✣❸■ ❍➴❈ ✣⑨ ◆➂◆●
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✲✲✲✲✲✯✲✲✲✲✲

P❍Ị ❚❘➴◆● ❍×◆●

Ù◆● ❉Ö◆● ❈Õ❆ ✣❆ ❚❍Ù❈ ✣➮■ ❳Ù◆● ❚❘❖◆● ▼❐❚ ❙➮
❇⑨■ ❚❖⑩◆ P❍✃ ❚❍➷◆●
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

✣➔ ◆➤♥❣ ✲ ✷✵✷✵


✣❸■ ❍➴❈ ✣⑨ ◆➂◆●
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✲✲✲✲✲✯✲✲✲✲✲

P❍Ị ❚❘➴◆● ❍×◆●

Ù◆● ❉Ö◆● ❈Õ❆ ✣❆ ❚❍Ù❈ ✣➮■ ❳Ù◆● ❚❘❖◆● ▼❐❚ ❙➮
❇⑨■ ❚❖⑩◆ P❍✃ ❚❍➷◆●
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
◆●⑨◆❍✿ P❍×❒◆● P❍⑩P ❚❖⑩◆ P

ữớ ữợ ồ ✣Ù❈ ❚❍⑨◆❍

✣➔ ◆➤♥❣ ✲ ✷✵✷✵

✸

▼ư❝ ❧ư❝
▲í✐ ❝❛♠ ✤♦❛♥
▲í✐ ❝↔♠ ì♥
▼ð ✤➛✉
✶ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❤❛✐ ❜✐➳♥ ✈➔ ù♥❣ ❞ö♥❣

✶✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❤❛✐
❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷ Ù♥❣ ❞ö♥❣ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❤❛✐ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❜❛ ❜✐➳♥ ✈➔ ù♥❣ ❞ö♥❣

✷✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❜❛
❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷ Ù♥❣ ❞ö♥❣ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❜❛ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✶
✶
✸
✼
✼
✶✸

✹✻
✹✻
✺✼

✼✵
✼✶




é
ị

Pữỡ tr ữỡ tr➻♥❤ ✈➔ ♥❤ú♥❣ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ❧➔ ♠ët tr♦♥❣
♥❤ú♥❣ ❝❤õ ✤➲ q✉❛♥ trå♥❣ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ð ❜➟❝ ❚r✉♥❣ ❤å❝ P❤ê t❤ỉ♥❣
✈➔ ❧➔ ❝❤õ ✤➲ ✤÷đ❝ ❇ë ●✐→♦ ❞ư❝ ✈➔ ✣➔♦ t↕♦ ✤÷❛ ✈➔♦ tr♦♥❣ ❝→❝ ❦ý t❤✐ ố
tứ trữợ tợ tữớ t ừ t ỵ tt ữỡ tr
ữỡ tr õ q t tt ỵ tt t❤ù❝✳ ◆❣♦➔✐ r❛✱ r➜t
♥❤✐➲✉ ❜➔✐ t♦→♥ ð ❜➟❝ ❚r✉♥❣ ❤å❝ P❤ê t❤ỉ♥❣ ✤÷đ❝ ❣✐↔✐ r➜t ❤✐➺✉ q✉↔ ♥➳✉ ❝❤ó♥❣ t❛
ù♥❣ ử ỵ tt tự ử t tự ✤è✐ ①ù♥❣✳ ✣➦❝ ❜✐➺t✱ ✤➙② ❝ơ♥❣ ❧➔
❝ỉ♥❣ ❝ư s➢❝ ❜➨♥ t❤÷í♥❣ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❝→❝ ❦ý t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ❝→❝ ❝➜♣✳
❱ỵ✐ ♠ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ✤❛ t❤ù❝ ✤è✐ ự ự ử ụ ữ ữợ sỹ
ữợ ừ t❤➛② ❣✐→♦ ❚r➛♥ ✣ù❝ ❚❤➔♥❤✱ ❝❤ó♥❣ tỉ✐ ✤➣ q✉②➳t ✤à♥❤ ❝❤å♥ ♥❣❤✐➯♥
❝ù✉ ✤➲ t➔✐✿ ✏Ù♥❣ ❞ö♥❣ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ tr♦♥❣ ♠ët sè ❜➔✐ t♦→♥ ♣❤ê
t❤ỉ♥❣✑ ✳ ❈❤ó♥❣ tỉ✐ ❤② ✈å♥❣ t↕♦ ✤÷đ❝ ♠ët t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ tèt ❝❤♦ ♥❤ú♥❣
♥❣÷í✐ q✉❛♥ t➙♠ ✤➳♥ ♠ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ tr♦♥❣ ♠ët sè ❜➔✐
t♦→♥ ♣❤ê t❤æ♥❣✳
✷✳ ▼Ö❈ ✣➑❈❍ ◆●❍■➊◆ ❈Ù❯

◆❣❤✐➯♥ ❝ù✉ ♥❤➡♠ t➻♠ ❤✐➸✉ ✈➔ ❧➔♠ rã ❝→❝ ✈➜♥ ✤➲ s❛✉✿

✭✶✮ ❑❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❤❛✐ ❜✐➳♥✱ ❜❛ ❜✐➳♥✳
✭✷✮ Ù♥❣ ❞ö♥❣ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ tr♦♥❣ ♠ët sè ❜➔✐ t♦→♥ ♣❤ê t❤ỉ♥❣ ♥❤÷ ❣✐↔✐
♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ❤❛② ❜➔✐ t♦→♥ ❝ü❝ trà✳
✸✳ ✣➮■ ❚×Đ◆● ❱⑨ P❍❸▼ ❱■ ◆●❍■➊◆ ❈Ù❯


✺

✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ❧➔ ❝→❝ ❝❤✉②➯♥ ✤➲ ✈➲ ✤❛ t❤ù❝✱ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣
tr➻♥❤ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❤❛✐ ❜✐➳♥✱ ❜❛ ❜✐➳♥✳
P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ❧➔ ♠è✐ ❧✐➯♥ q✉❛♥ ❣✐ú❛ ❝→❝ ✤è✐ t÷đ♥❣ tr➯♥❀ ❝→❝ ù♥❣ ❞ư♥❣
✤➸ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥✳

