Cơ sở groebner và một áp dụng cho phân tích nguyên sơ

✐ ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ r➺♥❣ ❝➳❝ ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ❤♦➭♥ t♦➭♥ tr✉♥❣ t❤ù❝ ✈➭ ❦❤➠♥❣ trï♥❣ ❧➷♣ ✈í✐ ➤Ị t➭✐ ❦❤➳❝✳ ◆❣✉å♥ t➭✐ ❧✐Ư✉ sư ❞ơ♥❣ ❝❤♦ ✈✐Ư❝ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥ ➤➲ ➤➢ỵ❝ sù ➤å♥❣ ý ❝đ❛ ❝➳ ♥❤➞♥ ✈➭ tỉ ❝❤ø❝✳ ❈➳❝ t❤➠♥❣ t✐♥✱ t➭✐ ❧✐Ư✉ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ➤➲ ➤➢ỵ❝ ❣❤✐ râ ♥❣✉å♥ ❣è❝✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✾ ♥➝♠ ✷✵✶✸ ❍ä❝ ✈✐➟♥ ◆❣✉②Ô♥ ❚❤ï② ❚r❛♥❣ ❳➳❝ ♥❤❐♥ ❳➳❝ ♥❤❐♥ ❝ñ❛ ❚r➢ë♥❣ ❦❤♦❛ ❝❤✉②➟♥ ♠➠♥ ❝ñ❛ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❚❙✳ ❚r➬♥ ◆❣✉②➟♥ ❆♥ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✐✐ ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ ❞➢í✐ sù ❝❤Ø ❜➯♦ ✈➭ ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝đ❛ ❚❙✳ ❚r➬♥ ◆❣✉②➟♥ ❆♥✳ ❚❤➬② ➤➲ ❞➭♥❤ ♥❤✐Ị✉ t❤ê✐ ❣✐❛♥ ❤➢í♥❣ ❞➱♥ ✈➭ ❣✐➯✐ ➤➳♣ ❝➳❝ t❤➽❝ ♠➽❝ ❝❤♦ t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❧➭♠ ❧✉❐♥ ✈➝♥✳ ❚➠✐ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ s➞✉ s➽❝ ➤Õ♥ t❤➬②✳ ❚➠✐ ①✐♥ ❣ư✐ tí✐ ❝➳❝ t❤➬② ❝➠ ❑❤♦❛ ❚♦➳♥✱ ❑❤♦❛ ❙❛✉ ➤➵✐ ❤ä❝ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ✲ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥ ❝ò♥❣ ♥❤➢ ❝➳❝ t❤➬② ❝➠ ➤➲ t❤❛♠ ❣✐❛ ❣✐➯♥❣ ❞➵② ❦❤ã❛ ❤ä❝ ✷✵✶✶✲✷✵✶✸✱ ❧ê✐ ❝➯♠ ➡♥ s➞✉ s➽❝ ♥❤✃t ✈Ò ❝➠♥❣ ❧❛♦ ❞➵② ỗ tr sốt q trì ụ t ủ ♥❤➭ tr➢ê♥❣✳ ❚➠✐ ①✐♥ ❝➯♠ ➡♥ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❈➠♥❣ ♥❣❤Ư t❤➠♥❣ t✐♥ ✈➭ ❚r✉②Ị♥ t❤➠♥❣ ✲ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥✱ ♥➡✐ t➠✐ ➤❛♥❣ ❝➠♥❣ t➳❝✱ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ ❝❤♦ t➠✐ ❤♦➭♥ t❤➭♥❤ ❦❤ã❛ ❤ä❝ ♥➭②✳ ❚➠✐ ①✐♥ ❝➯♠ ➡♥ ❣✐❛ ➤×♥❤✱ ❜➵♥ ❜❒ ✈➭ ♥❣➢ê✐ t❤➞♥ ➤➲ q✉❛♥ t➞♠✱ t➵♦ ➤✐Ị✉ ❦✐Ư♥✱ ➤é♥❣ ✈✐➟♥✱ ❝ỉ ✈ị ➤Ĩ t➠✐ ❝ã t❤Ĩ ❤♦➭♥ t❤➭♥❤ ♥❤✐Ư♠ ✈ơ ❝đ❛ ♠×♥❤✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✾ ♥➝♠ ✷✵✶✸ ❍ä❝ ✈✐➟♥ ◆❣✉②Ơ♥ ❚❤ï② ❚r❛♥❣ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✐✐✐ ▼ơ❝ ❧ơ❝ ❚r❛♥❣ ▲ê✐ ❝❛♠ ➤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ▲ê✐ ❝➯♠ ➡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ▼ô❝ ❧ô❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐✐ ▼ë ➤➬✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❈❤➢➡♥❣ ✶✳ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳ P❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷✳ ❈❤✐Ò✉ ✈➭ ➤é ❝❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ❈❤➢➡♥❣ ✷✳ ❈➡ së ●r♦❡❜♥❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✶✳ ❚❤ø tù tõ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷✳ ❈➡ së ●r♦❡❜♥❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✸✳ ❚❤✉❐t t♦➳♥ ❇✉❝❤❜❡r❣❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ❈❤➢➡♥❣ ✸✳ P❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝ñ❛ ❝➳❝ ✐➤➟❛♥ tr♦♥❣ K[x, y] t❤❡♦ ❝➡ së ●r♦❡❜♥❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✶✳ ❈➡ së ●r♦❡❜♥❡r ❝ñ❛ ✈➭♥❤ K[x, y] ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✷✼ ✸✳✷✳ ❚Ý♥❤ t♦➳♥ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ❑Õt ❧✉❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✶ ▼ë ➤➬✉ ◆➝♠ ✶✾✻✹✱ ❍✐r♦♥❛❦❛ ➤➲ ❣✐í✐ t❤✐Ö✉ ❦❤➳✐ ♥✐Ö♠ ❝➡ së ❝❤✉➮♥ t➽❝ ❝❤♦ ✐➤➟❛♥ ❝➳❝ ỗ ũ từ ì tứ ột s ❇✉❝❤❜❡r❣❡r ➤➲ ➤Þ♥❤ ♥❣❤Ü❛ ➤é❝ ❧❐♣ ♠ét ❦❤➳✐ ♥✐Ư♠ t➢➡♥❣ tù ❝❤♦ ✐➤➟❛♥ ❝➳❝ ➤❛ t❤ø❝ ♠➭ ➠♥❣ ❣ä✐ ❧➭ ❝➡ së ●r♦❡❜♥❡r✱ t➟♥ ♥❣➢ê✐ t❤➬② ❤➢í♥❣ ❞➱♥ ❝đ❛ ❇✉❝❤❜❡r❣❡r✱ ữ ò r ột tt t tí ❝➡ së ●r♦❡❜♥❡r✱ ❧➭ t❤✉❐t t♦➳♥ ❇✉❝❤❜❡r❣❡r✳ ❈➡ së ●r♦❡❜♥❡r ♥❤❛♥❤ ❝❤ã♥❣ trë t❤➭♥❤ tr✉♥❣ t➞♠ ❝ñ❛ ➜➵✐ sè ♠➳② tÝ♥❤ ✭❈♦♠♣✉t❡r ❆❧❣❡❜r❛✮ ✈➭ ❧➭ ❝➠♥❣ ❝ơ ❤÷✉ ❤✐Ư✉ tr♦♥❣ r✃t ♥❤✐Ị✉ ❜➭✐ t♦➳♥ ❝đ❛ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ✈➭ ❍×♥❤ ❤ä❝ ➤➵✐ sè✳ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ✈Ị ❝➡ së ●r♦❡❜♥❡r ✈➭ ♠ét ➳♣ ❞ơ♥❣ ❝đ❛ ❝➡ së ●r♦❡❜♥❡r ➤Ĩ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ♠ét ✐➤➟❛♥ tr♦♥❣ ✈➭♥❤ ➤❛ t❤ø❝ K[x, y]✱ ✈í✐ K ❧➭ ♠ét tr➢ê♥❣✱ t❤❡♦ ❜➭✐ ❜➳♦ ✧■❞❡❛❧ ❜❛s❡s ❛♥❞ ♣r✐♠❛r② ❞❡❝♦♠♣♦s✐t✐♦♥✿ ❝❛s❡ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✧ ❝đ❛ ▲❛③❛r❞ ❬✸❪✳ ❈ị♥❣ ❝➬♥ ♣❤➯✐ ♥ã✐ t❤➟♠ r➺♥❣✱ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝ñ❛ ♠ét ✐➤➟❛♥ ❧➭ ♠ét ❜➭✐ t♦➳♥ q✉❛♥ trä♥❣ tr♦♥❣ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ✈➭ ❍×♥❤ ❤ä❝ ➜➵✐ sè✱ ➤➷❝ ❜✐Ưt ❧➭ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ ✐➤➟❛♥ tr♦♥❣ ✈➭♥❤ ➤❛ t❤ø❝ ✈í✐ ❤Ö sè tr➟♥ ♠ét tr➢ê♥❣✳ ▲✉❐♥ ✈➝♥ ❜❛♦ ❣å♠ ❜❛ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ♠ét tr×♥❤ ❜➭② ♠ét sè ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥❤➢ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ ✐➤➟❛♥ tr➟♥ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ❝❤✐Ị✉ ❝đ❛ ✈➭♥❤✱ ➤é ❝❛♦ ❝đ❛ ✐➤➟❛♥✳ ❈❤➢➡♥❣ ❤❛✐ tr×♥❤ ❜➭② ❝❤✐ t✐Õt ✈Ị ❝➡ së ●r♦❡❜♥❡r ✈➭ t❤✉❐t t♦➳♥ ❇✉❝❤❜❡r❣❡r ➤Ĩ t×♠ ❝➡ së rr t tt ữ ủ trì ❜➭② ♠ét t❤✉❐t t♦➳♥ ❝đ❛ ▲❛③❛r❞ ✈Ị ♠ét ➳♣ ❞ơ♥❣ ❝đ❛ ❝➡ së ●r♦❡❜♥❡r tr♦♥❣ ✈✐Ư❝ t×♠ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝ñ❛ ♠ét ✐➤➟❛♥ tr♦♥❣ ✈➭♥❤ ➤❛ t❤ø❝ ❤❛✐ ❜✐Õ♥ Số hóa trung tâm học liệu K[x, y] ✈í✐ K ❧➭ ♠ét tr➢ê♥❣✳ http://www.lrc-tnu.edu.vn/ ✷ ❈❤➢➡♥❣ ✶ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ✈➭ t♦➭♥ ❜é ❧✉❐♥ ✈➝♥✱ t❛ ❧✉➠♥ ❣✐➯ t❤✐Õt A ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ❝ã ➤➡♥ ✈Þ✳ ✶✳✶ P❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ✶✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ s➡ ❝ñ❛ ✭✐✮ ❈❤♦ Q ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ A✳ ❚❛ ♥ã✐ r➺♥❣ Q ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ A ♥Õ✉✿ Q ❧➭ ✐➤➟❛♥ t❤❐t sù ❝đ❛ A❀ ✭✐✐✮ ❱í✐ a, b ❜✃t ❦ú t❤✉é❝ A ♠➭ ab ∈ Q❀ a ∈ Q✱ tå♥ t➵✐ n ∈ N s❛♦ ❝❤♦ bn ∈ Q✳ ✶✳✶✳✷ ❱Ý ❞ô✳ ✭✐✐✮ ❚r♦♥❣ ✭✐✮ ▼ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ s➡✳ Z✱ ✐➤➟❛♥ 4Z ❧➭ ♥❣✉②➟♥ s➡ ♥❤➢♥❣ ❦❤➠♥❣ ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè✳ ❧➭ ✐➤➟❛♥ P ❧➭ ✐➤➟❛♥ ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ A❀ Q1 , Q2 , , Qn ✭n ≥ n ✲ ♥❣✉②➟♥ s➡ ❝đ❛ A✳ ❑❤✐ ➤ã✱ ∩ Qi ❝ị♥❣ ❧➭ P ✲ ♥❣✉②➟♥ s➡✳ i=1 1✮ ❧➭ ❝➳❝ ❈❤♦ ♥❣✉②➟♥ tè ❝ñ❛ A Q ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ s➡ ❝ñ❛ ✈➭ t❛ ♥ã✐ r➺♥❣ ♥❣✉②➟♥ tè ♥❤á ♥❤✃t ❝đ❛ ✶✳✶✳✹ ❇ỉ ➤Ị✳ ✐➤➟❛♥ P ❈❤♦ ✶✳✶✳✺ ▼Ư♥❤ ➤Ị✳ tè✐ ➤➵✐ ❝đ❛ Q ❧➭ P A✳ ❑❤✐ ➤ã✱ P = √ Q ✶✳✶✳✸ ❇æ ➤Ị✳ ✲ ♥❣✉②➟♥ s➡✳ ❍➡♥ ♥÷❛✱ A ❝❤ø❛ Q✳ P ❈❤♦ Q ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ A s❛♦ ❝❤♦ √ Q=m ❧➭ ♠ét ✐➤➟❛♥ A✳ ❑❤✐ ➤ã✱ Q ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ s➡ ❤❛② ✐➤➟❛♥ m ✲ ♥❣✉②➟♥ s➡ ❝ñ❛ A✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸ ✶✳✶✳✻ ➜Þ♥❤ ♥❣❤Ü❛✳ ❈❤♦ I ❧➭ ✐➤➟❛♥ t❤ù❝ sù ❝đ❛ A✳ ▼ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ I ❧➭ ♠ét ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ I ♥❤➢ ❧➭ ❣✐❛♦ ❝đ❛ ❤÷✉ ❤➵♥ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ s➡ ❝ñ❛ A✳ ▼ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ I = Q1 ∩ Q2 ∩ ∩ Qn , ✈í✐ √ Qi = Pi ❤❛② Qi ❧➭ Pi ✲ ♥❣✉②➟♥ s➡✱ i = 1, 2, , n ➤➢ỵ❝ ❣ä✐ ❧➭ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ tè✐ t✐Ĩ✉ ❝đ❛ ✭✐✮ I ♥Õ✉ t❤á❛ ♠➲♥ ❤❛✐ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ P1 , P2 , , Pn ❧➭ n ✐➤➟❛♥ ♥❣✉②➟♥ tè ➤➠✐ ♠ét ❦❤➳❝ ♥❤❛✉ ❝ñ❛ A✱ ✭✐✐✮ n ∀j = 1, , n t❛ ❝ã Qj ❚❛ ♥ã✐ ∩ Qi ✳ i=1 i=j ❧➭ ✐➤➟❛♥ ♣❤➞♥ tÝ❝❤ ➤➢ỵ❝ ❝đ❛ I A ♥Õ✉ ♥ã ❝ã ♠ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡✳ ✶✳✶✳✼ ➜Þ♥❤ ❧ý ợ ủ ị ý t tứ t I ❧➭ ♠ét ✐➤➟❛♥ ♣❤➞♥ tÝ❝❤ A✳ ●✐➯ sö I = Q1 ∩ Q2 ∩ ∩ Qn ✈í✐ √ Qi = Pi ❀ i = 1, 2, , n ✈➭ I = Q1 ∩ Q2 ∩ ∩ Qn ✈í✐ ➤ã✱ Qi = Pi ❀ i = 1, 2, , n n=n ✈➭ t❛ ❝ã✿ ✶✳✶✳✽ ➜Þ♥❤ ♥❣❤Ü❛✳ ∩ Qn , √ ❧➭ ❤❛✐ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ tè✐ t✐Ĩ✉ ❝đ❛ I ✳ ❑❤✐ {P1 , P2 , , Pn } = {P1 , P2 , , Pn }✳ ●✐➯ sư I ❧➭ ✐➤➟❛♥ ♣❤➞♥ tÝ❝❤ ➤➢ỵ❝ ❝đ❛ A, ✈➭ I = Q1 ∩ Q2 ∩ Qi = Pi ✱ i = 1, , n ❧➭ ♠ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ tè✐ t✐Ĩ✉ ❝đ❛ I ❑❤✐ ➤ã✱ t❐♣ {P1 , , Pn } ➤é❝ ❧❐♣ ✈í✐ ❝➳❝❤ ❝❤ä♥ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ tè✐ t✐Ĩ✉ ❝đ❛ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝đ❛ ✶✳✶✳✾ ➜Þ♥❤ ♥❣❤Ü❛✳ t✐Ĩ✉ ❝đ❛ ●✐➯ sư I ✈➭ ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ I ✱ ❦ý ❤✐Ư✉ Ass(I)✳ I ❧➭ ✐➤➟❛♥ ♣❤➞♥ tÝ❝❤ ➤➢ỵ❝ ❝đ❛ A ❈➳❝ ♣❤➬♥ tư tè✐ Ass(I) ➤➢ỵ❝ ❣ä✐ ❧➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè tè✐ t✐Ĩ✉ ❝đ❛ I ❤❛② ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝➠ ❧❐♣✱ ♥❣✉②➟♥ tè ♥❤ó♥❣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ò ợ ọ ủ I Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✹ ✭➜Þ♥❤ ý t tứ ị ý ợ ❝ñ❛ I ❧➭ ♠ét ✐➤➟❛♥ ♣❤➞♥ tÝ❝❤ A✱ Ass(I) = {P1 , , Pn }✳ ●✐➯ sö I = Q1 ∩ Q2 ∩ ∩ Qn √ ✈í✐ Qi = Pi ❀ i = 1, 2, , n ✈➭ I = Q1 ∩ Q2 ∩ ∩ Qn Qi = Pi ❀ i = 1, 2, , n ✈í✐ ➤ã✱ ỗ i Pi tí s➡ tè✐ t✐Ĩ✉ ❝đ❛ ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè tè✐ t✐Ĩ✉ ❝đ❛ I ♣❤➬♥ ♥❣✉②➟♥ s➡ ø♥❣ ✈í✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝➠ ❧❐♣ ❝đ❛ ❜ë✐ t❤× I Qi = Qi I ✳ ❑❤✐ ❤❛② t❤➭♥❤ ❧➭ ①➳❝ ➤Þ♥❤ ❞✉② ♥❤✃t I ✱ ❦❤➠♥❣ ♣❤ô t❤✉é❝ ✈➭♦ ❝➳❝❤ ❝❤ä♥ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ tè✐ t✐Ĩ✉✳ ✶✳✶✳✶✶ ➜Þ♥❤ ❧ý✳ ▼ä✐ ✐➤➟❛♥ t❤ù❝ sù tr♦♥❣ ✈➭♥❤ ◆♦❡t❤❡r ➤Ò✉ ❝ã ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡✱ ❞♦ ➤ã ❝ã ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ tè✐ t✐Ĩ✉✳ ✶✳✶✳✶✷ ➜Þ♥❤ ❧ý ✈➭♥❤ ➤❛ t❤ø❝ ✶✳✶✳✶✸ ❱Ý ❞ơ✳ ❝➳❝ ❜✐Õ♥ ✭➜Þ♥❤ ❧ý ❝➡ së ❍✐❧❜❡rt✮✳ ●✐➯ sö A ❧➭ ✈➭♥❤ ◆♦❡t❤❡r✳ ❑❤✐ ➤ã✱ A[x1 , , xn ] ❝ò♥❣ ❧➭ ✈➭♥❤ ◆♦❡t❤❡r✳ ●✐➯ sö K ❧➭ ♠ét tr➢ê♥❣ ✈➭ A = K[x, y] ❧➭ ✈➭♥❤ ➤❛ t❤ø❝ ❝ñ❛ x, y ❚❛ ❝ã✿ xy, y = x, y ∩ y = x, y ∩ y ❧➭ ❤❛✐ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ tè✐ t✐Ĩ✉ ❝đ❛ I = xy, y ✳ ▲➵✐ ❝ã✱ x, y = x, y = x, y ♥➟♥ Ass(I) = { y , x, y }✱ tr♦♥❣ ➤ã y ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝➠ ❧❐♣ ✭tè✐ t✐Ĩ✉✮ ❝đ❛ I ✈➭ x, y ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ♥❤ó♥❣ ❝đ❛ I ▼Ư♥❤ ➤Ị s❛✉ ➤➢❛ r❛ ố ệ ữ s ị ❤ã❛✳ ❈❤ó ý✱ ✈í✐ S = A \ P ❧➭ t❐♣ ➤ã♥❣ ♥❤➞♥ ❝ñ❛ A✱ I ❧➭ ✐➤➟❛♥ ❝ñ❛ A✱ t❛ ❦ý ❤✐Ö✉ IP = S −1 I ✳ ✶✳✶✳✶✹ ▼Ư♥❤ ➤Ị✳ ●✐➯ sư Q ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ s➡ ❝đ❛ A ✈í✐ √ Q = P, P ∈ Spec(A)✳ ❑❤✐ ➤ã✿ ✭✐✮ ◆Õ✉ ✭✐✐✮ ◆Õ✉ P P P ⊆P t❤× t❤× PP = QP = AP ✳ PP ∩ A = P Số hóa trung tâm học liệu ✈➭ QP = Q http://www.lrc-tnu.edu.vn/ ✺ ✶✳✷ ❈❤✐Ò✉ ✈➭ ➤é ❝❛♦ ✶✳✷✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❈❤✐Ị✉ ❑r✉❧❧✮ ▼ét ❞➲② ❣✐➯♠ t❤ù❝ sù ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè P0 ⊃ P1 ⊃ P2 ⊃ ⊃ Pn ❝đ❛ ✈➭♥❤ A ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ①Ý❝❤ ♥❣✉②➟♥ tè ❝ã ➤é ❞➭✐ ❧➭ n✳ ❈❐♥ tr➟♥ ❝ñ❛ ➤é ❞➭✐ t✃t ❝➯ ❝➳❝ ①Ý❝❤ ♥❣✉②➟♥ tè tr♦♥❣ A ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ A✱ ❤❛② ❝❤✐Ị✉ ❝đ❛ ✈➭♥❤ A✳ ❑Ý ❤✐Ư✉ ❧➭ dim A✳ ✶✳✷✳✷ ➜Þ♥❤ ♥❣❤Ü❛✳ ●✐➯ sư P ❧➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝đ❛ A✳ ❈❤✐Ị✉ ❝đ❛ ✐➤➟❛♥ P ❧➭ ❝❤✐Ị✉ ❝đ❛ ✈➭♥❤ A/P ✱ ❦ý ❤✐Ư✉ dim P ●✐➯ sư I ❧➭ ♠ét ✐➤➟❛♥ ❜✃t ❦ú ❝đ❛ A t❤× dim I = sup {dimP | P ∈ V (I)} , tr♦♥❣ ➤ã V (I) ❧➭ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ A ❝❤ø❛ I ✳ ✶✳✷✳✸ ➜Þ♥❤ ♥❣❤Ü❛✳ ●✐➯ sư P ❧➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝đ❛ A✳ ❈❤✐Ị✉ ❞➭✐ ❧í♥ ♥❤✃t ❝đ❛ ♠ä✐ ❞➲② ❣✐➯♠ t❤ù❝ sù ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè P ①✉✃t ♣❤➳t tõ ✐➤➟❛♥ ❝ñ❛ = P0 ⊃ P1 ⊃ P2 ⊃ ⊃ Pr P ✱ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤é ❝❛♦ ❝đ❛ P ✱ ❦Ý ❤✐Ư✉ ❧➭ ht P ●✐➯ sư I ❧➭ ♠ét A✳ ➜é ❝❛♦ ❝đ❛ ✐➤➟❛♥ I ✱ ❦Ý ❤✐Ư✉ ht I ✱ ➤➢ỵ❝ ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ ht I = inf {ht P | P ∈ V (I)}✳ ▼Ư♥❤ ➤Ị s❛✉ ♥❤➽❝ ❧➵✐ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤✐Ị✉ ✈➭ ➤é ❝❛♦✳ ✶✳✷✳✹ ▼Ư♥❤ ➤Ị✳ ✈➭ ♥Õ✉ ✭✐✮ ●✐➯ sö K ❧➭ ♠ét tr➢ê♥❣✳ ❑❤✐ ➤ã✱ dim K[x1 , , xn ] = n m ❧➭ ✐➤➟❛♥ tè✐ ➤➵✐ ❝ñ❛ K[x1 , , xn ] t❤× ht m = n ✭✐✐✮ ◆Õ✉ (A, m) ❧➭ ột ị tì dim A = ht m ✭✐✐✐✮ ❈❤♦ P ❧➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ ✈➭♥❤ A✱ ❦❤✐ ➤ã dim AP = ht P AP = ht P ✳ ✭✐✈✮ ❚r♦♥❣ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t D✱ ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ã ➤é ❝❛♦ ❧➭ ✐➤➟❛♥ ❝❤Ý♥❤✳ ✶✳✷✳✺ ➜Þ♥❤ ❧ý I ✭➜Þ♥❤ ❧ý ✐➤➟❛♥ ❝❤Ý♥❤ ❝đ❛ ❑r✉❧❧✮✳ ●✐➯ sư A ❧➭ ♠ét ✈➭♥❤ ◆♦❡t❤❡r✱ ❧➭ ✐➤➟❛♥ t❤ù❝ sù ❝đ❛ ✶✳✷✳✻ ➜Þ♥❤ ♥❣❤Ü❛✳ ✐➤➟❛♥ ♥❣✉②➟♥ tè A s✐♥❤ ❜ë✐ n ♣❤➬♥ tö✳ ❑❤✐ ➤ã ht(I) ≤ n ❈❤♦ Q ⊂ P ❧➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ A✳ ▼ét ❞➲② ❝➳❝ Q = P0 ⊂ P1 ⊂ Pn = P s❛♦ ❝❤♦ Pi = Pi+1 , ∀i✱ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❞➲② ♥❣✉②➟♥ tè ❜➲♦ ❤♦➭ ❣✐÷❛ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝❤❒♥ ❣✐÷❛ Số hóa trung tâm học liệu Q ✈➭ P ♥Õ✉ ✈í✐ ♠ä✐ i✱ ❦❤➠♥❣ tå♥ t➵✐ Pi ✈➭ Pi+1 http://www.lrc-tnu.edu.vn/ ✻ ❚❛ ♥ã✐ r➺♥❣ ✈➭♥❤ A ❧➭ ❝❛t❡♥❛r② ♥Õ✉ ✈í✐ ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè Q ⊂ P ❝đ❛ A ❧✉➠♥ tå♥ t➵✐ ♠ét ❞➲② ♥❣✉②➟♥ tè ❜➲♦ ❤♦➭ ❣✐÷❛ Q ✈➭ P ✈➭ ♠ä✐ ❞➲② ♥❣✉②➟♥ tè ❜➲♦ ❤♦➭ ❣✐÷❛ Q ✈➭ P ➤Ị✉ ❝ã ❝❤✉♥❣ ➤é ❞➭✐✳ ◆Õ✉ A ❧➭ ♠ét ✈➭♥❤ ❝❛t❡♥❛r② ✈➭ P ∈ Spec(A) t❤× t❛ ❝ã dim A = dim A/P + ht P ✳ ❱Ý ❞ô✱ ✈➭♥❤ K[x1 , , xn ] ❧➭ ♠ét ✈➭♥❤ ❝❛t❡♥❛r②✳ ➜➷❝ ❜✐Öt✱ K[x, y] ❧➭ ✈➭♥❤ ❝❛t❡♥❛r② ♥➟♥ ♥Õ✉ P ∈ Spec(K[x, y]) ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝❤✐Ị✉ t❤× ht P = 1✳ ❉♦ ➤ã✱ t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✷✳✹(iv) t❤× P ❧➭ ✐➤➟❛♥ ❝❤Ý♥❤✳ ❚õ ➤ã✱ t❛ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ ❦Õt q✉➯ s❛✉✳ ✶✳✷✳✼ ▼Ư♥❤ ➤Ị✳ K[x, y]✱ K ●✐➯ sư = I = f0 , , fk = K[x, y]✱ ❧➭ ♠ét tr➢ê♥❣✳ ❑❤✐ ➤ã✱ ✈í✐ f0 , , fk ∈ U CLN {fi } = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ dim I = ❈❤ø♥❣ ♠✐♥❤✳ ✭⇒✮ ❱× I = ♥➟♥ dim I = ❤♦➷❝ dim I = ◆Õ✉ dim I = t❤× tå♥ t➵✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè P s❛♦ ❝❤♦ I ⊆ P ✈➭ dim P = 1✳ ❱× K[x, y] ❧➭ ♠✐Ị♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t ♥➟♥ ❉♦ ➤ã q|fi ✈í✐ ♠ä✐ i ❱× ✈❐② P ❧➭ ✐➤➟❛♥ ❝❤Ý♥❤✳ ❉♦ ➤ã P = q ✱ q ∈ K[x, y]✳ = 0, , k ✳ ➜✐Ò✉ ♥➭② ✈➠ ❧ý ✈í✐ ❣✐➯ t❤✐Õt U CLN {fi } = 1✳ dim I = ✭⇐✮ ●✐➯ sö U CLN {fi } = 1✳ ❑❤✐ ➤ã tå♥ t➵✐ ➤❛ t❤ø❝ ❜✃t ❦❤➯ q✉② d ∈ K[x, y] t❤á❛ ♠➲♥ d|fi ✈í✐ ♠ä✐ i = 0, , k ➜✐Ị✉ ♥➭② ❦Ð♦ t❤❡♦ I ⊆ d ✳ ❱× d ❜✃t ❦❤➯ q✉② ♥➟♥ d ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè✳ ▼➷t ❦❤➳❝ dim I = ♥➟♥ d ❧➭ ✐➤➟❛♥ tè✐ ➤➵✐✳ ❉♦ ➤ã ➤ã ht d = 2✳ ❱➠ ❧ý ✈í✐ ➜Þ♥❤ ❧ý ✐➤➟❛♥ ❝❤Ý♥❤ ❝ñ❛ ❑r✉❧❧✳ ❉♦ U CLN {fi } = 1✳ ●✐➯ sư I ❧➭ ✐➤➟❛♥ ❝❤✐Ị✉ ❝đ❛ A ◆❤❐♥ ①Ðt r➺♥❣✱ ❝❤Ø ❝ã ✐➤➟❛♥ tè✐ ➤➵✐ ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝❤ø❛ I ✳ ❉♦ ➤ã✱ ♥Õ✉ A ❧➭ ✈➭♥❤ ◆♦❡t❤❡r t❤× I ❝❤Ø ❝ã ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❧➭ ❝➳❝ ✐➤➟❛♥ tè✐ ➤➵✐ ✈➭ ❦❤➠♥❣ ❝ã ✐➤➟❛♥ ♥❣✉②➟♥ tè ♥❤ó♥❣✳ ❍❛② ♥ã✐ ❝➳❝❤ ❦❤➳❝✱ I ❜✐Ĩ✉ ❞✐Ơ♥ ❞✉② ♥❤✃t t❤➭♥❤ ❣✐❛♦ ❝ñ❛ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ s➡✳ ➜➷❝ ❜✐Ưt✱ ♠ét ✐➤➟❛♥ ❝đ❛ ✈➭♥❤ K[x, y] ❝ã ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❞✉② ♥❤✃t ♥Õ✉ ❝➳❝ ➤❛ t❤ø❝ tr♦♥❣ t❐♣ s✐♥❤ ❝đ❛ ♥ã ♥❣✉②➟♥ tè ❝ï♥❣ ♥❤❛✉✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✼ ❈❤➢➡♥❣ ✷ ❈➡ së ●r♦❡❜♥❡r ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ t❛ ♥❣❤✐➟♥ ❝ø✉ ❝➡ së ●r♦❡❜♥❡r ❣✐í✐ ❤➵♥ tr♦♥❣ ✈➭♥❤ ➤❛ t❤ø❝ A = K[x1 , x2 , , xn ] = K[x]✱ tr♦♥❣ ➤ã K ❧➭ ♠ét tr➢ê♥❣✳ ❈➳❝ ✐➤➟❛♥ tr♦♥❣ A ❝ã t❤Ĩ ➤➢ỵ❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❜ë✐ ♠ét t❐♣ s✐♥❤ ❣å♠ ❤÷✉ ❤➵♥ ❝➳❝ ➤❛ t❤ø❝✱ ❣ä✐ ❧➭ ❝➡ së✳ ❚❛ ❝ã t❤Ó ❝❤Ø r❛ r➺♥❣✱ ✈í✐ ❜✃t ❦ú ♠ét t❐♣ s✐♥❤ ❤÷✉ ❤➵♥ ♥➭♦ ❝đ❛ ♠ét ✐➤➟❛♥ I tr♦♥❣ A ➤Ị✉ ❝ã t❤Ĩ ❞ï♥❣ t❤✉❐t t♦➳♥ ➤Ĩ ❜✐Õ♥ ➤ỉ✐ t❤➭♥❤ ♠ét ❝➡ së ●r♦❡❜♥❡r ❝đ❛ I ✳ ✷✳✶ ❚❤ø tù tõ ✷✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ t❤ø❝ ❝đ❛ ●✐➯ sư ≤ ❧➭ ♠ét t❤ø tù t♦➭♥ ♣❤➬♥ tr➟♥ t❐♣ T t✃t ❝➯ ❝➳❝ ➤➡♥ K[x]✳ ❚❤ø tù ≤ ➤➢ỵ❝ ❣ä✐ ❧➭ t❤ø tù tõ ♥Õ✉ ♥ã t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿ ✭✐✮ ❱í✐ ♠ä✐ ✭✐✐✮ ◆Õ✉ m ∈ T, ≤ m, m1 , m2 , m ∈ T ♠➭ m1 ≤ m2 t❤× mm1 ≤ mm2 ✷✳✶✳✷ ❇ỉ ➤Ị✳ ▼ét t❤ø tù t♦➭♥ ♣❤➬♥ ≤ tr➟♥ T ❧➭ t❤ø tù tèt ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ♠ä✐ ❞➲② ➤➡♥ t❤ø❝ t❤ù❝ sù ❣✐➯♠✿ m1 > m2 > m3 > ➤Ò✉ ❞õ♥❣ ✭s❛✉ ❤÷✉ ❤➵♥ ♣❤➬♥ tư✮✳ ❈❤ø♥❣ ♠✐♥❤✳ ◆Õ✉ ≤ ❦❤➠♥❣ ❧➭ t❤ø tù tèt t❤× tå♥ t➵✐ t❐♣ ❝♦♥ B ⊆ T s❛♦ ❝❤♦ B ❦❤➠♥❣ ❝ã ♣❤➬♥ tö ♥❤á ♥❤✃t✳ ▲✃② Số hóa trung tâm học liệu m1 ❧➭ ♣❤➬♥ tư tï② ý t❤✉é❝ B ✳ ❱× B ❦❤➠♥❣ ❝ã http://www.lrc-tnu.edu.vn/ ✸✵ ✸✳✶✳✻ ❇ỉ ➤Ị✳ ◆Õ✉ Gi+1 = LC(fi ) LC(fi+1 ) t❤× Gi+1 fi+1 ∈ fi , fi1 , , f0 ỗ i = 0, , k − 1✳ ❈è ➤Þ♥❤ i ❈❤ø♥❣ ♠✐♥❤✳ ∈ {0, , k}✳ ❚õ ❇ỉ ➤Ị ✸✳✶✳✺ t❛ ➤➢ỵ❝ Gi+1 = LC(fi ) LC(fi+1 ) ∈ K[x]✳ ❑❤✐ ➤ã✱ h = Gi+1 fi+1 − y d(fi+1 )−d(fi ) fi ❧➭ ♠ét ♣❤➬♥ tư ❝đ❛ I ✳ ❱× ❦Ð♦ t❤❡♦ F ❧➭ ♠ét ❝➡ së ●r♦❡❜♥❡r ❝đ❛ I ♥➟♥ ➜Þ♥❤ ❧ý ✷✳✷✳✶✶ F h → 0✳ ❱× ✈❐②✱ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✷✳✽✱ ❈❤➢➡♥❣ ✷✱ t❛ ❝ã✿ t 0=h− j=0 cj Tj fj lc (fj ) tr ó ỗ j cj ∈ K ✱ fj ∈ F ✱ Tj ∈ K[x, y] ✈➭ ♠❛①{lt(Tj fj )} = lt(h)✳ ❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝ñ❛ h✱ lt(h) < y d(fi+1 ) ≤ lt(fi+1 ) ❱× ✈❐②✱ ✈í✐ ♠ä✐ j > i✱ cj = ì F ợ s ế t ề t ủ ❝➳❝ ➤➡♥ t❤ø❝ ❦❤ë✐ ➤➬✉✳ ❚❤❡♦ ➤ã✱ t❛ ➤➢ỵ❝✱ i Gi+1 fi+1 = y d(fi+1 )−d(fi ) fi + j=0 ❑❤✐ ➤ã s✉② r❛✿ ✸✳✶✳✼ ❇ỉ ➤Ị✳ ❈❤ø♥❣ ♠✐♥❤✳ cj Tj fj lc (fj ) Gi+1 fi+1 ∈ fi , fi−1 , , f0 ✳ ❱í✐ ♠ä✐ ❈❤♦ i = 0, , k, t❛ ❝ã primpart(f0 ) ❝❤✐❛ ❤Õt fi ✳ P = primpart(f0 )✳ ❍✐Ó♥ ♥❤✐➟♥ P ❝❤✐❛ ❤Õt f0 ✳ ●✐➯ sö P ❝❤✐❛ ❤Õt fj ✱ ✈í✐ ♠ä✐ j ≤ n < k ✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ j ≤ n✱ fj = hj P ✱ hj ∈ K[x, y]✳ ❚❤❡♦ ❇ỉ ➤Ị ✸✳✶✳✻✱ tå♥ t➵✐ kj , , k0 ∈ K[x, y] s❛♦ ❝❤♦ Gj+1 fj+1 = kj fj + kj−1 fj−1 + + k0 f0 ❑Õt ❤ỵ♣ ✈í✐ ❣✐➯ t❤✐Õt q✉② ♥➵♣✱ t❛ ➤➢ỵ❝✿ Gj+1 fj+1 = P (kj hj + kj−1 hj−1 + + k0 h0 ) ❉♦ ➤ã✱ P ❝❤✐❛ ❤Õt Gj+1 fj+1 ✳ ❱× P ❧➭ ♥❣✉②➟♥ s➡ tr♦♥❣ K[x, y] ✈➭ Gj+1 ♥➟♥ ∈ K[x] P ❝❤✐❛ ❤Õt fj+1 ✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸✶ ✸✳✶✳✽ ❇ỉ ➤Ị✳ U CLN {fi } = t❤× ✈í✐ ♠ä✐ i = 1, , k ✱ LC(fi ) ❝❤✐❛ ❤Õt ◆Õ✉ fi ✳ ❈❤ø♥❣ ♠✐♥❤✳ ◆Õ✉ U CLN {fi } = t❤× tõ ❇ỉ ➤Ị t ợ t ị ĩ LC(f0 ) P = primpart(f0 ) = 1✳ ❑❤✐ ➤ã✱ f0 ∈ K[x] ✈➭ = f0 ✳ ●✐➯ sö LC(fj ) ❝❤✐❛ ❤Õt fj ✈í✐ ♠ä✐ j ≤ n < k ✳ ❑❤✐ ➤ã✱ ❇ỉ ➤Ị ✸✳✶✳✺ ❝❤Ø r❛ r➺♥❣ t➵✐ LC(fj ) ❝❤✐❛ ❤Õt fi ✈í✐ ♠ä✐ i ≤ j ✳ ❱× ✈❐② tå♥ hi ∈ K[x, y] s❛♦ ❝❤♦ fi = hi LC(fj )✳ ❑❤✐ ➤ã ❇ỉ ➤Ị ✸✳✶✳✻ ❝❤Ø r❛✿ Gj+1 fj+1 = kj fj +kj−1 fj−1 + +k0 f0 = LC(fj ).