ ❤ä❝✳ ✲ ❙❡♠✐♥❛r ❜é ♠➠♥ ➜➵✐ sè✲❍×♥❤ ❤ä❝✲❚➠♣➠✱ ❦❤♦❛ ❚♦➳♥✲❈➡✲❚✐♥ ❤ä❝✳ ✲ ❍é✐ ♥❣❤Þ ❚♦➳♥ ❤ä❝ ❚♦➭♥ q✉è❝ ❧➬♥ t❤ø ✻✱ ❍✉Õ ✾✴✷✵✵✷✳ ộ ị sốì ọ trt r ♦♥ ➤ ➭ ▲➵t ✶✶✴✷✵✵✸✳ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛ ❛♥❞ ❈♦♠❜✐♥❛t♦r✐❝s✱ ❆❧❧❛❤❛❜❛❞✱ ■♥❞✐❛ ✶✷✴✷✵✵✸✳ ✲ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛ ❛♥❞ ■♥t❡r❛❝t✐♦♥ ✇✐t❤ ❈♦♠❜✐♥❛t♦r✐❝s ❛♥❞ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡tr②✱ ■❈❚P✱ ❚r✐❡st✱ ■t❛❧② ✻✴✷✵✵✹✳ ✲ ộ ị P sốì ọ t ố ❍å ❈❤Ý ▼✐♥❤ ✶✶✴✷✵✵✺✳ ❙❝❤♦♦❧ ❛♥❞ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ❍❛♥♦✐ ✶✴✷✵✵✻✳ ✲ ❚❤❡ ❙❡❝♦♥❞ ❏❛♣❛♥✲❱✐❡t♥❛♠ ❏♦✐♥t ❙❡♠✐♥❛r ♦♥ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ▼❡✐❥✐ ❯♥✐✈❡rs✐t②✱ ❚♦❦②♦✱ ❏❛♣❛♥ ✸✴✷✵✵✻✳ ✲ ❙❡♠✐♥❛r ♦♥ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ❋❛❝✉❧t② ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❉✉✐s❜✉r❣✲❊ss❡♥✱ ❊ss❡♥✱ ●❡r♠❛♥② ✻✴✷✵✵✻✳ ✾✹ ❯♥✐✈❡rs✐t② ♦❢ ❈➳❝ ❝➠♥❣ tr×♥❤ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❧✉❐♥ ➳♥ ✶✳ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ❉✳ ❚✳ ❈✉♦♥❣ ✭✷✵✵✸✮✱ ❞❞✲❙❡q✉❡♥❝❡s P♦✐♥❝❛rÐ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❑♦s③✉❧ ❝♦♠♣❧❡①✱ ❏✳ ▼❛t❤✳ ✷✳ ♣❛rt✐❛❧ ❙❤♦rt ❝♦♠♠✉♥✐❝❛t✐♦♥✱ ❊✉❧❡r✲ ❱✐❡t♥❛♠ ✸✶✭✸✮✱ ♣♣✳ ✸✺✸✲✸✺✽✳ ◆✳ ❚✳ ❈✉♦♥❣✱ ❉✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ❍✳ ▲✳ ❚r✉♦♥❣ ✭✷✵✵✻✮✱ ❖♥ ❛ ♥❡✇ ✐♥✈❛r✐❛♥t ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❧♦❝❛❧ r✐♥❣s✱ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ✸✳ ❛♥❞ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ Pr♦❝✳ ❚❤❡ ✷♥❞ ❏❛♣❛♥✲❱✐❡t♥❛♠ ❏♦✐♥t ❙❡♠✐♥❛r ♦♥ ▼❡✐❥✐ ❯♥✐✈❡rs✐t②✱ ❚♦❦②♦✱ ❏❛♣❛♥✱ ♣♣✳ ✷✻✲✸✼✳ ❉✳ ❚✳ ❈✉♦♥❣ ✭✷✵✵✼✮✱ ❞❞✲❙❡q✉❡♥❝❡s P♦✐♥❝❛rÐ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❑♦s③✉❧ ❝♦♠♣❧❡①✱ ❛♥❞ ❏✳ ❆❧❣❡❜r❛ ❆♣♣❧✳ ♣❛rt✐❛❧ ❊✉❧❡r✲ ✻✭✷✮✱ ♣♣✳ ✷✵✼✲ ✷✸✶✳ ✹✳ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ❉✳ ❚✳ ♠♦❞✉❧❡s✱ ❑♦❞❛✐✳ ▼❛t❤✳ ❏✳ ✺✳ ❈✉♦♥❣ ◆✳ ❚✳ ❛♥❞ ❉✳ ❈✉♦♥❣ ✭✷✵✵✼✮✱ s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✸✵✱ ♣♣✳ ✹✵✾✲✹✷✽✳ ❚✳ ❈✉♦♥❣ ✭✷✵✵✼✮✱ ❣❡♥❡r❛❧✐③❡❞ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✱ ✻✳ ❖♥ ❖♥ t❤❡ ❏✳ ❆❧❣❡❜r❛ str✉❝t✉r❡ ♦❢ s❡q✉❡♥t✐❛❧❧② ✸✶✼✱ ♣♣✳ ✼✶✹✲✼✹✷✳ ❉✳ ❚✳ ❈✉♦♥❣ ✭✷✵✵✼✮✱ ♣✲❙t❛♥❞❛r❞ s②st❡♠s ♦❢ ♣❛r❛♠❡t❡rs✱ ❧♦❝❛❧✐③❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ♣r❡♣r✐♥t✳ ✾✺ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ◆✳ ❚✳ ❈➢ê♥❣ ✭✶✾✾✺✮✱ ▲ý t❤✉②Õt ❦✐Ĩ✉ ➤❛ t❤ø❝ ✈➭ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ tr♦♥❣ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ✈➭ ø♥❣ ❞ô♥❣✱ ▲✉❐♥ ➳♥ t✐Õ♥ sÜ ❦❤♦❛ ❤ä❝✱ ❍➭ ◆é✐✳ ❚✐Õ♥❣ ❆♥❤ ❬✷❪ ▼✳ ❆✉s❧❛♥❞❡r ♣❧✐❝✐t②✱ ❬✸❪ ❛♥❞ ❆♥♥✳ ▼❛t❤✳ ❉✳ ❆✳ ❇✉❝❤s❜❛✉♠ ✭✶✾✺✽✮✱ ❈♦❞✐♠❡♥s✐♦♥ ❛♥❞ ♠✉❧t✐✲ ✻✽✱ ♣♣✳ ✻✷✺✲✻✺✼✳ ▼✳ ❇r♦❞♠❛♥♥ ✭✶✾✼✽✮✱ ❆ ♣❛rt✐❝✉❧❛r ❝❧❛ss ♦❢ r❡❣✉❧❛r ❞♦♠❛✐♥s✱ ❏✳ ❆❧❣❡❜r❛ ✺✹✱ ♣♣✳ ✸✻✻✲✸✼✸✳ ❬✹❪ ▼✳ ❇r♦❞♠❛♥♥✱ ❈✳ ❘♦tt❤❛✉s ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣ ✭✷✵✵✵✮✱ ❖♥ ❛♥♥✐❤✐❧❛t♦rs ❛♥❞ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ❏✳ P✉r❡ ❆♣♣❧✳ ❆❧❣❡❜r❛ ✶✺✸✱ ♣♣✳ ✶✾✼✲✷✷✼✳ ❬✺❪ ▼✳ ❇r♦❞♠❛♥♥ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣ ✭✶✾✾✽✮✱ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣②✿ ❆♥ ❆❧❣❡❜r❛✐❝ ■♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ●❡♦♠❡tr✐❝ ❆♣♣❧✐❝❛t✐♦♥s✱ ❬✻❪ ❲✳ ❇r✉♥s ❛♥❞ ❏✳ ❍❡r③♦❣ ✭✶✾✾✸✮✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❈♦❤❡♥ ▼❛❝❛✉❧❛② r✐♥❣s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✼❪ ❉✳ ❚✳ ❈✉♦♥❣ ✭✷✵✵✼✮✱ ♣✲❙t❛♥❞❛r❞ s②st❡♠s ♦❢ ♣❛r❛♠❡t❡rs✱ ❧♦❝❛❧✐③❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ♣r❡♣r✐♥t✳ ❬✽❪ ◆✳ ❚✳ ❈✉♦♥❣ ✭✶✾✾✵✮✱ ❖♥ t❤❡ ❧❡♥❣t❤ ♦❢ ♣♦✇❡rs ♦❢ s②st❡♠s ♦❢ ♣❛r❛♠❡t❡rs ✐♥ ❧♦❝❛❧ r✐♥❣✱ ◆❛❣♦②❛ ▼❛t❤✳ ❏✳ ✶✷✵✱ ♣♣✳ ✼✼✲✽✽✳ ✾✻ ❬✾❪ ◆✳ ❚✳ ❈✉♦♥❣ ✭✶✾✾✷✮✱ ❖♥ t❤❡ ❧❡❛st ❞❡❣r❡❡ ♦❢ ♣♦❧②♥♦♠✐❛❧s ❜♦✉♥❞✐♥❣ ❛❜♦✈❡ t❤❡ ❞✐❢❢❡r❡♥❝❡s ❜❡t✇❡❡♥ ❧❡♥❣t❤s ❛♥❞ ♠✉❧t✐♣❧✐❝✐t✐❡s ♦❢ ❝❡rt❛✐♥ s②st❡♠s ♦❢ ♣❛r❛♠❡t❡rs ✐♥ ❧♦❝❛❧ r✐♥❣✱ ❬✶✵❪ ◆✳ ❚✳ ❈✉♦♥❣ ✭✶✾✾✺✮✱ ✐❞❡❛❧s ✐♥ ❧♦❝❛❧ r✐♥❣s✱ ❬✶✶❪ ◆✳ ❚✳ ❈✉♦♥❣ ❬✶✷❪ ✶✷✺✱ ♣♣✳ ✶✵✺✲✶✶✹✳ ♣✲❙t❛♥❞❛r❞ s②st❡♠s ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣✲st❛♥❞❛r❞ ❆❝t❛ ▼❛t❤✳ ❱✐❡t♥❛♠ ✭✶✾✾✽✮✱ ◆♦❡t❤❡r✐❛♥ s❝❤❡♠❡✱ ◆❛❣♦②❛ ▼❛t❤✳ ❏✳ ❘❡♠❛r❦s ♦♥ Pr♦❝✳ ❆✳ ▼✳ ❙✳ t❤❡ ✷✵✭✶✮✱ ♣♣✳ ✶✹✺✲✶✻✶✳ ♥♦♥✲❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧♦❝✉s ✶✷✻✱ ♣♣✳ ✶✵✶✼✲✶✵✷✷✳ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ❉✳ ❚✳ ❈✉♦♥❣ ✭✷✵✵✸✮✱ ❞❞✲❙❡q✉❡♥❝❡s ❛♥❞ ♣❛rt✐❛❧ ❊✉❧❡r✲ P♦✐♥❝❛rÐ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❑♦s③✉❧ ❝♦♠♣❧❡①✱ ❙❤♦rt ❝♦♠♠✉♥✐❝❛t✐♦♥✱ ♥❛♠ ❏✳ ▼❛t❤✳ ❬✶✸❪ ♦❢ ❱✐❡t✲ ✸✶✭✸✮✱ ♣♣✳ ✸✺✸✲✸✺✽✳ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ❉✳ ❚✳ ❈✉♦♥❣ ✭✷✵✵✼✮✱ ❞❞✲❙❡q✉❡♥❝❡s ❛♥❞ ♣❛rt✐❛❧ ❊✉❧❡r✲ P♦✐♥❝❛r❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❑♦s③✉❧ ❝♦♠♣❧❡①✱ ❏✳ ❆❧❣❡❜r❛ ❆♣♣❧✳ ✻✭✷✮✱ ♣♣✳ ✷✵✼✲✷✸✶✳ ❬✶✹❪ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ❉✳ ❚✳ ❈✉♦♥❣ ✭✷✵✵✼✮✱ ❖♥ s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✱ ❬✶✺❪ ❑♦❞❛✐✳ ▼❛t❤✳ ❏✳ ✸✵✱ ♣♣✳ ✹✵✾✲✹✷✽✳ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ❉✳ ❚✳ ❈✉♦♥❣ ✭✷✵✵✼✮✱ ❖♥ t❤❡ str✉❝t✉r❡ ♦❢ s❡q✉❡♥t✐❛❧❧② ❣❡♥❡r❛❧✐③❡❞ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✱ ❬✶✻❪ ♦♥ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ◆✳ ❚✳ ❈✉♦♥❣ ❝❤❛r❛❝t❡r✐st✐❝s ❩✳ ❬✶✽❪ ✸✶✼✱ ♣♣✳ ✼✶✹✲✼✹✷✳ ◆✳ ❚✳ ❈✉♦♥❣✱ ❉✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ❍✳ ▲✳ ❚r✉♦♥❣ ✭✷✵✵✻✮✱ ❖♥ ❛ ♥❡✇ ✐♥✈❛r✐❛♥t ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❧♦❝❛❧ r✐♥❣s✱ ❬✶✼❪ ❏✳ ❆❧❣❡❜r❛ ❛♥❞ ♦❢ ❱✳ ❚✳ ❝❡rt❛✐♥ Pr♦❝✳ ❚❤❡ ✷♥❞ ❏❛♣❛♥✲❱✐❡t♥❛♠ ❏♦✐♥t ❙❡♠✐♥❛r ▼❡✐❥✐ ❯♥✐✈❡rs✐t②✱ ❚♦❦②♦✱ ❏❛♣❛♥✱ ♣♣✳ ✷✻✲✸✼✳ ❑❤♦✐ ✭✶✾✾✻✮✱ s②st❡♠s ♦❢ ❖♥ t❤❡ ♣❛rt✐❛❧ ♣❛r❛♠❡t❡rs ✐♥ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❧♦❝❛❧ r✐♥❣s✱ ▼❛t❤✳ ✷✷✷✱ ♣♣✳ ✸✽✸✲✸✾✵✳ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ▲✳ ❚✳ ◆❤❛♥ ✭✷✵✵✸✮✱ Ps❡✉❞♦ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❛♥❞ ♣s❡✉❞♦ ❣❡♥❡r❛❧✐③❡❞ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✱ ✾✼ ❏✳ ❆❧❣❡❜r❛ ✷✻✼✱ ♣♣✳ ✶✺✻✲✶✼✼✳ ❬✶✾❪ ❙✳ ❋❛r✐❞✐ ✭✷✵✵✹✮✱ ❙✐♠♣❧✐❝✐❛❧ P✉r❡ ❆♣♣❧✳ ❆❧❣❡❜r❛ ❬✷✵❪ tr❡❡s ❛r❡ s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ❏✳ ✶✾✵✱ ♣♣✳ ✶✷✶✲✶✸✻✳ ❈✳ ❆✳ ❋r❛♥❝✐s❝♦ ❛♥❞ ❍✳ ❍✳ ❚❛✐ ✭✷✵✵✻✮✱ ❲❤✐s❦❡rs ❛♥❞ s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ❣r❛♣❤s✱ ♣r❡♣r✐♥t✳ ❬✷✶❪ ❙✳ ●♦t♦ ✭✶✾✽✶✮✱ ❆♣♣r♦①✐♠❛t❡❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② r✐♥❣s✱ ❏✳ ❆❧❣❡❜r❛ ✼✻✱ ♣♣✳ ✷✶✹✲✷✷✺✳ ❬✷✷❪ ❙✳ ●♦t♦ ❛♥❞ ❑✳ ❨❛♠❛❣✐s❤✐ ✭✶✾✽✺✮✱ ❚❤❡ t❤❡♦r② ♦❢ ✉♥❝♦♥❞✐t✐♦♥❡❞ str♦♥❣ ❞✲s❡q✉❡♥❝❡s ❛♥❞ ♠♦❞✉❧❡s ♦❢ ❢✐♥✐t❡ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✱ ♣r❡♣r✐♥t ✭✉♥♣✉❜✲ ❧✐s❤❡❞✮✳ ❬✷✸❪ ❆✳ ●r♦t❤❡♥❞✐❡❝❦ ✭✶✾✻✼✮✱ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣②✱ ▲❡❝t✳ ◆♦t❡s ✐♥ ▼❛t❤✳ ✹✶✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥✲❍❡✐❞❡❧❜❡r❣✲◆❡✇ ❨♦r❦✳ ❬✷✹❪ ❘✳ ❍❛rts❤♦r♥❡ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡tr②✱ ✭✶✾✼✼✮✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥✲ ❍❡✐❞❡❧❜❡r❣✲◆❡✇ ❨♦r❦✳ ❬✷✺❪ ❘✳ ❍❛rts❤♦r♥❡ ✭✶✾✻✻✮✱ ❘❡s✐❞✉❡s ❛♥❞ ❞✉❛❧✐t②✱ ▲❡❝t✳ ◆♦t❡s ✐♥ ▼❛t❤✳ ✷✵✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥✲❍❡✐❞❡❧❜❡r❣✲◆❡✇ ❨♦r❦✳ ❬✷✻❪ ▼✳ ❍❡rr♠❛♥♥✱ ❜❧♦✇✐♥❣ ✉♣✱ ❬✷✼❪ ❏✳ ❍❡r③♦❣ ❙✳ ■❦❡❞❛ ❛♥❞ ❯✳ ❖r❜❛♥③ ✭✶✾✽✽✮✱ ❊q✉✐♠✉❧t✐♣❧✐❝✐t② ❛♥❞ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥✲❍❡✐❞❡❧❜❡r❣✲◆❡✇ ❨♦r❦✳ ❛♥❞ ❉✳ P♦♣❡s❝✉ ✭✷✵✵✻✮✱ ❋✐♥✐t❡ ❢✐❧tr❛t✐♦♥s ♦❢ ♠♦❞✉❧❡s ❛♥❞ s❤❡❧❧❛❜❧❡ ♠✉❧t✐❝♦♠♣❧❡①❡s✱ ♣r❡♣r✐♥t✳ ❬✷✽❪ ❏✳ ❍❡r③♦❣ ❛♥❞ ❊✳ ❙❜❛r❛ ✭✷✵✵✷✮✱ ❙❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s ❛♥❞ ❧♦❝❛❧ ✭▼✉♠❜❛✐✱ ❝♦❤♦♠♦❧♦❣②✱ ✷✵✵✵✮✱ ❆❧❣❡❜r❛✱ ❛r✐t❤♠❡t✐❝ ❛♥❞ ❣❡♦♠❡tr②✱ ✸✷✼✲✸✹✵✱ P❛rt ❚❛t❛ ■♥st✳ ❋✉♥❞✳ ❘❡s✳ ❙t✉❞✳ ▼❛t❤✳✱ ■✱ ■■ ✶✻✱ ❚❛t❛ ❈✳ ❍✉♥❡❦❡ ✭✶✾✽✷✮✱ ❚❤❡ t❤❡♦r② ♦❢ ❞✲s❡q✉❡♥❝❡s ❛♥❞ ♣♦✇❡rs ♦❢ ✐❞❡❛❧s✱ ❆❞✈✳ ■♥st✳ ❋✉♥❞✳ ❘❡s✳✱ ❇♦♠❜❛②✳ ❬✷✾❪ ✐♥ ▼❛t❤✳ ✹✻✱ ♣♣✳ ✷✹✾✲✷✼✾✳ ✾✽ ❬✸✵❪ ❚✳ ❑❛✇❛s❛❦✐ ✭✷✵✵✵✮✱ ❖♥ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ❬✸✶❪ ❚✳ ❑❛✇❛s❛❦✐ ✭✷✵✵✷✮✱ ❬✸✷❪ ♦❢ ◆♦❡t❤❡r✐❛♥ s❝❤❡♠❡s✱ ✸✺✷✭✻✮✱ ♣♣✳ ✷✺✶✼✲✷✺✺✷✳ ❖♥ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ▼❛❝❛✉❧❛②❢✐❝❛t✐♦♥ ❛r✐t❤♠❡t✐❝ ▼❛❝❛✉❧❛②❢✐❝❛t✐♦♥ ♦❢ ❧♦❝❛❧ r✐♥❣s✱ ✸✺✹✭✶✮✱ ♣♣✳ ✶✷✸✲✶✹✾✳ ❚✳ ❑❛✇❛s❛❦✐ ✭✷✵✵✻✮✱ ❖♥ ❋❛❧t✐♥❣s✬ ❆♥♥✐❤✐❧❛t♦r ❚❤❡♦r❡♠✱ ❏❛♣❛♥✲❱✐❡t♥❛♠ ❏♦✐♥t ❙❡♠✐♥❛r ♦♥ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ Pr♦❝✳ ❚❤❡ ✷♥❞ ▼❡✐❥✐ ❯♥✐✈❡r✲ s✐t②✱ ❚♦❦②♦✱ ❏❛♣❛♥✱ ♣♣✳ ✶✻✾✲✶✼✹✳ ❬✸✸❪ ❉✳ ❑✐r❜② ❛♥❞ ❏✲▲✳ ●❛r❝✐❛ ❘♦✐❣ ✭✶✾✽✻✮✱ ❖♥ t❤❡ ❑♦s③✉❧ ❤♦♠♦❧♦❣② ♠♦❞✉❧❡s ❢♦r t❤❡ ♣♦✇❡rs ♦❢ ❛ ♠✉❧t✐♣❧✐❝✐t② s②st❡♠✱ ❬✸✹❪ ❍✳ ▼❛ts✉♠✉r❛ ✭✶✾✽✻✮✱ ▼❛t❤❡♠❛t✐❦❛ ❈♦♠♠✉t❛t✐✈❡ ❘✐♥❣ ❚❤❡♦r②✱ ✸✸✱ ♣♣✳ ✾✻✲✶✵✶✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ▲♦❝❛❧ ❘✐♥❣s✱ ❬✸✺❪ ▼✳ ◆❛❣❛t❛ ✭✶✾✻✷✮✱ ❬✸✻❪ ❏✲▲✳ ●❛r❝✐❛ ❘♦✐❣ ✭✶✾✽✻✮✱ ❖♥ ♣♦❧②♥♦♠✐❛❧ ❜♦✉♥❞s ❢♦r t❤❡ ❑♦s③✉❧ ❤♦♠♦❧✲ ♦❣② ♦❢ ❝❡rt❛✐♥ ♠✉❧t✐♣❧✐❝✐t② ■♥t❡rs❝✐❡♥❝❡ ◆❡✇ ❨♦r❦✳ s②st❡♠s✱ ❏✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ✸✸✭✷✮✱ ♣♣✳ ✹✶✶✲✹✶✻✳ ❬✸✼❪ P✳ ❙❝❤❡♥③❡❧ ❢✐❧t❡r❡❞ ♠♦❞✉❧❡s✳ ❋✐♦r❡♥t✐♥✐✱ ❬✸✽❪ ✭✶✾✾✽✮✱ ❖♥ t❤❡ ❞✐♠❡♥s✐♦♥ ❢✐❧tr❛t✐♦♥ ❛♥❞ ❈♦❤❡♥✲▼❛❝❛✉❧❛② Pr♦❝✳ ♦❢ t❤❡ ❋❡rr❛r❛ ▼❡❡t✐♥❣ ✐♥ ❤♦♥♦r ♦❢ ▼❛r✐♦ ❯♥✐✈❡rs✐t② ♦❢ ❆♥t✇❡r♣✱ ❲✐❧r✐❥❦✱ ❇❡❧❣✐✉♠✱ ♣♣✳ ✷✹✺✲✷✻✹✳ ❘✳ P✳ ❙t❛♥❧❡② ✭✶✾✾✻✮✱ ❈♦♠❜✐♥❛t♦r✐❝s tt r ăa t r u ¨ ❙t ❝r❛❞ ❛♥❞ ✉s❡r ❇♦st♦♥✳ ❲✳ ❱♦❣❡❧ ✭✶✾✽✻✮✱ ❇✉❝❤s❜❛✉♠ ❘✐♥❣s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥✳ ❬✹✵❪ ▼✳ ❚♦✉s✐ ❛♥❞ ❙✳ ❨❛ss❡♠✐ ✭✷✵✵✺✮✱ ❙❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s ✉♥❞❡r ❜❛s❡ ❝❤❛♥❣❡✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✾✾ ✸✸✱ ♣♣✳ ✸✾✼✼✲✸✾✽✼✳ ❬✹✶❪ ◆✳ ❱✳ ❚r✉♥❣ ❧♦❝❛❧ r✐♥❣s✱ ❬✹✷❪ ◆✳ ❱✳ ✭✶✾✽✶✮✱ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ▼❛t❤✳ Pr♦❝✳ ❈❛♠❜r✐❞❣❡ P❤✐❧♦s✳ ❙♦❝✳ ❚r✉♥❣ ✭✶✾✽✸✮✱ ❈❛♠❜✳ P❤✐❧✳ ❙♦❝✳ ❬✹✸❪ ❆ ❆❜s♦❧✉t❡❧② s✉♣❡r❢✐❝✐❛❧ ✉♥♠✐①❡❞ ✽✾✭✷✮✱ ♣♣✳ ✷✸✼✲✷✸✾✳ s❡q✉❡♥❝❡s✱ ▼❛t❤✳ Pr♦❝✳ ✾✸✱ ♣♣✳ ✸✺✲✹✼✳ ◆✳ ❱✳ ❚r✉♥❣ ✭✶✾✽✻✮ ✱ ❚♦✇❛r❞ ❛ t❤❡♦r② ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✱ ◆❛❣♦②❛ ▼❛t❤✳ ❏✳ ✶✵✷✱ ♣♣✳ ✶✲✹✾✳ ❚✐Õ♥❣ P❤➳♣ ❬✹✹❪ ❉✳ ❋❡rr❛♥❞ ❛♥❞ ▼✳ ❘❛②♥❛✉❞ ✭✶✾✼✵✮✱ ❋✐❜r❡s ❢♦r♠❡❧❧❡s ❞✬✉♥ ❛♥♥❡❛✉ ❧♦❝❛❧ ◆♦❡t❤❡r✐❡♥✱ ❬✹✺❪ ❆♥♥✳ ❙❝✳ ❊❝♦❧❡ ◆♦r♠✳ ❙✉♣✳ ❏✳ P✳ ❙❡rr❡ ✭✶✾✼✺✮✱ ✸✱ ♣♣✳ ✷✾✺✲✸✶✶✳ ❆❧❣❒❜r❡ ▲♦❝❛❧❡✳ ▼✉❧t✐♣❧✐❝✐tÐs✱ ✸r❞ ❡❞✳✱ ▲❡❝t✳ ◆♦t❡s ✐♥ ▼❛t❤✳ ✶✶✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥✲❍❡✐❞❡❧❜❡r❣✲◆❡✇ ❨♦r❦✳ ❚✐Õ♥❣ ❬✹✻❪ ◆✳ ❚✳ ➤ø❝ ❈✉♦♥❣✱ P✳ ❙❝❤❡♥③❡❧ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ▼♦❞✉❧♥✱ ă U ts r ▼❛t❤✳ ◆❛❝❤r✳ ✭✶✾✼✽✮✱ ❱❡r❛❧❧❣❡♠❡✐♥❡rt❡ ✽✺✱ ♣♣✳ ✺✼✲✼✸✳ ❜❡r ▼❛❝❛✉❧❛②❢✐③✐❡r✉♥❣✱ ▼❛t❤✳ ts ă U ❆r❝❤✳ ▼❛t❤✳ ✭❇❛s❡❧✮ ❬✹✾❪ ❜❡r ❞✐❡ ❛♥♥✉❧❛t♦r❡♥ ❧♦❦❛❧❡r ❑♦❤♦♠♦❧♦❣✐❡♥❣r✉♣♣❡♥✱ ✸✵✱ ♣♣✳ ✹✼✸✲✹✼✻✳ P✳ ❙❝❤❡♥③❡❧ ✭✶✾✽✷✮✱ ❉✉❛❧✐s✐❡r❡♥❞❡ ❑♦♠♣❧❡①❡ ✐♥ ❞❡r ❧♦❦❛❧❡♥ ❆❧❣❡❜r❛ ✉♥❞ ❇✉❝❤s❜❛✉♠✲❘✐♥❣❡✱ ▲❡❝t✳ ◆♦t❡s ✐♥ ❍❡✐❞❡❧❜❡r❣✲◆❡✇ ❨♦r❦✳ ✶✵✵ ▼❛t❤✳ ✾✵✼✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥✲ Thank you for evaluating AnyBizSoft PDF Splitter A watermark is added at the end of each output PDF file To remove the watermark, you need to purchase the software from http://www.anypdftools.com/buy/buy-pdf-splitter.html