✹✳ ◆❍■➏▼ ❱Ư ◆●❍■➊◆ ❈Ù❯

◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ❧➔ t➻♠ ❤✐➸✉ ✈➲ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❤❛✐ ❜✐➳♥✱ ❜❛ ❜✐➳♥❀ ❝→❝
❞↕♥❣ ❜➔✐ t➟♣ ù♥❣ ử

Pì PP
Pữỡ ự ỵ tt ✣å❝ t➔✐ ❧✐➺✉✱ ♣❤➙♥ t➼❝❤✱ s♦ s→♥❤✱ tê♥❣
❤ñ♣ ✈➔ sû ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ s✉② ❧✉➟♥ ❝õ❛ t♦→♥ ❤å❝✳

✻✳ ❈❻❯ ể ế
ố ử ỗ ữỡ

ữỡ ✶✿ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❤❛✐ ❜✐➳♥ ✈➔ ù♥❣ ❞ư♥❣
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤è✐
①ù♥❣ ❤❛✐ ❜✐➳♥✱ s❛✉ ✤â →♣ ❞ư♥❣ ❝❤ó♥❣ ✤➸ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ✈➲ ❣✐↔✐ ♣❤÷ì♥❣
tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ♣❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû✱ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣
t❤ù❝✳

✶✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❤❛✐ ❜✐➳♥
▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣
❤❛✐ ❜✐➳♥✳
✶✳✷✳ Ù♥❣ ❞ư♥❣ ❝õ❛ ✤❛ t❤ù❝ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❤❛✐ ❜✐➳♥
▼ö❝ ♥➔② ❞➔♥❤ ✤➸ tr➻♥❤ ❜➔② ♠ët sè ❜➔✐ t♦→♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤❀ ❤➺ ♣❤÷ì♥❣
tr➻♥❤❀ ❜➔✐ t♦→♥ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû❀ rót ❣å♥ ❜✐➸✉ t❤ù❝❀ t➼♥❤ ❝❤✐❛ ❤➳t✳

❈❤÷ì♥❣ ✷✿ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❜❛ ❜✐➳♥ ✈➔ ù♥❣ ❞ö♥❣


✻

❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤è✐
①ù♥❣ ❜❛ ❜✐➳♥✱ s❛✉ ✤â →♣ ❞ư♥❣ ❝❤ó♥❣ ✤➸ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ✈➲ ❣✐↔✐ ♣❤÷ì♥❣
tr➻♥❤❀ ❤➺ ♣❤÷ì♥❣ tr➻♥❤❀ ♣❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû❀ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣
t❤ù❝✳
✷✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❜❛ ❜✐➳♥
▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t✱ ✈➼ ❞ö ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣
❜❛ ❜✐➳♥✳
✷✳✷✳ Ù♥❣ ❞ö♥❣ ❝õ❛ ✤❛ t❤ù❝ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❜❛ ❜✐➳♥
▼ö❝ ♥➔② ❞➔♥❤ ✤➸ tr➻♥❤ ❜➔② ♠ët sè ❜➔✐ t♦→♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤❀ ❤➺ ♣❤÷ì♥❣
tr➻♥❤❀ ❜➔✐ t♦→♥ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû❀ rót ❣å♥ ❜✐➸✉ t❤ù❝❀ t➼♥❤ ❝❤✐❛ ❤➳t✳


✼

❈❤÷ì♥❣ ✶

✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❤❛✐ ❜✐➳♥ ✈➔ ù♥❣
❞ư♥❣

❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣
❤❛✐ ❜✐➳♥✱ s❛✉ ✤â →♣ ❞ö♥❣ ❝❤ó♥❣ ✤➸ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ✈➲ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱
❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ♣❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû✱ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✳

✶✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣
❤❛✐ ❜✐➳♥
▼ư❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣
❤❛✐ ❜✐➳♥✳

✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ▼ët ✤ì♥ t❤ù❝ f (x, y) ❝õ❛ ❝→❝ ❜✐➳♥ ✤ë❝ ❧➟♣ x, y ✭tr÷í♥❣

❤đ♣ ❝❤✉♥❣ ♥❤➜t ❝â t❤➸ ❧➔ ❝→❝ sè ♣❤ù❝✮ ✤÷đ❝ ❤✐➸✉ ❧➔ ❜✐➸✉ t❤ù❝ ❝â ❞↕♥❣✿
f (x, y) = akl xk y l ,

tr♦♥❣ ✤â akl = 0 ❧➔ ♠ët sè ✭❤➡♥❣ sè✮✱ k, l ❧➔ ♥❤ú♥❣ sè ♥❣✉②➯♥ ❦❤ỉ♥❣ ➙♠✳ ❙è akl
✤÷đ❝ ❣å✐ ❧➔ ❤➡♥❣ sè✱ ❝á♥ k + l ✤÷đ❝ ❣å✐ ❧➔ ❜➟❝ ❝õ❛ ✤ì♥ t❤ù❝ f (x, y) ✈➔ ✤÷đ❝ ❦➼
❤✐➺✉ ❧➔
deg [f (x, y)] = deg axk y l = k + l.

❈→❝ sè k, l t÷ì♥❣ ù♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❜➟❝ ❝õ❛ ỡ tự ố ợ x, y. ữ
❝õ❛ ✤ì♥ t❤ù❝ ❤❛✐ ❜✐➳♥ ❜➡♥❣ tê♥❣ ❝õ❛ ❝→❝ ❜➟❝ ❝õ❛ ❝→❝ ✤ì♥ t❤ù❝ t❤❡♦
tø♥❣ ❜✐➳♥✳ ❈❤➥♥❣ ❤↕♥✿ 3x2 y ✈➔ 32 x2 y 3 ❧➔ ❝→❝ ✤ì♥ t❤ù❝ t❤❡♦ x, y ợ tữỡ
ự

♥❣❤➽❛ ✭❬✸❪✮✳ ❍❛✐ ✤ì♥ t❤ù❝ ❝õ❛ ❝→❝ ❜✐➳♥ x, y ữủ ồ ỗ

tữỡ tỹ ú õ ❤➺ sè ❦❤→❝ ♥❤❛✉✳ ◆❤÷ ✈➟②✱ ❤❛✐ ✤ì♥ t❤ù❝ ✤÷đ❝ ❣å✐

ỗ ú õ
Axk y l , Bxk y l , (A = B).

✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ●✐↔ ksûl Axk yl ✈➔ Bxmyn ❧➔ ❤❛✐m ✤ì♥
t❤ù❝ ❝õ❛ ❝→❝ ❜✐➳♥
n

x, y. ❚❛ ♥â✐ r➡♥❣ ✤ì♥ t❤ù❝ Ax y trë✐ ❤ì♥ ✤ì♥ t❤ù❝ Bx y
❜✐➳♥ x, y, ♥➳✉ k > m, ❤♦➦❝ k = m ✈➔ l > n.

t❤❡♦ t❤ù tü ❝õ❛ ❝→❝

❈❤➥♥❣ ❤↕♥✿ ✣ì♥ t❤ù❝ x4y2 t❤❡♦ t❤ù tü x, y ❧➔ trë✐ ❤ì♥ ✤ì♥ t❤ù❝ x2y7 ✱ ❝á♥
✤ì♥ t❤ù❝ x4y6 ❧➔ trë✐ ❤ì♥ ✤ì♥ t❤ù❝ x4y5 ✳
✶✳✶✳✹ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ▼ët ❤➔♠ sè P (x, y) ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤❛ t❤ù❝ t❤❡♦ ❝→❝
❜✐➳♥ sè ①✱ ②✱ õ õ t ữủ ữợ tờ ❝õ❛ ❤ú✉ ❤↕♥ ❝→❝ ✤ì♥
t❤ù❝✳ ◆❤÷ ✈➟②✱ ✤❛ t❤ù❝ P (x, y) t❤❡♦ ❝→❝ ❜✐➳♥ sè x, y ❧➔ ❤➔♠ sè ❝â ❞↕♥❣
akl xk y l .

P (x, y) =
k+l≤m

❇➟❝ ợ t ừ ỡ tự tr tự ữủ ❣å✐ ❧➔ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝✳
✶✳✶✳✺ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ✣❛ t❤ù❝ P (x, y) ✤÷đ❝ ❣å✐ ❧➔ ✤è✐ ①ù♥❣ ♥➳✉ ♥â ❦❤æ♥❣
t❤❛② ✤ê✐ ❦❤✐ ✤ê✐ ❝❤é ❝õ❛ x ✈➔ y✱ ♥❣❤➽❛ ❧➔
P (x, y) = P (y, x).

❈❤➥♥❣ ❤↕♥✿
P (x, y) = x2 + xy + y 2 , Q(x, y) = x2 y + xy 2

❧➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝õ❛ ❝→❝ ❜✐➳♥ x, y✳
✶✳✶✳✻ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❈→❝ ✤❛ t❤ù❝ σj

(j = 1, 2)✱

tr♦♥❣ ✤â

σ1 = x + y, σ2 = xy

✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì sð ❝õ❛ ❝→❝ ❜✐➳♥ x, y✳
✶✳✶✳✼ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ f (x, y) ✤÷đ❝ ❣å✐ ❧➔ t❤✉➛♥ ♥❤➜t ❜➟❝
♠ ♥➳✉ ✈ỵ✐ ∀t = 0 t❛ ❝â
m
f (tx, ty) = t f (x, y).

✶✳✶✳✽ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❈→❝ ✤❛ t❤ù❝ sk = xk + yk (k = 1 , 2 , ...) ✤÷đ❝ ❣å✐ ❧➔

❝→❝ tê♥❣ ❧ô② t❤ø❛ ❜➟❝ ❦

❝õ❛ ❝→❝ ❜✐➳♥ x, y✳

✶✳✶✳✾ ✣à♥❤ ❧➼ ✭❬✸❪✮✳ ▼é✐ tê♥❣ ❧ô② t❤ø❛ sm = xm + ym õ t ữủ

ữợ ởt tự ❜➟❝ ♠ ❝õ❛ σ1 ✈➔ σ2 ✳


✾

❈❤ù♥❣ ♠✐♥❤✳

❚❛ ❝â

σ1 sk−1 = (x + y)(xk−1 + y k−2 ) = xk + y k + xy(xk−2 + y k−2 ) = sk + σ2 sk−2 .

◆❤÷ ✈➟②

✭✶✳✶✮
❈ỉ♥❣ t❤ù❝ ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ❝ỉ♥❣ t❤ù❝ ◆❡✇t♦♥ ♥â ❝❤♦ ♣❤➨♣ t➼♥❤ sk t❤❡♦ sk−1✈➔
sk−2 ✳ ❱ỵ✐ m = 1, m = 2, ✣à♥❤ ❧➼ ✶✳✶✳✾ ✤ó♥❣ ✈➻
sk = σ1 sk−1 − σ2 sk−2 .

s1 = x + y = σ 1 ,
s2 = x2 + y 2 = (x + y)2 − 2xy = σ1 2 − 2σ2 .

●✐↔ sû ✤à♥❤ ❧➼ ✤➣ ✤ó♥❣ ❝❤♦ ♠❁❦✳ ❑❤✐ ✤â sk−2, sk−1 ❧➛♥ ❧÷đt ❧➔ ❝→❝ ✤❛ t❤ù❝ ❜➟❝
❦✲✷✱ ❦✲✶ ❝õ❛ σ1, σ2 ✳ ❚❤❡♦ ❝æ♥❣ t❤ù❝ ✭✷✳✶✮ t❛ s✉② r❛ sk ❧➔ ✤❛ t❤ù❝ ❜➟❝ ❦ ❝õ❛ σ1✈➔
σ2 ✳ ❚❤❡♦ ♥❣✉②➯♥ ❧➼ q✉② ♥↕♣ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝
✭✶✳✶✮ ✈➔ ❝→❝ ❜✐➸✉ t❤ù❝ ❝õ❛ s1, s2 ð ❝❤ù♥❣ ♠✐♥❤ tr➯♥✱ ❞➵ ❞➔♥❣ ♥❤➟♥ ✤÷đ❝ ❝→❝ ❜✐➸✉
t❤ù❝ s❛✉
s1
s2
s3
s4
s5

= x + y = σ1 ,
= σ1 2 − 2σ2 ,
= σ1 3 − 3σ1 σ2 ,
= σ1 4 − 4σ1 2 σ2 + 2σ2 2 ,

= σ1 5 − 5σ1 3 σ2 + 5σ1 σ2 2 .