(kj hj +kj−1 hj−1 + +k0 h0 ) ✈í✐ k0 , k1 , , kj ∈ K[x, y]✳ ❉♦ ➤ã✱ LC (fj ) fj+1 = LC (fj ) l LC (fj+1 ) tr♦♥❣ ➤ã l ∈ K[x, y]✳ ❑❤✐ ➤ã LC(fj+1 ) ❝❤✐❛ ❤Õt fj+1 ✈í✐ ♠ä✐ j < k ✳ ✸✳✶✳✾ ❇ỉ ➤Ị✳ ❈❤ø♥❣ ♠✐♥❤✳ ◆Õ✉ U CLN {fi } = t❤× LC(fk ) = 1✳ ❱× ỗ j = 0, , k LC(fj+1 ) ❝❤✐❛ ❤Õt LC(fj ) ✈➭ LC(fj ) ❝❤✐❛ ❤Õt fj ♥➟♥ ❦❤✐ ➤ã LC(fk ) ❝❤✐❛ ❤Õt fj ✈í✐ ♠ä✐ j ≤ k ✳ ❑❤✐ ➤ã U CLN {fi } = ❦Ð♦ t❤❡♦ LC(fk ) = 1✳ ❙✉② r❛ ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❈❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✸✳✶✳✸ trờ ợ sử ị ý ú U CLN {fi } = 1✳ ❳Ðt U CLN {fi } = 1✳ ❑❤✐ ➤ã tå♥ t➵✐ D ∈ K[x, y] s❛♦ ỗ i = 1, 2, , k, t ❝ã✿ fi = DFi ✈➭ U CLN (F0 , F1 , , Fk ) = 1✳ ❚❤❡♦ ❣✐➯ t❤✐Õt {F0 , F1 , , Fk } ❧➭ ❝➡ së ●r♦❡❜♥❡r tè✐ t✐Ĩ✉ ❝đ❛ J ✈➭ ➤Þ♥❤ ❧ý ➤ó♥❣ ✈í✐ = F0 , F1 , , Fk {F0 , F1 , , Fk }✳ ❚❛ ❝ã F0 = P H0 G1 Gk+1 ✈í✐ P = primpart(F0 ) s✉② r❛ H0 |content(F0 ) ∈ K[x] ✈➭ ✈× H0 ❧➭ ♠♦♥✐❝ ✐♥ y ♥➟♥ H0 = 1✳ ❍➡♥ ♥÷❛✱ P = ✈➭ Gk+1 = ➤➢ỵ❝ s✉② r❛ tõ U CLN {Fi } = ✈➭ ❤➡♥ ♥÷❛ ✈í✐ ♠ä✐ i = 0, 1, , k ✱ t❛ ❝ã✿ Fi = P Hi Gi+1 Gk+1 , Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸✷ tr♦♥❣ ➤ã ♠ä✐ Gi ✈➭ Hi ➤➢ỵ❝ ➤Þ♥❤ ♥❣❤Ü❛ ♥❤➢ tr♦♥❣ ➜Þ♥❤ ❧ý ✸✳✶✳✸✳ ▲➢✉ ý r➺♥❣ ✈í✐ i = 1, , k ✱ t❛ ❝ã✿ fi = D.Hi Gi+1 Gk ➜➷❝ ❜✐Öt✱ f0 ➤ã = D.G1 Gk tr♦♥❣ ➤ã G1 , , Gk ∈ K[x]✱ s✉② r❛ D = pd tr♦♥❣ p = primpart(f0 ) ✈➭ d ∈ K[x]✳ ❱× fk = D.Hk = pdHk ✱ tr♦♥❣ ➤ã Hk ❧➭ ♠♦♥✐❝ ✐♥ y ✱ d = content(fk ) ♥➟♥ d = Gk+1 ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ i = 0, 1, , k, fi = pHi Gi+1 Gk+1 tr ó ỗ Hi y ✱ ✈➭ ✈í✐ ♠ä✐ i = 0, 1, , k − 1, LC (fi ) LC (D) LC (Fi ) = = Gi+1 , LC (fi+1 ) LC (D) LC (Fi+1 ) ❉♦ ➤ã ➤Þ♥❤ ❧ý ➤ó♥❣ ✈í✐ {fi }✳ ❇➞② ❣✐ê✱ t❛ ❝❤ø♥❣ ♠✐♥❤ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ U CLN {fi } = 1✱ P = primpart(f0 )✱ Gk+1 = content(fk ) ỗ i = 0, 1, , k ✱ Hi = fi LC(fi ) ✳ ❑❤✐ ➤ã✱ P = = Gk+1 ➤➢ỵ❝ s✉② r❛ tõ ❇ỉ ➤Ị ✸✳✶✳✼ ✈➭ ✸✳✶✳✾✳ ❍➡♥ ♥÷❛✱ t❤❡♦ ❇ỉ ➤Ị ✸✳✶✳✽✱ ❧➭ ♠♦♥✐❝ ✐♥ y ✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ fi = Hi LC(fi ) = Hi Hi ∈ K[x, y] i ≤ k✱ LC (fi ) LC (fi+1 ) LC (fk−2 ) LC (fk−1 ) LC (fi+1 ) LC (fi+2 ) LC (fk−1 ) LC (fk ) ❇➺♥❣ ❝➳❝❤ ❣✐➯♥ ➢í❝ ❝➳❝ ♥❤➞♥ tư ❝❤✉♥❣ ✈➭ ❜✐Õ♥ ➤ỉ✐ t❛ ➤➢ỵ❝ LC(fk ) = ✈➭ t❤❡♦ ❇ỉ ➤Ị ✸✳✶✳✾✱ t❛ ➤➢ỵ❝✿ fi = P Hi Gi+1 Gk+1 tr ó ỗ i = 0, , k − t❤× Gi+1 = LC(fi ) LC(fi+1 ) ∈ K[x] t❤❡♦ ❇ỉ ➤Ị ✸✳✶✳✺✳ ❱Ý ❞ơ s❛✉ sÏ ❝❤Ø r❛ ❝✃✉ tró❝ ❝đ❛ ♠ét ❝➡ së ●r♦❡❜♥❡r tè✐ tể ợ t tr ị ý tr♦♥❣ ➤ã t❛ ❣✐➯ sư ✸✳✶✳✶✵ ❱Ý ❞ơ✳ ❈❤♦ K = R✱ t❐♣ ❝➳❝ sè t❤ù❝✳ F = {f0 , f1 , f2 , f3 }✱ tr♦♥❣ ➤ã f0 = x7 , f1 = y x5 − yx6 , f2 = y x − y x3 , f3 = y − y Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸✸ ❍✐Ĩ♥ ♥❤✐➟♥✱ = G4 = content(f3 ) = P = primpart(f0 ) ✈➭ Gi+1 = LC(fi ) LC(fi+1 ) s✉② r❛ G1 = x2 , G2 = x4 , G3 = x ❉♦ ➤ã✱ H0 = 1, H1 = y − yx, H2 = y − y x2 , H3 = y − y ✸✳✷ ❚Ý♥❤ t♦➳♥ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ✸✳✷✳✶ ❇ỉ ➤Ị✳ ❈❤♦ I = f0 , , fk ♠ét ❝➡ së ●r♦❡❜♥❡r tè✐ t✐Ĩ✉ ❝đ❛ ❝➳❝ ➤➡♥ t❤ø❝ ❦❤ë✐ ➤➬✉ ✈➭ I ✳ ❈❤♦ F ✸✳✶✳✸✱ fi ❈è ➤Þ♥❤ F = {f0 , , fk } ⊆ K[x, y] ❧➭ ➤➢ỵ❝ s➽♣ ①Õ♣ t❤❡♦ ❝❤✐Ị✉ t➝♥❣ ❝đ❛ U CLN {fi } = tr ị ý tì ọ ứ tr ó ế Gj Hj ợ ị ĩ j = 1, , k ✱ t❛ ❝ã I ⊆ Hj , Gj ✳ j ∈ {1, , k} ✈➭ ❝❤ä♥ i ∈ {0, , k}✳ ❚❤❡♦ ➜Þ♥❤ ❧ý = P Hi Gi+1 Gk+1 ✳ ❱× ✈❐②✱ ♥Õ✉ i ≤ j t❤× fi ∈ Hj , Gj ✳ ◆Õ✉ i > j ✱ t❛ ❝❤ø♥❣ ♠✐♥❤ Hi ∈ Hj , Gj ✳ ◗✉② ♥➵♣ t❤❡♦ n✱ tr♦♥❣ ➤ã i = j + n✳ ❳Ðt n = 1✳ ❱× Hj+1 = fj+1 LC(fj+1 ) ✈➭ Gj+1 = LC(fj ) LC(fj+1 ) ♥➟♥✱ Gj+1 fj+1 = LC(fj ).Hj+1 ❚❤❡♦ ❦Õt q✉➯ ❝đ❛ ❇ỉ ➤Ị ✸✳✶✳✻✱ LC(fj ).Hj+1 = kj fj + + k0 f0 ✈í✐ k0 , k1 , , kj ∈ K[x, y]✳ ❈❤✐❛ ❝➯ ❤❛✐ ✈Õ ❝ñ❛ ➤➻♥❣ t❤ø❝ tr➟♥ ❝❤♦ LC(fj ) ✈➭ t❤❡♦ ❝➳❝❤ ①➞② ❞ù♥❣ ❝ñ❛ F ✱ t❛ ❝ã✿ Hj+1 = kj Hj +kj−1 Hj−1 Gj +kj−2 Hj−2 Gj−1 Gj + +k1 H1 G2 Gj +k0 H0 G1 Gj ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ m ≤ j✱ Hj+1 ∈ Hj , Hj−1 , , Hm+1 , Hm , Gm ❙✉② r❛✱ Hj+1 ∈ Hj , Gj ✳ ●✐➯ sư ✈í✐ l < n✱ Hj+l ∈ Hj , Gj ✳ ❳Ðt Hj+n ✳ ❈ã✱ Hj+n ∈ Hj+n−1 , Hj+n−2 , , Hj+1 , Hj , Gj , Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸✹ ✈➭ t❤❡♦ ❣✐➯ t❤✐Õt q✉② ♥➵♣✱ Hj+n−1 , Hj+n−2 , , Hj+1 ∈ Hj , Gj ❉♦ ➤ã✱ Hj+n ∈ Hj , Gj ✈í✐ ♠ä✐ n ∈ N✳ ❙✉② r❛✱ Hi ∈ Hj , Gj ✈í✐ ♠ä✐ i ≥ j ✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ i ✈➭ j ✱ fi ∈ Hj , Gj ✳ ➜Þ♥❤ ❧ý s❛✉ ❝❤♦ t❤✃② sù ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ ❝➳❝ ✐➤➟❛♥ tè✐ ➤➵✐ ❝❤ø❛ I = f0 , , fk ✳ ❈➳❝ ✐➤➟❛♥ tè✐ ➤➵✐ ♥➭② sÏ ❧➭ ❝➡ së ❝❤♦ q✉➳ tr×♥❤ tÝ♥❤ t♦➳♥ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ❝đ❛ I ✸✳✷✳✷ ➜Þ♥❤ ❧ý✳ ❈❤♦ ➜Þ♥❤ ❧ý ✸✳✶✳✸ ✈➭ F = {f0 , , fk } ⊆ K[x, y] U CLN {fi } = 1✳ ❑❤✐ ➤ã✱ ✭✐✮ ▼ét ✐➤➟❛♥ tè✐ ➤➵✐ ❝❤ø❛ t✃t ❝➯ ❝➳❝ ♥❤✃t ♠ét ❝➷♣ Gi tr♦♥❣ fi ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✐➤➟❛♥ ➤ã ❝❤ø❛ Ýt Gi ✱ Hi ✳ ✭✐✐✮ ▼ét ✐➤➟❛♥ tè✐ ➤➵✐ ❝❤ø❛ ❝ñ❛ ➤➢ỵ❝ ①➞② ❞ù♥❣ ♥❤➢ tr♦♥❣ F ➤➢ỵ❝ s✐♥❤ ❜ë✐ ♠ét ♥❤➞♥ tö ❜✃t ❦❤➯ q✉② K[x] ✈➭ ♠ét ♥❤➞♥ tö ❜✃t ❦❤➯ q✉② v(x, y) ❝ñ❛ Hi ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮ ●ä✐ ♠♦➤✉♥ u(x) u(x) m ❧➭ ♠ét ✐➤➟❛♥ tè✐ ➤➵✐ tr♦♥❣ K[x, y] s❛♦ ❝❤♦ F ⊆ m✳ ❚❤❡♦ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✸✳✶✳✸✱ t❛ ❝ã✿ f0 = G1 Gk+1 ∈ m ❦Ð♦ t❤❡♦ tå♥ t➵✐ j ≤ k + s❛♦ ❝❤♦ Gj ∈ m✳ ●ä✐ l ❧➭ sè ❧í♥ ♥❤✃t s❛♦ ❝❤♦ Gl ∈ m✳ ❑❤✐ ➤ã✱ fl = P Hl Gl+1 Gk+1 s✉② r❛ Hl ∈ m✳ ợ ế Gl , Hl m tì từ ❇ỉ ➤Ị ✸✳✷✳✶ t❛ ➤➢ỵ❝ F ⊆ m✳ ✭✐✐✮ t➵✐ ❈❤♦ m ❧➭ ♠ét ✐➤➟❛♥ tè✐ ➤➵✐ ❝ñ❛ K[x, y] ❝❤ø❛ F ✱ ❦❤✐ ➤ã t❤❡♦ ✭✐✮✱ tå♥ i ∈ {1, , k} s❛♦ ❝❤♦ Gi , Hi ⊆ m✳ ❉♦ ➤ã✱ m ❝❤ø❛ ♠ét ♣❤➬♥ tö ❜✃t ❦❤➯ q✉② u(x) ❝ñ❛ Gi ✳ ❳Ðt ♣❤Ð♣ ❝❤✐Õ✉ s❛✉✿ φ : K [x] [y] → (K [x] / u (x) ) [y] ➜➷t L = K[x]/ u(x) ✱ s✉② r❛ φ(m) ❧➭ ♠ét ✐➤➟❛♥ tè✐ ➤➵✐ ❝ñ❛ L[y] ✈➭ φ(Hi ) ∈ φ(m)✳ ❑❤✐ ➤ã✱ φ(m) ❝❤ø❛ ♠ét ♣❤➬♥ tö ❜✃t ❦❤➯ q✉② α ❝đ❛ φ(Hi )✳ ❈❤ä♥ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸✺ v(x, y) ∈ K[x, y] s❛♦ ❝❤♦ φ(v(x, y)) = α✳ ❱× ✈❐② ♥➟♥ u(x) ⊆ m, v(x, y) ∈ m✳ ❍➡♥ ♥÷❛✱ L[y]/ φ(v(x, y)) ❧➭ ♠ét tr➢ê♥❣ ✈× L[y] ❧➭ ♠ét ♠✐Ò♥ ♥❣✉②➟♥ ❝➳❝ ✐➤➟❛♥ ❝❤Ý♥❤ ✈➭ φ(v(x, y)) ❧➭ tè✐ ➤➵✐✳ ❚❤❡♦ ➤Þ♥❤ ❧ý ➤➻♥❣ ❝✃✉✱ t❛ ❝ã L[y]/ φ(v(x, y)) ➤➻♥❣ ❝✃✉ ✈í✐ K[x, y]/ u(x), v(x, y) ✳ ❉♦ ➤ã✱ u(x), v(x, y) ❧➭ tè✐ ➤➵✐ ✈➭ ❜➺♥❣ m✳ ❚❛ ➤Ĩ ý r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ tr♦♥❣ ➜Þ♥❤ ❧ý ✸✳✷✳✷ ❧➭ ❝➳❝ fi ❦❤➠♥❣ ❝ã ➢í❝ ❝❤✉♥❣✱ s✉② r❛ I = f0 , f1 , , fk ❧➭ ♠ét ✐➤➟❛♥ ❝❤✐Ị✉ ❝đ❛ K[x, y] ❉♦ ➤ã✱ I ❦❤➠♥❣ ó tố ú ữ ì ỗ t ♣❤➬♥ ♥❣✉②➟♥ tè ❝ñ❛ I ❧➭ ♠ét ✐➤➟❛♥ tè✐ ➤➵✐ ó ó tể ợ tí t tr ị ❧ý ✸✳✷✳✷✳ ❱Ý ❞ơ s❛✉ ➤➞② sư ❞ơ♥❣ ➜Þ♥❤ ❧ý ✸✳✷✳✷ ➤Ĩ ①➳❝ ➤Þ♥❤ ❝➳❝ ✐➤➟❛♥ tè✐ ➤➵✐ ❝❤ø❛ ✐➤➟❛♥ ❝❤✐Ò✉ s✐♥❤ ❜ë✐ F = {f0 , f1 , f2 , f3 } ⊆ R[x, y] ❝đ❛ ❱Ý ❞ơ ✸✳✶✳✶✵✳ ✸✳✷✳✸ ❱Ý ❞ô✳ ❈❤♦ I = f0 , f1 , f2 , f3 tr♦♥❣ ➤ã✱ f0 = x7 , f1 = y x5 − yx6 , f2 = y x − y x3 , f3 = y − y ●ä✐ G1 = x2 , G2 = x4 , G3 = x, H1 = y − yx, H2 = y − y x2 ✈➭ H3 = y − y ✳ ❑❤✐ ➤ã✱ ❝➳❝ ✐➤➟❛♥ tè✐ ➤➵✐ ❝❤ø❛ I ❧➭✿ m1 = x, y ✈➭ m2 = x, y − , tr♦♥❣ ➤ã G1 , H1 , G2 , H2 , G3 , H3 ⊂ m1 ✈➭ G3 , H3 ⊂ m2 ✳ ❚❤✉❐t t♦➳♥ s❛✉ tÝ♥❤ t♦➳♥ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ❝đ❛ ✐➤➟❛♥ I ❝ã sè ❝❤✐Ị✉ ❧➭ ❦❤➠♥❣✳ ❚❤✉❐t t♦➳♥ ❈❤♦ f0 , f1 , , fk ❧➭ ♠ét ✐➤➟❛♥ ❝❤✐Ị✉ ❝đ❛ K[x, y]✱ ➤➢ỵ❝ ❝❤♦ ❜ë✐ ♠ét ❝➡ së ●r♦❡❜♥❡r tè✐ t✐Ó✉ ①➞② ❞ù♥❣ ♥❤➢ tr♦♥❣ ➜Þ♥❤ ❧ý ✸✳✶✳✸ ✈í✐ ✈➭ ❝❤♦ P Gk+1 = 1✱ u(x), v(x, y) ❧➭ ♠ét ✐➤➟❛♥ tè✐ ➤➵✐ ❝❤ø❛ fi ợ tí t tr ị ý ✿ f0 , f1 , , fk ❀ u(x), v(x, y) ✳ ■♥♣✉t ✿ ▼ét ❝➡ së ❝ñ❛ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ❝đ❛ I t➢➡♥❣ ø♥❣ ✈í✐ ❖✉t♣✉t ✿ ✶✳ ❘❊P▲❆❈❊ f0 ❜ë✐ ❇❡❣✐♥ Số hóa trung tâm học lieäu u(x), v(x, y) ✳ u(x)m http://www.lrc-tnu.edu.vn/ ✸✻ m ❧➭ ❧í♥ ♥❤✃t s❛♦ ❝❤♦ u(x)m |f0 ✳ tr♦♥❣ ➤ã ✷✳ ❇✐Ĩ✉ ❞✐Ơ♥ Hk ❞➢í✐ ❞➵♥❣ w(x, y)v(x, y)n + z(x, y)u(x) tr♦♥❣ ➤ã v(x, y) ✈➭ w(x, y) ❧➭ ♥❣✉②➟♥ tè ❝ï♥❣ ♥❤❛✉ ♠➠➤✉♥ u(x)✳ s ≡ v(x, y)n ♠➠➤✉♥ u(x) ✈➭ t ≡ w(x, y) ♠➠➤✉♥ u(x) ❚×♠ s❛♦ ❝❤♦ Hk ≡ st ♠➠➤✉♥ u(x)m ✳ ❘❊P▲❆❈❊ Hk ❜ë✐ s✳ ✸✳ ❘❊▼❖❱❊ ♠ä✐ fi tr♦♥❣ ➤ã ➜Ó ①❡♠ ∈ {f1 , , fk−1 } u(x)m |fi ❤♦➷❝ s|fi ✳ s ✈➭ t ➤➢ỵ❝ tÝ♥❤ t♦➳♥ t❤Õ ♥➭♦ tr♦♥❣ ❜➢í❝ ✷ ❝đ❛ t❤✉❐t t♦➳♥ tr➟♥✱ t❛ ①❡♠ ①Ðt ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉✳ ✸✳✷✳✹ ▼Ư♥❤ ➤Ị✳ u(x), v(x, y) tr♦♥❣ K[x]✱ ❈❤♦ A = K[x, y]✱ ❧➭ ♠ét ✐➤➟❛♥ tè✐ ➤➵✐ ❝ñ❛ ✈➭ v(x, y) H ≡ v(x, y)n w(x, y) u✱ tr ≡ w ❈❤ø♥❣ ♠✐♥❤✳ A K ❧➭ ♠ét tr➢ê♥❣✳ ❈❤♦ tr♦♥❣ ➤ã ❧➭ ❜✃t ❦❤➯ q✉② tr➟♥ tr➢ê♥❣ ♠➠➤✉♥ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ ♠➠➤✉♥ tr♦♥❣ ➤ã ♠➠➤✉♥ u(x)❀ v r✱ tå♥ t➵✐ ✈➭ ❧➭ ❜✃t ❦❤➯ q✉② L = K[x]/ u(x) ✳ ▲✃② ♥❣✉②➟♥ tè ❝ï♥❣ ♥❤❛✉ ♠➠➤✉♥ sr (x, y) u ✈➭ H ≡ sr tr ◗✉② ♥➵♣ t❤❡♦ w u(x) m = ✈➭ ♠➠➤✉♥ tr (x, y) s❛♦ ❝❤♦ u✳ sr ≡ v n ur ✳ r✳ ◆Õ✉ r = 1✱ t❤❡♦ ❣✐➯ t❤✐Õt t❛ ❝ã s1 (x, y) = v(x, y)n ✈➭ t1 (x, y) = w(x, y)✳ ●✐➯ sư ➤ó♥❣ ✈í✐ r✱ ❦❤✐ ➤ã H ≡ sr tr ♠➠➤✉♥ ur tr♦♥❣ ➤ã sr ≡ v n ♠➠➤✉♥ u✱ tr ≡ w ♠➠➤✉♥ u✳ ❉♦ ➤ã✱ H − sr tr ∈ ur ✳ ❈❤♦ H − sr tr = ur z, tr♦♥❣ ➤ã z ∈ K[x, y]✳ ❱× sr ≡ v n ♠➠➤✉♥ u ✈➭ tr ≡ w ♠➠➤✉♥ u✱ sr = φ(sr ) ✈➭ tr = φ(tr ) ❧➭ ♥❣✉②➟♥ tè ❝ï♥❣ tr L[y] tr ó ợ ị ĩ tr ứ ị ý (ii) ì tồ t α ✈➭ β s❛♦ ❝❤♦ sr α + tr β = tr♦♥❣ ➤ã α ∈ L [y] ✈➭ β L [y] ó tể ợ tì t tt t ❊✉❝❧✐❞❡✳ ➜➷t φ (α) = α✱ φ (β) = β ✱ sr+1 = sr + βzur ; tr+1 = tr + αzur ✳ ❍✐Ó♥ ♥❤✐➟♥✱ sr+1 ≡ sr ♠➠➤✉♥ u ✈➭ tr+1 ≡ tr ♠➠➤✉♥ u✳ ❑❤✐ ➤ã✱ sr+1 tr+1 = sr tr + (sr α + tr β) zur + αβz u2r Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸✼ ❚❤➟♠ ♥÷❛✱ t❛ ❝ã sr α ❉♦ ➤ã✱ ❱× ✈❐②✱ + tr β = ♥➟♥ sr α + tr β = + au ✈í✐ a ∈ K[x, y]✳ sr+1 tr+1 = sr tr + (1 + au) zur + αβz u2r = sr tr + zur + azur+1 + αβz u2r = H + azur+1 + αβz u2r azur+1 + αβz u2r ∈ ur+1 , H ≡ sr+1 tr+1 ♠➠➤✉♥ ur+1 ✳ ▲➢✉ ý r➺♥❣ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ tr➟♥ ❝➳❝ ❜➢í❝ q✉② ♥➵♣ ➤➢ỵ❝ sư ❞ơ♥❣ ♥❤➢ tr♦♥❣ ❜➢í❝ ❤❛✐ ❝đ❛ ➜Þ♥❤ ❧ý ✸✳✶✳✸ ➤Ĩ ❞➬♥ ❞➬♥ t❤✐Õt ❧❐♣ sm ✈➭ tm tr♦♥❣ ➤ã ❧➭ sè ❧í♥ ♥❤✃t s❛♦ ❝❤♦ m um |f0 ✳ ➜Þ♥❤ ❧ý s❛✉ ❝❤Ø r❛ r➺♥❣ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ ✐➤➟❛♥ ❝❤✐Ị✉ tr♦♥❣ K[x, y] ❝ã t❤Ĩ ➤➢ỵ❝ tÝ♥❤ t♦➳♥ t❤❡♦ t❤✉❐t t♦➳♥ tr➟♥✳ ✸✳✷✳✺ ➜Þ♥❤ ❧ý✳ ❈❤♦ I = f0 , f1 , , fk tr♦♥❣ ➤ã K [x, y] ❧➭ ♠ét ❝➡ së ●r♦❡❜♥❡r tè✐ t✐Ĩ✉ ❝đ❛ I ❧ý ✸✳✶✳✸ ✈➭ F = {f0 , f1 , , fk } ⊆ ợ ự tr ị U CLN {fi } = 1✳ ❑❤✐ ➤ã ø♥❣ ❞ơ♥❣ ❝đ❛ t❤✉❐t t♦➳♥ ❝❤♦ ỗ m ợ tí t tr ị ý ❧➭ ❝❤Ø r❛ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ❝ñ❛ m tr♦♥❣ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ ❈❤ø♥❣ ♠✐♥❤✳ ❱× I✳ I ❧➭ ♠ét ✐➤➟❛♥ ❝❤✐Ò✉ tr♦♥❣ ✈➭♥❤ ◆♦❡t❤❡r A = K[x, y] ♥➟♥ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ tè tè✐ t✐Ĩ✉ ❝đ❛ I ❧➭ ❝➳❝ ✐➤➟❛♥ tè✐ ➤➵✐✱ t❛ ❝❤Ø r❛ r➺♥❣ I ❝ã ♠ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❦❤➠♥❣ t❤Ĩ rót ❣ä♥ ➤➢ỵ❝ ❞✉② ♥❤✃t ♥❤➢ s❛✉✿ I = Q1 ∩ Q2 Qr , tr ó ỗ i = 1, , r✱ Qi = mi ❧➭ ♠ét ✐➤➟❛♥ tè✐ ➤➵✐ ❝❤ø❛ I ✳ ❑❤✐ ➤ã mi = u, v ✱ tr♦♥❣ ➤ã ✈í✐ ♠ä✐ j ✱ u ❧➭ ♠ét ♥❤➞♥ tư ❜✃t ❦❤➯ q✉② ❝đ❛ Gj ✈➭ v ❧➭ ♠ét ♣❤➬♥ tư ❜✃t ❦❤➯ q✉② ❝đ❛ (K [x] / u ) [y]✱ ➤➢ỵ❝ t❤✐Õt ❧❐♣ ♥❤➢ tr♦♥❣ ➜Þ♥❤ ❧ý ✸✳✷✳✷✳ ➜✐Ị✉ ➤ã ❝❤Ø r❛ r➺♥❣ ♦✉t♣✉t t❤ø i✱ ❈è ➤Þ♥❤ Qi ❧➭ mi ✲ ♥❣✉②➟♥ s➡ ✈➭ ❜➺♥❣ Qi ✳ i ✈➭ ❝❤♦ mi = m = u, v ❧➭ ✐♥♣✉t t❤ø i ✈➭ Qi = Q ❧➭ ♦✉t♣✉t t➢➡♥❣ ø♥❣✳ ❑❤✐ ➤ã✱ Q = um , f1 , , fk−1 , s , tr♦♥❣ ➤ã m ✈➭ s ➤➢ỵ❝ ♠➠ t➯ ♥❤➢ tr♦♥❣ t❤✉❐t t♦➳♥✳ ◆➟♥ t❛ ❝ã✿ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸✽ s✉② r❛ √ v nm um ∈ Q ✈➭ s = v n + qu ∈ Q √ = (s − qu)m ∈ Q✳ ❱× Q ⊇ m ✈➭ ✈× m ❧➭ ✐➤➟❛♥ tè✐ ➤➵✐ ♥➟♥ Q = m ❦Ð♦ t❤❡♦ Q ❧➭ m ✲ ♥❣✉②➟♥ s➡✳ ➜Ó ❝❤ø♥❣ ♠✐♥❤ ♣❤➢➡♥❣ ❤ã❛ t➵✐ Q = Qi ✱ tr➢í❝ t✐➟♥ t❛ ❝❤Ø r❛ r➺♥❣ I, Q ✈➭ Qi ❝ã ❝ï♥❣ ➤Þ❛ m✳ ❚❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✳✶✹(i)✱t❛ ❝ã (Qj )m = Rm ✈í✐ ♠ä✐ j = i✳ ❑❤✐ ➤ã✱ Im = (Q1 )m ∩ ∩ (Qr )m = (Qi )m ▲➵✐ ❝ã I ⊆ Q ✈× um |f0 ✈➭ fk = Hk = st + pum tr♦♥❣ ➤ã t ợ ị ĩ tr tt t ì Im ♠✐♥❤ ⊆ Qm ✳ ➜Ó ❝❤ø♥❣ Qm ⊆ Im ✱ t❛ ❝➬♥ ❝❤Ø r❛ r➺♥❣ um , s ∈ Im ✈× râ r➭♥❣ f1 , f2 , , fk−1 t❤✉é❝ Im ✳ ❱× m ❧➭ ❧í♥ ♥❤✃t s❛♦ ❝❤♦ um |f0 ∈ K[x] ♥➟♥ s✉② r❛ um ∈ Im ✳ ❍➡♥ ♥÷❛✱ st = Hk − pum ∈ Im , ✈➭ t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ t✱ t ∈ m✳ ❑❤✐ ➤ã✱ st t = s ∈ Im ✈➭ Im = Qm = (Qi )m ✳ ❉♦ ➤ã✱ t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✳✶✹(ii)✱ Qi = A ∩ (Qi )m = A ∩ Qm = Q ❈➳❝ ✈Ý ❞ô s❛✉ ❝❤Ø r❛ ø♥❣ ❞ơ♥❣ ❝đ❛ t❤✉❐t t♦➳♥✳ ❚r♦♥❣ ❱Ý ❞ơ ✸✳✷✳✻✱ ➤Þ♥❤ ♥❤➢ tr♦♥❣ ❝➳❝ ❱Ý ❞ô ✸✳✶✳✶✵ ✈➭ ✸✳✷✳✸✱ tr♦♥❣ ➤ã F ①➳❝ K = R ❧➭ tr➢ê♥❣ ❝➳❝ sè t❤ù❝✳ ✸✳✷✳✻ ❱Ý ❞ô✳ ❈❤♦ I = x7 , y x5 − yx6 , y x − y x3 , y − y ✈➭ ❣ä✐ G1 = x2 ; G2 = x4 ; G3 = x, H0 = 1; H1 = y − yx; H2 = y − y x2 ; H3 = y − y ❑❤✐ ➤ã✱ m1 = x, y ✈➭ m2 = x, y − Soá hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸✾ ❈➳❝ t❤➭♥❤ s ủ I ợ tì ❞ô♥❣ t❤✉❐t t♦➳♥ ❤❛✐ ❧➬♥✳ ■✳ ■♥♣✉t✿ m1 = x, y ✳ ✶✳ ❑❤➠♥❣ t❤❛② ➤ỉ✐ ✈× f0 ✷✳ ❚❤❛② t❤Õ = x7 ✳ H3 ❜ë✐ y ✈× x7 ❦❤➠♥❣ ❝❤✐❛ ❤Õt H3 = y (y − 1)✳ ✸✳ ❑❤➠♥❣ t❤❛② ➤ỉ✐ ✈× ❝➯ f1 ✈➭ f2 ➤Ị✉ ❦❤➠♥❣ ❝❤✐❛ ❤Õt ❝❤♦ f0 ❤♦➷❝ H3 ♠í✐✳ Q1 = x7 , f1 , f2 , y ✳ ❖✉t♣✉t✿ ■■✳ ■♥♣✉t✿ m2 = x, y − ✳ ✶✳ ❑❤➠♥❣ t❤❛② ➤æ✐✳ ✷✳ ❚❤❛② t❤Õ H3 ❜ë✐ y − ✈× x7 ❦❤➠♥❣ ❝❤✐❛ ❤Õt H3 = y (y − 1)✳ ✸✳ ❑❤➠♥❣ t❤❛② ➤ỉ✐ ✈× ❝➯ f1 ✈➭ f2 ➤Ò✉ ❦❤➠♥❣ ❝❤✐❛ ❤Õt ❝❤♦ f0 ❤♦➷❝ H3 ♠í✐✳ ❖✉t♣✉t✿ Q2 = x7 , f1 , f2 , y − ✳ ❑❤✐ ➤ã✱ I = x7 , y x5 − yx6 , y x − y x3 , y ∩ x7 , y x5 − yx6 , y x − y x3 , y − ❧➭ ♠ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ I ✳ ✸✳✷✳✼ ❱Ý ❞ơ✳ ❈❤♦ I = f0 , f1 , f2 ∈ R[x, y]✱ tr♦♥❣ ➤ã✱ f0 = x3 + x, f1 = yx2 + y − x2 − 1, f2 = y − 2x2 − ❑❤✐ ➤ã✱ G1 = x, G2 = x2 + 1, H1 = y − 1, H2 = y − 2x2 − 1✳ ➜Þ♥❤ ❧ý ✸✳✷✳✷ s✉② r❛ m1 = x, y − , m2 = x2 + 1, y − x , m3 = x2 + 1, y + x tr♦♥❣ ➤ã m1 ❝❤ø❛ G1 , H1 ✱ ❝➯ m2 ✈➭ m3 ➤Ò✉ ❝❤ø❛ G2 , H2 ✳ ◗✉➳ tr×♥❤ tÝ♥❤ t♦➳♥ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ➤➢ỵ❝ ❞✐Ơ♥ t➯ ♥❤➢ s❛✉✿ ■✳ ■♥♣✉t✿ m1 = x, y − ✳ ✶✳ ❚❤❛② t❤Õ f0 ❜ë✐ ✷✳ ❚❤❛② t❤Õ x✳ H2 ❜ë✐ y − ✈× H2 = (y − 1)(y + 1) − 2x2 ✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✹✵ ✸✳ ▲♦➵✐ ❜á f1 ✈× f1 ❖✉t♣✉t✿ = (y − 1)(x2 + 1)✳ Q1 = x, y − ✳ ■■✳ ■♥♣✉t✿ m2 = x2 + 1, y − x ✳ ✶✳ ❚❤❛② t❤Õ f0 ❜ë✐ ✷✳ ❚❤❛② t❤Õ x2 + 1✳ H2 ❜ë✐ y − x ✈× H2 = (y − x) (y + x) − x2 + ✳ ✸✳ ▲♦➵✐ ❜á f2 ✳ ❖✉t♣✉t✿ Q2 = x2 + 1, y − x ✳ ■■■✳ ■♥♣✉t✿ m3 = x2 + 1, y + x ✳ ❖✉t♣✉t✿ ❱× ✈❐②✱ Q3 = m3 t➢➡♥❣ tù ♣❤➬♥ ■■✳ I = x, y − ∩ x2 + 1, y − x ∩ x2 + 1, y + x ❧➭ ♠ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ I ✳ ✸✳✷✳✽ ❱Ý ❞ơ✳ ❈❤♦ I = f0 , f1 , f2 ∈ R[x, y]✱ tr♦♥❣ ➤ã f0 = yx2 x3 − 2x2 + x , f1 = yx2 xy + 4xy + 4x , f2 = yx2 y + 12x2 y − 24xy − 16x2 + 32x I ❧➭ ♠ét ✐➤➟❛♥ ♠ét ❝❤✐Ị✉ ✈× I ⊂ y ✳ ❈ã D = U CLN {fi } = yx2 ✳ ➜➷t gi = fi D ✳ ❑❤✐ ➤ã✱ J = g0 , g1 , g2 ❧➭ ♠ét ✐➤➟❛♥ ❝❤✐Ị✉ ✈➭ ✈× ✈❐②✱ t❤✉❐t t♦➳♥ ❝ã t❤Ĩ ➳♣ ❞ơ♥❣ ❝❤♦ ❝➳❝ ✐➤➟❛♥ tè✐ ➤➵✐ ❝❤ø❛ ➜è✐ ✈í✐ J✳ J ✱ ❝ã G1 = (x − 1)2 , G2 = x, H1 = (y − 2)2 , H2 = g2 = y +12x2 y−24xy−16x2 +32x ❑❤✐ ➤ã✱ ➜Þ♥❤ ❧ý ✸✳✷✳✷ ❝❤Ø r❛ m1 = x − 1, y − ⊃ G1 , H1 , m2 = x, y ⊃ G2 , H2 ◗✉➳ tr×♥❤ tÝ♥❤ t♦➳♥ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ❝ñ❛ ■✳ ■♥♣✉t✿ J ♥❤➢ s❛✉✿ m1 = x − 1, y − ✳ ✶✳ ❚❤❛② t❤Õ g0 ❜ë✐ ✷✳ ❚❤❛② t❤Õ (x − 1)2 ✳ H2 ❜ë✐ (y − 2)2 ✈× H2 = (y − 2)2 (y + 4) + (12y − 16) (x − 1)2 ✳ Soá hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✹✶ ✸✳ ▲♦➵✐ ❜á g1 ✈× ❖✉t♣✉t✿ ■■✳ ■♥♣✉t✿ (y − 2)2 |g1 = x(y − 2)2 ✳ Q1 = (x − 1)2 , (y − 2)2 ✳ m2 = x, y ✳ ✶✳ ❚❤❛② t❤Õ f0 ❜ë✐ ✷✳ ❚❤❛② t❤Õ H2 ❜ë✐ y ✈× H2 = y + x.