❱✐➺❝ t➼♥❤ tê♥❣ ❝→❝ ❧ơ② t❤ø❛ sk t❤❡♦ ❝ỉ♥❣ t❤ù❝ ❧➦♣ ổ ữủ t
t t trữợ tờ sk ✈➔ sk−1✳ ✣æ✐ ❦❤✐ t❛ ❝➛♥ ❝â ❜✐➸✉ t❤ù❝ ❝õ❛ sk
❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ σ1 ✈➔ σ2✳ ❈ỉ♥❣ t❤ù❝ t÷ì♥❣ ù♥❣ ✤÷đ❝ t➻♠ r❛ ♥➠♠ ✶✼✼✾ ❜ð✐
♥❤➔ t♦→♥ ❤å❝ ❆♥❤ ❊✳❲❛r✐♥❣✳
✶✳✶✳✶✵ ✣à♥❤ ❧➼ ✭❬✸❪✮✳ ✭❈ỉ♥❣ t❤ù❝ ❲❛r✐♥❣✮✳ ❚ê♥❣ ❧ơ② t❤ø❛ sk ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ q✉❛
❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì sð σ1 , σ2 t❤❡♦ ❝ỉ♥❣ t❤ù❝✿
[k/2]

sk
(−1)m (k − m − 1)! k−2m m
=
σ1
σ2 ,
k
m!
(k
−
2m)!
m=0
tr♦♥❣ ✤â [k/2] ❦➼ ❤✐➺✉ ❧➔ ♣❤➛♥ ♥❣✉②➯♥ ❝õ❛ ❦✴✷✳
❈❤ù♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤
♥↕♣✳ ❱ỵ✐ k = 1, k = 2 ❝ỉ♥❣ t❤ù❝ t÷ì♥❣

✭✶✳✷✮

❝ỉ♥❣ t❤ù❝ ✭✶✳✷✮ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉②
ù♥❣ ❝â ❞↕♥❣

1
1
s1 = σ1 , s2 = σ12 − σ2 .
2
2




ữ ợ k = 1, k = 2 ổ t❤ù❝ ✭✶✳✷✮ ✤ó♥❣✳ ●✐↔ sû ❝ỉ♥❣ t❤ù❝ ❲❛r✐♥❣ ✤➣
✤ó♥❣ ❝❤♦ s1, s2, ..., sk−1✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❝æ♥❣ t❤ù❝ ✤â ✤ó♥❣ ❝❤♦ sk ❝❤ó♥❣ t❛ sû
❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ✭✶✳✶✮✳ ❚❛ ❝â
1
1
sk = [σ1 sk−1 − σ2 sk−2 ]
k
k
k
(−1)m (k − m − 2)! k−2m−1 m
=
σ1 .
σ1
σ2 −
k−1
m!
(k
−
2m
−
2)!

m
k−2
−
σ2 .
k
=
1
−
k

1
k

(−1) (k − m − 2)!(k − 1) k−2m m
σ1
σ2
m! (k − 2m − 1)!

m

n

n
m

(−1)n (k − n − 3)! k−2n−1 n
σ
σ2
n! (k − 2n − 2)! 1

(−1)n (k − n − 3)!(k − 2) k−2n−2 n+1
σ1
σ2 .
n! (k − 2n − 2)!

❚r♦♥❣ tê♥❣ t❤ù ❤❛✐ t❤❛② n + 1 ❜ð✐ m✳ ❑❤✐ ✤â ❤❛✐ tê♥❣ ❝â t❤➸ ❦➳t ❤đ♣ t❤➔♥❤
♠ët ♥❤÷ s❛✉✿
1
1
sk =
k
k
1
−
k
=

1
k

m

m

(−1)m−1 (k − m − 2)!(k − 2) k−2m m
σ1
σ2
(m − 1)!(k − 2m)!

(−1)m (k − m − 2)!

❙û ❞ö♥❣ ❝→❝ ❝æ♥❣ t❤ù❝
t❛ ❝â

m

(−1)m (k − m − 2)!(k − 1) k−2m m
σ1
σ2 −
m! (k − 2m − 1)!

k−1
k−2
+
σ1k−2m σ2m .
m!(k − 2m − 1)! (m − 1)!(k − 2m)!

1
m
1
k − 2m
=
,
=
(m − 1)! m! (k − 2m − 1)! (k − 2m)!
k−1
k−2
k(k − m − 1)
+
=

.
m!(k − 2m − 1)! (m − 1)!(k − 2m)!
m!(k − 2m)!

❈✉è✐ ❝ò♥❣✱ ✈➻

(k − m − 2)!(k − m − 1) = (k − m − 1)!,

♥➯♥ t❛ ❝â ❝æ♥❣ t❤ù❝ ❝➛♥ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✿


✶✶

sk
k

[k/2]

=
m=0

m

(−1) (k−m−1)! k−2m m
σ2 .
m!(k−2m)! σ1

✣à♥❤ ❧➼ ✶✳✶✳✶✵ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❲❛r✐♥❣ ❞➵ ❞➔♥❣ ♥❤➟♥ ✤÷đ❝
❝→❝ ❜✐➸✉ t❤ù❝ ❝õ❛ sk = xk + yk t❤❡♦ σ1 = x + y, σ2 = xy s❛✉ ✤➙②✿ s1 = σ1,
s2 = σ1 2 − 2σ2 ,

s3 = σ1 3 − 3σ1 σ2 ,
s4 = σ1 4 − 4σ1 2 σ2 + 2σ2 2 ,
s5 = σ1 5 − 5σ1 3 σ2 + 5σ1 σ2 2 ,
s6 = σ1 6 − 6σ1 4 σ2 + 9σ1 2 σ2 2 − 2σ2 3 ,
s7 = σ1 7 − 7σ1 5 σ2 + 14σ1 3 σ2 2 − 7σ1 σ2 3 ,
s8 = σ1 8 − 8σ1 6 σ2 + 20σ1 4 σ2 2 − 16σ1 2 σ2 3 + 2σ2 4 ,
s9 = σ1 9 − 9σ1 7 σ2 + 27σ1 5 σ2 2 − 30σ1 3 σ2 3 + 9σ1 σ2 4 ,
s10 = σ1 10 − 10σ1 8 σ2 + 35σ1 6 σ2 2 − 50σ1 4 σ2 3 + 25σ1 2 σ2 4 − 2σ2 5 ,

✶✳✶✳✶✶ ✣à♥❤ ❧➼ ✭❬✸❪✮✳ ✭✣à♥❤ ❧➼ ❝ì ❜↔♥✮✳ ▼å✐ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ P (x, y) ❝õ❛ ❝→❝

❜✐➳♥ x, y ✤➲✉ õ t ữủ ữợ tự p(1, σ2) t❤❡♦ ❝→❝ ❜✐➳♥
σ1 = x + y, σ2 = xy, ♥❣❤➽❛ ❧➔
P (x, y) = p(σ1 , σ2 ).
✭✶✳✸✮
❈❤ù♥❣ rữợ t t trữớ ủ ỡ tự tr õ ❧ơ② t❤ø❛ ❝õ❛ x ✈➔ y
❝ị♥❣ ❜➟❝✱ ♥❣❤➽❛ ❧➔ ✤ì♥ t❤ù❝ ❞↕♥❣ xk yk . ❍✐➸♥ ♥❤✐➯♥ ❧➔
axk y k = a(xy)k = aσ2 k .

❚✐➳♣ t❤❡♦✱ ①➨t ✤ì♥ t❤ù❝ ❞↕♥❣ bxk yl (k = l)✳ ❱➻ ✤❛ t❤ù❝ ❧➔ ✤è✐ ①ù♥❣✱ ♥➯♥ ❝â sè
❤↕♥❣ ❞↕♥❣ bxl yk ✳ ✣➸ ①→❝ ✤à♥❤✱ t❛ ❣✐↔ sû k < l ✈➔ ①➨t tê♥❣ ❝õ❛ ❤❛✐ ✤ì♥ t❤ù❝ tr➯♥
b(xk y l + xl y k ) = bxk y l (xl−k + y l−k ) = bσ2 k sl−k .

❚❤❡♦ ❝æ♥❣ t❤ù❝ ❲❛r✐♥❣ 4sl−k ❧➔ ♠ët ✤❛ t❤ù❝ ❝õ❛ ❝→❝ ❜✐➳♥ σ1, σ2✱ ♥➯♥ ♥❤à t❤ù❝
♥â✐ tr➯♥ ❧➔ ♠ët ✤❛ t❤ù❝ ❝õ❛ σ1, σ2✳ ❱➻ ♠å✐ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❧➔ tê♥❣ ❝õ❛ ❝→❝ sè
❤↕♥❣ axk yk ✈➔ b(xk yl + xl yk )✱ ♥➯♥ ♠å✐ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ✤➲✉ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝ ð
❞↕♥❣ ✤❛ t❤ù❝ t❤❡♦ ❝→❝ ❜✐➳♥ σ1 ✈➔ σ2✳ ✣à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

✶✳✶✳✶✷ ✣à♥❤ ❧➼ ✭❬✸❪✮✳ ✭❚➼♥❤ ❞✉② ♥❤➜t✮✳ ◆➳✉ ❝→❝ ✤❛ t❤ù❝ ϕ(σ1, σ2) ✈➔ ψ(σ1, σ2)

❦❤✐ t❤❛② σ1 = x + y, σ2 = xy ❝❤♦ t❛ ❝ò♥❣ ♠ët ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ P (x, y)✱ t❤➻
❝❤ó♥❣ ♣❤↔✐ trị♥❣ ♥❤❛✉✱ ♥❣❤➽❛ ❧➔ ϕ(σ1, σ2) ≡ ψ(σ1, σ2)✳
❈❤ù♥❣ ♠✐♥❤✳ ✣➦t φ(σ1, σ2) = ϕ(σ1, σ2) − ψ(σ1, σ2)✳ ❑❤✐ ✤â t❤❡♦ ❣✐↔ t❤✐➳t t❛ ❝â✿
φ(x + y, xy) = ϕ(x + y, xy) − ψ(x + y, xy) = P (x, y) − P (x, y) = 0.


✶✷

❚❛ s➩ ❝❤ù♥❣ tä r➡♥❣ φ(σ1, σ2) ≡ 0✳ ❉➵ t❤➜② r➡♥❣✱ s❛✉ ❦❤✐ ♠ð ♥❣♦➦❝ t❤➻ ❜✐➸✉
t❤ù❝
k
l
❧➔ ♠ët ✤❛ t❤ù❝ ❝õ❛ ❝→❝
x, y ❧➔ xk+l y l ✳

f (x, y) := (x + y) (xy)
❜✐➳♥ x, y ✈➔ ❝â sè ❤↕♥❣ trë✐

♥❤➜t t❤❡♦ t❤ù tü ❝→❝ ❜✐➳♥

●✐↔ sû φ(σ1, σ2) ❝â ❞↕♥❣
Akl σ1k σ2l .