(12xy − 24y − 16x + 32)✳ ✸✳ ▲♦➵✐ ❜á g1 ✈× ❖✉t♣✉t✿ ❑❤✐ ➤ã✱ x ✈× g0 = x.(x − 1)2 ✳ x|g1 = x(y − 2)2 ✳ Q2 = x, y ✳ J = (x − 1)2 , (y − 2)2 ∩ x, y ❧➭ ♠ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝ñ❛ J ✳ ❈ã D = y ∩ x2 ❧➭ ♠ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝ñ❛ D ✳ ❑❤✐ ➤ã✱ t❛ ❝ã✿ I= y ∩ x2 (x − 1)2 , (y − 2)2 ∩ x, y , ❧➭ ♠ét ♣❤➞♥ tÝ❝❤ ❝ñ❛ I ✳ ▲➢✉ ý r➺♥❣ tr♦♥❣ ✈Ý ❞ô tr➟♥✱ x ✈➭ y ❧➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ♥❤ó♥❣ ❝đ❛ I ✈× x, y ❝❤ø❛ ❝➯ x ✈➭ y ✳ ❑❤✐ ➤ã ♣❤➞♥ tÝ❝❤ ❝✉è✐ ❝ï♥❣ ➤➢ỵ❝ ❜✐Ĩ✉ ❞✐Ơ♥ ♥❤➢ ❦Õt q✉➯ ❝đ❛ ❤❛✐ ♣❤Ð♣ ❣✐❛♦ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ s➡✳ ▲➢✉ ý r➺♥❣ ♥Õ✉ A = y ∩ x2 ∩ (x − 1)2 , (y − 2)2 ∩ x, y ✈➭ B= t❤× y ∩ x2 (x − 1)2 , (y − 2)2 ∩ x, y , A = B ✈× x2 y(x − 1)2 ∈ A ♥❤➢♥❣ x2 y(x − 1)2 ∈ B ✳ ❚r♦♥❣ ✈Ý ❞ô s❛✉ ➤➞②✱ t❛ ①Ðt ♠ét ✐➤➟❛♥ ♠ét ❝❤✐Ị✉ ❦❤➠♥❣ ❝ã t❤➭♥❤ ♣❤➬♥ ♥❤ó♥❣✳ ✸✳✷✳✾ ❱Ý ❞ô✳ ❈❤♦ I = f0 , f1 , f2 ✱ tr♦♥❣ ➤ã fi = (y − 1)2 gi ✈➭ gi ợ ị ĩ tr í ụ g0 = x3 −2x2 +x, g1 = xy +4xy+4x, g2 = y +12x2 y−24xy−16x2 +32x ◆❤➢ ➤➲ ❜✐Õt tr♦♥❣ ❱Ý ❞ơ ✸✳✷✳✽✱ t❛ ❝ã ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ J = (x − 1)2 , (y − 2)2 ∩ x, y ❧➭ J = g0 , g1 , g2 ✳ ➜➷t D = U CLN {fi } = (y − 1)2 ✳ Số hóa trung tâm học lieäu http://www.lrc-tnu.edu.vn/ ✹✷ ❑❤✐ ➤ã✱ D = (y − 1)2 ✈➭ D ❦❤➠♥❣ ♣❤➯✐ ❧➭ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ♥❤ó♥❣ ❝đ❛ I ✳ ❱× ✈❐② I = (y − 1)2 ∩ (x − 1)2 , (y − 2)2 ∩ x, y ❱➭ ❞♦ ➤ã t❛ t❤✃② r➺♥❣ t❤✉❐t t♦➳♥ ❝ã t❤Ĩ ➤➢ỵ❝ ❞ï♥❣ ➤Ĩ ♣❤➞♥ tÝ❝❤ ✐➤➟❛♥ tr♦♥❣ K[x, y] s❛✉ ❦❤✐ tÝ♥❤ t♦➳♥ ❝➡ së ●r♦❡❜♥❡r tè✐ t✐Ĩ✉ ❝đ❛ ✐➤➟❛♥ ➤ã✳ P❤➞♥ tÝ❝❤ ♥➭② sÏ ❧➭ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❞✉② ♥❤✃t ➤è✐ ✈í✐ ❝➳❝ ✐➤➟❛♥ ❝❤✐Ị✉ ❝đ❛ K[x, y]✳ ❚✉② ♥❤✐➟♥ ➤è✐ ✈í✐ ❝➳❝ ✐➤➟❛♥ ❝❤✐Ị✉ ♠ét✱ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝ã t❤Ĩ ❦❤➠♥❣ ❞✉② ♥❤✃t ✈➭ ❦❤➠♥❣ ♣❤➯✐ ❧➭ ♣❤➞♥ tÝ❝❤ ❝❤✉➮♥✱ ♠➭ ❧➭ ♣❤➞♥ tÝ❝❤ ❜❛♦ ❣å♠ tÝ❝❤ ❝ñ❛ ❝➳❝ ❣✐❛♦ ❝ñ❛ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ s➡✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✹✸ ❑Õt ❧✉❐♥ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ ➤➲ tr×♥❤ ❜➭② ❤Ư t❤è♥❣ ❦✐Õ♥ t❤ø❝ ✈Ị ❝➡ së ●r♦❡❜♥❡r✱ t❤✉❐t t♦➳♥ ❇✉❝❤❜❡r❣❡r ➤Ĩ t×♠ ❝➡ së ●r♦❡❜♥❡r✱ ❝➡ së ●r♦❡❜♥❡r tè✐ t✐Ĩ✉ ✈➭ ❝➡ së ●r♦❡❜♥❡r rót ❣ä♥✳ ▲✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ❝✃✉ tró❝ ❝đ❛ ❝➡ së ●r♦❡❜♥❡r tè✐ t✐Ĩ✉ ❝đ❛ ❝➳❝ ✐➤➟❛♥ tr♦♥❣ ✈➭♥❤ ➤❛ t❤ø❝ ❤❛✐ ❜✐Õ♥ K[x, y] ✈í✐ K ❧➭ ♠ét tr➢ê♥❣✳ ▲✉❐♥ ✈➝♥ tr×♥❤ ❜➭② t❤✉❐t t♦➳♥ ❝đ❛ ▲❛③❛r❞ ✈Ị ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ ♠ét ✐➤➟❛♥ ❝❤✐Ị✉ ❦❤➠♥❣ tr♦♥❣ ✈➭♥❤ K[x, y]✱ tõ ➤ã tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ ✐➤➟❛♥ ❝❤✐Ị✉ ♠ét tr♦♥❣ ✈➭♥❤ Số hóa trung tâm học liệu K[x, y]✳ http://www.lrc-tnu.edu.vn/ ✹✹ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ▲➟ ❚✉✃♥ ❍♦❛ ✭✷✵✵✸✮✱ ➜➵✐ số tí sở ră or ọ q✉è❝ ❣✐❛ ❍➭ ◆é✐✳ ❬✷❪ ❉✳❈♦① ✭✶✾✾✶✮✱ ❚✳▲✐tt❧❡ ❛♥❞ ❉✬❖s❤❡❛✱ ■❞❡❛❧✱ ❱❛r✐rt✐❡s✱ ❛♥❞ ❆❧❣♦r✐t❤♠s✱ ❙♣r✐♥❣❡r ❱❡r❧❛❣✳ ❬✸❪ ▲❛③❛r❞✱ ❉✳ ✭✶✾✽✺✮✱ ✧■❞❡❛❧ ❜❛s❡s ❛♥❞ ♣r✐♠❛r② ❞❡❝♦♠♣♦s✐t✐♦♥✿ ❝❛s❡ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✧✱ ❏✳ ❙②♠❜♦❧✐❝ ❈♦♠♣✉t❛t✐♦♥ ✶✱ ♣♣✳ ✷✻✶✲✷✼✵✳ ❬✹❪ ▼❛ts✉♠✉r❛✱ ❍✳ ✭✶✾✽✻✮✱ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r②✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✺❪ ❑r❡✉③❡r✱ ▼✳ ❛♥❞ ❘♦❜❜✐❛♥♦✱ ▲✳ ✭✷✵✵✵✮✱ ❈♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡✲ ❜r❛ ✶✱ ❙♣r✐♥❣❡r✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ... ❝❛♦ ❝đ❛ ✐➤➟❛♥✳ ❈❤➢➡♥❣ ❤❛✐ tr×♥❤ ❜➭② ❝❤✐ t✐Õt ✈Ị ❝➡ së ●r♦❡❜♥❡r ✈➭ t❤✉❐t t♦➳♥ ❇✉❝❤❜❡r❣❡r ➤Ĩ tì sở rr t tt ữ ủ ❈❤➢➡♥❣ ❜❛ tr×♥❤ ❜➭② ♠ét t❤✉❐t t♦➳♥ ❝đ❛ ▲❛③❛r❞ ✈Ị ♠ét ➳♣ ❞ơ♥❣ ❝đ❛ ❝➡ së ●r♦❡❜♥❡r... tr➟♥✱ ♥Õ✉ ❝➬♥ t❤✐Õt✱ t❛ ➤➢ỵ❝ Gr ∈ At s❛♦ ❝❤♦ λ(Gr ) = f ✈➭ deg(Gr ) = deg(F )✳ ❑❤✐ ➤ã t ị ý tì F sở rr ❝đ❛ I ✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ị ý tết ợ ❜❛ ➤✐Ị✉ ❦✐Ư♥ t➢➡♥❣... lt(fj ) ✈í✐ ♠ä✐ j = i✳ ✷✳✸✳✽ ➜Þ♥❤ ❧ý✳ ●✐➯ sư ❜❐❝ tỉ♥❣ t❤Ĩ tr➟♥ ❝➳❝ ➤➡♥ t❤ø❝ ❝đ❛ ❑❤✐ ➤ã ✈í✐ ỗ I ủ sở rr rút ọ ủ A A ➤Ị✉ tå♥ t➵✐ ❞✉② ♥❤✃t ✭s❛✐ ❦❤➳❝ ➤➡♥ ✈Þ✮ ♠ét ❝➡ I✳ ❇✃t ❦ú ♠ét ❝➡ së ●r♦❡❜♥❡r✱ F