φ(σ1 , σ2 ) =
k,l

✣➸ t➻♠ sè ❤↕♥❣ trë✐ ♥❤➜t✱ ❝❤ó♥❣ t❛ s➩ ❝❤å♥ tr♦♥❣ φ(σ1, σ2) ❝→❝ sè ❤↕♥❣ ❝â k + l
❧➔ ❧ỵ♥ ♥❤➜t✳ ❚✐➳♣ t❤❡♦✱ tr♦♥❣ ❝→❝ sè ❤↕♥❣ ♥â✐ tr➯♥✱ ❝❤å♥ r❛ ❝→❝ sè ❤↕♥❣ ✈ỵ✐ ❣✐→
trà ❧ỵ♥ ♥❤➜t ❝õ❛ l✳ ❱➼ ❞ö✱ ♥➳✉
φ(σ1 , σ2 ) = −σ1 4 σ2 − 4σ1 2 σ2 3 + 2σ1 σ2 4 − 6σ1 σ2 2 + 10σ2 3 − 7σ1 + 5σ2 + 1

t❤➻ sè ❤↕♥❣ ✤✉ì❝ ❝❤å♥ s➩ ❧➔ 2σ1σ24.

◆❤÷ ✈➟②✱ ❣✐↔ sû ❝❤å♥ ✤÷đ❝ ✤ì♥ t❤ù❝ Aσ1mσ2n✳ ❑❤✐ ✤â✱ ♥➳✉ t❤❛② σ1 =
x + y, σ2 = xy ✱ t❤➻ sè ❤↕♥❣ trë✐ ♥❤➜t ❝õ❛ φ s➩ ❧➔ Axm+n y n ✳ ❚❤➟t ✈➟②✱ ❣✐↔
sû Bσ1k σ2l ❧➔ ✤ì♥ t❤ù❝ tũ ỵ ợ Axm+nyn õ t ồ t❛ ❝â
❤♦➦❝ m + n > l + l✱ ❤♦➦❝ m + n = k + l✱ ♥❤÷♥❣ n > l✳ ❚r♦♥❣ ❝↔ ❤❛✐ tr÷í♥❣ ❤đ♣
t❤➻ Axm+nyn trë✐ ❤ì♥ Bxk+l yl ✳
◆❤÷ ✈➟② ❝❤ó♥❣ t❛ ✤➣ ❝❤ù♥❣ tä r➡♥❣ Axm+nyn ❧➔ ✤ì♥ t❤ù❝ trë✐ ♥❤➜t ❝õ❛
φ(x+y, xy)✱ ♥➯♥ φ(x+y, xy) = 0, ∀x, y ✱ ♥➳✉ φ(σ1 , σ2 )✳ ❱➟②✱ t❛ ❝â φ(σ1 , σ2 ) ≡ 0✳
✣à♥❤ ❧➼ ✤✉ì❝ ❝❤ù♥❣ ♠✐♥❤✳
✣➸ ♠✐♥❤ ❤å❛✱ ①➨t ✈➼ ❞ư s❛✉ ✤➙②✳
✶✳✶✳✶✸ ❱➼ ❞ö ✭❬✸❪✮✳ ❇✐➸✉ ❞✐➵♥ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣
f (x, y) = x5 + x4 y + x3 y 3 + xy 4 + y 5

❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❲❛r✐♥❣ t❛ ❝â
f (x, y) = (x5 + y 5 ) + xy(x3 + y 3 ) + (xy)3
= s5 + σ 2 s3 + σ 2 2
= (σ1 5 − 5σ1 3 σ2 + 5σ1 σ2 2 ) + σ2 (σ1 3 − 3σ1 σ2 ) + σ2 3
= σ1 5 − 4σ1 3 σ2 + 2σ1 σ2 2 + σ2 3 .


✶✸

✶✳✷

Ù♥❣ ❞ö♥❣ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❤❛✐ ❜✐➳♥

✶✳✷✳✶ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ❤❛✐ ➞♥

●✐↔ sû P (x, y) ✈➔ Q(x, y) ❧➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✳ ❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤✿
P (x, y) = 0
Q(x, y) = 0.

✭✶✳✹✮

❇➡♥❣ ❝→❝❤ ✤➦t x + y = σ1, xy = 2 tr ỡ s ỵ ỡ t ữ ❤➺ ✭✶✳✹✮
✈➲ ❞↕♥❣✿
p(σ1 , σ2 ) = 0
✭✶✳✺✮
q(σ1 , σ2 ) = 0.
❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ t❤÷í♥❣ ✤ì♥ ❣✐↔♥ ❤ì♥ ❤➺ ✭✶✳✹✮ ✈➔ t❛ ❝â t❤➸ ❞➵ ❞➔♥❣ t➻♠
✤÷đ❝ ♥❣❤✐➺♠ (σ1, σ2)✳ ❙❛✉ ❦❤✐ t➻♠ ✤÷đ❝ ❝→❝ ❣✐→ trà ❝õ❛ σ1, σ2✱ ❝➛♥ ♣❤↔✐ t➻♠ ❝→❝
❣✐→ trà ❝õ❛ ❝→❝ ➞♥ sè x ✈➔ y ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✹✮✳ ✣✐➲✉ ♥➔② ❝â t❤➸ t❤ü❝ ❤✐➺♥
✤÷đ❝ ♥❤í ✤à♥❤ ❧➼ s❛✉ ✤➙②✳

✶✳✷✳✶ ✣à♥❤ ❧➼ ✭❬✸❪✮✳ ●✐↔ sû σ1 ✈➔ σ2 ❧➔ ❝→❝ sè t❤ü❝ ♥➔♦ ✤â✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤
❜➟❝ ❤❛✐

✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤

z 2 − σ1 z + σ2 = 0
x + y = σ1
xy = σ2 .

✭✶✳✻✮
✭✶✳✼✮

❧✐➯♥ ❤➺ ✈ỵ✐ ♥❤❛✉ ♥❤÷ s❛✉✿ ♥➳✉ z1 , z2 ❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮ t❤➻
❤➺ ✭✶✳✼✮ ❝â ♥❣❤✐➺♠

x = z1
y = z2 ;

x = z2
y = z1

✈➔ ♥❣♦➔✐ r❛ ❦❤æ♥❣ ❝á♥ ❝â ♥❣❤✐➺♠ ♥➔♦ ❦❤→❝✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ x = a, y = b ❧➔
♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✼✮ t❤➻ ❝→❝ sè a, b ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮✳
❈❤ù♥❣ ♠✐♥❤✳

❱✐❡t✿

◆➳✉ z1, z2 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮✱ t❤➻ t❤❡♦ ❝ỉ♥❣ t❤ù❝
z 1 + z 2 = σ1 , z1 z 2 = σ2 ,


✶✹

s✉② r❛

x2 = z 2
y =2 z 1

x1 = z 1
y1 = z 2

❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✼✮✳ ❱➜♥ ✤➲ ❦❤ỉ♥❣ ❝á♥ ❝â ♥❣❤✐➺♠ ♥➔♦ ❦❤→❝ s➩ ✤÷đ❝ s✉②
r❛ tứ s ũ ừ ỵ s ữủ ự ữợ
sỷ 4x = a, y = b ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✼✮✱ ♥❣❤➽❛ ❧➔

a + b = σ1 , ab = σ1 σ2 .

❑❤✐ ✤â t❛ ❝â

z 2 − σ1 z + σ2 = z 2 − (a + b)z + ab = (z − a)(z − b).

✣✐➲✉ ✤â ❝❤ù♥❣ tä r➡♥❣ ❝→❝ sè a, b ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ ✭✶✳✻✮✳
✣à♥❤ ❧➼ ữủ ự
ố ũ ú ỵ r ✈➔ ✤õ ✤➸ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮ ❝â ♥❣❤✐➺♠ ❧➔
∆ = σ12 − 4σ2 ≥ 0.

✣➸ ♠✐♥❤ ❤å❛ ①➨t ✈➼ ❞ö s❛✉ ✤➙②✳
✶✳✷✳✷ ❱➼ ❞ư ✭❬✸❪✮✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤

✭✶✳✽✮

x3 + y 3 = 35
x + y = 5.

✣➦t x + y = σ1, xy = σ2✳ ❚❛ ❝â
❉♦ ✤â t❛ ❝â ❤➺

x3 + y 3 = σ1 3 − 3σ1 σ2 .
σ1 3 − 3σ1 σ2 = 35
σ1 = 5.

❚ø ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t➻♠ ✤÷đ❝ σ2
♣❤÷ì♥❣ tr➻♥❤

= 6✳

x+y =5
xy = 6.

❑❤✐ ✤â x, y s➩ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺


✶✺

❚❤❡♦ ✣à♥❤ ❧➼ ✶✳✷✳✶✱ x ✈➔ y ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
z 2 − 5z + 6 = 0

✈➔ t❛ t➻♠ ✤÷đ❝ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦
x2 = 3
y2 = 2.

x1 = 2
y1 = 3
✶✳✷✳✸ ❱➼ ❞ö

✭❬✸❪✮✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤
x3 − y 3 = 5
xy 2 − x2 y = 1.

❍➺ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❝❤÷❛ ♣❤↔✐ ❧➔ ❤➺ ✤è✐ ①ù♥❣✳ ◆➳✉ ✤➦t z = −y t❤➻ t❛ ❝â ❤➺
♣❤÷ì♥❣ tr➻♥❤
x3 + z 3 = 5
xz 2 + x2 z = 1.

❧➔ ❤➺ ✤è✐ ①ù♥❣ ✈ỵ✐ x ✈➔ 4z ✳ ✣➦t σ1 = x + z ✈➔ σ2 = xz t❛ ❝â ❤➺ ♣❤÷ì♥❣ tr➻♥❤

σ1 (σ12 − 3σ2 ) = 5
⇔
σ1 σ2
=1

σ1 σ2 = 1
σ13 − 3 = 5.

❍➺ ❝✉è✐ ❝ò♥❣ ❝â ♥❣❤✐➺♠ σ1 = 2, σ2 = 12 ✳ ❉♦ ✤â t❛ ❝â ❤➺ ♣❤÷ì♥❣ tr➻♥❤
●✐↔✐ ❤➺ t❛ ❝â ❝→❝ ♥❣❤✐➺♠

x+z =2
⇔
xz = 21

x1 =
y1 =
✶✳✷✳✹ ❱➼ ❞ư

♣❤÷ì♥❣ tr➻♥❤

√
2+ 2
2 √
−2+ 2
2

⇔

x−y =2

xy
= − 21 .
x2 =
y2 =

√
2− 2
2 √
−2− 2
.
2

✭❬✸❪✮✳ ✭❚❤✐ ❍❙● ❧ỵ♣ ✶✵ t❤➔♥❤ ♣❤è ❍➔ ◆ë✐✱ ✶✾✾✾✲✷✵✵✵✮✳ ❈❤♦ ❤➺
x2 + y 2 + x + y = 18
xy(x + 1)(y + 1) = m.

❛✮ ●✐↔✐ ❤➺ ✈ỵ✐ ♠❂✼✷✳
❜✮ ❚➻♠ t➜t ❝↔ ❝→❝ ❣✐→ trà ❝õ❛ t❤❛♠ sè ♠ ✤➸ ❝â ♥❣❤✐➺♠✳


✶✻

❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔ ❤➺ ✤è✐ ①ù♥❣ t❤❡♦ ❝→❝ ❜✐➳♥ x, y✳ ❚✉② ♥❤✐➯♥✱ ♥➳✉ t❛
✤➦t σ1 = x + y, σ2 = xy t❤➻ s➩ ❣➦♣ ❦❤â ❦❤➠♥ ❦❤✐ ♣❤↔✐ ✤÷❛ ✈➲ ❤➺ ❜➟❝ ✷ t❤❡♦
σ1 , 2 t õ t số ỵ r➡♥❣✱ ♥➳✉ ✈✐➳t ♣❤÷ì♥❣ tr➻♥❤ ✤➛✉ ❝õ❛ ❤➺ ð
❞↕♥❣
x(x + 1) + y(y + 1) = 18

t❤➻ ❤➺ ❝ô♥❣ ✤è✐ ①ù♥❣ t❤❡♦ ❝→❝ ❜✐➳♥

X = x(x + 1), Y = y(y + 1)

✈➔ ❝â ❞↕♥❣

X + Y = 18
XY = m.

❚❛ ❝â X, Y ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤
✭✶✳✾✮

t2 18t + m = 0.

ợ ữỡ tr➻♥❤ ✭✶✳✾✮ trð t❤➔♥❤
❚ø ✤â t❛ t➻♠ ✤÷đ❝

t2 − 18t + 72 = 0.
X = 12, 1
Y = 6, 1

X = 6, 1
Y = 12.

❚rð ❧↕✐ ❝→❝❤ ✤➦t ➞♥ t❛ ❝â ❤➺ s❛✉✿
x(x + 1) = 12
y(y + 1) = 6

x(x + 1) = 6
y(y + 1) = 12.

●✐↔✐ ❝→❝ ❤➺ ♥➔② t❛ t➻♠ ✤÷đ❝ ✽ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✤➣ ❝❤♦ ❧➔✿

x=3
y=2
x=2
y=3

❜✮ ❉➵ t❤➜② r➡♥❣

x=3
y = −3

x = −3
y=3

x = −4
y=2
x=2
y = −4

1
1
1
x(x + 1) = (x + )2 − ≥ − .
2
4
4

x = −4
y = −3

x = −3

y = −4.


✶✼

❉♦ ✤â tø ❝→❝❤ ✤➦t ➞♥ X, Y s✉② r❛ ✤✐➲✉ ❦✐➺♥ ❝õ❛ X, Y ❧➔✿ X ≥ − 41 , Y
❦➼ ❤✐➺✉

≥ − 14 .

f (t) = t2 − 18t + m.

❑❤✐ ✤â ❤➺ ❜❛♥ ✤➛✉ ❝â ♥❣❤✐➺♠ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✾✮ ❝â ❤❛✐ ♥❣❤✐➺♠
1
X1 , X2 t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿ X1 ≥ X2 ≥ − 4 . ❚ø ✤â t❛ ❝â ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤

∆′ ≥ 0
 81 − m ≥ 0
73
73
1
1.f (− 4 ) ≥ 0 ⇔ m + 76
≥ 0 ⇔ − ≤ m ≤ 81.


16
s
− 14
9 ≥ − 14

2 ≥



✶✳✷✳✺ ❱➼ ❞ö

✣➦t

√
4

✭❬✸❪✮✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❝➠♥ t❤ù❝
√
4

√
4

x = y, 97 − x = z ✳

97 − x +

√
4

x = 5.

❑❤✐ ✤â t❛ ❝â ❤➺

y+z =5

y 4 + z 4 = 97.

✣➦t σ1 = y + z, σ2 = yz ✳ ❚❛ ❝â ❤➺ ♣❤÷ì♥❣ tr➻♥❤
σ1 = 5
σ14 − 4σ12 σ2 + 2σ22 = 97.

ứ õ t õ ữỡ tr ố ợ σ2

σ22 − 50σ2 + 264 = 0.

P❤÷ì♥❣ tr➻♥❤ ♥➔② ❝â ❝→❝ ♥❣❤✐➺♠✿ σ1 = 6, σ2 = 44✳ ◆❤÷ ✈➟②✱ ❜➔✐ t♦→♥ ❞➝♥ ✤➳♥
❣✐↔✐ ❤❛✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤
y+z =5
yz = 6

y+z =5
yz = 44.

❍➺ t❤ù ♥❤➜t ❝â ❝→❝ ♥❣❤✐➺♠
y1 = 2
z2 = 3

y2 = 3
z2 = 2.

✤â t➻♠ ✤÷đ❝ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✿ x1 = 16, x2 = 81✳ ❉➵ t❤➜②
r➡♥❣ ❤➺ ❝á♥ ❧↕✐ ✈æ ♥❣❤✐➺♠✳
❱➟② ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔ x1 = 16, x2 = 81.

✶✽

✶✳✷✳✻ ❱➼ ❞ư✳

√

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤
x+

17 − x2 + x 17 − x2 = 9.
√
❦✐➺♥ ❝õ❛ x, y ❧➔ |x| ≤ 17, y ≥ 0✳

✣➦t y = 17 − x2✳ ✣✐➲✉
tr➯♥ t ữỡ tr tữỡ ữỡ ợ s

ợ ❝→❝ ✤✐➲✉ ❦✐➺♥

x + y + xy = 19
x2 + y 2 = 17.

✣➦t σ1 = x + y, σ2 = xy✳ ❑❤✐ ✤â t❛ ❝â ❤➺ ♣❤÷ì♥❣ tr➻♥❤
σ1 + σ2 = 9
σ12 − 2σ2 = 17.

❚ø ♣❤÷ì♥❣ tr➻♥❤ ✤➛✉ ❝õ❛ ❤➺ tr➯♥✱ t❛ ❝â σ2 = 9 − σ1✳ ❚❤❛② ❜✐➸✉ t❤ù❝ tr➯♥ ✈➔♦
♣❤÷ì♥❣ tr➻♥❤ ❝á♥ ❧↕✐ ❝õ❛ ❤➺✱ t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤
σ1 2 + 2σ1 − 35 = 0.

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t❛ ❝â ❝→❝ ♥❣❤✐➺♠ σ1 = 5, σ1 = −7✳ ❱ỵ✐ σ1 = 5✱ t❛ ❝â

σ2 = 4✱ ✈ỵ✐ 4σ1 = −7✱ t❤➻ 4σ2 = 16✳
❱ỵ✐ σ1 = 5, σ2 = 4 t❤➻ x, y ❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t❛ ✤÷đ❝

t2 − 5t + 4 = 0.

x=1
y=4

x=4
y = 1.

❱ỵ✐ σ1 = −7, σ2 = 16 t❤➻ x, y ❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
t2 + 7t + 16 = 0.

❉➵ t❤➜② r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ✤➙② ✈ỉ ♥❣❤✐➺♠✳ ❱➟② ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
✤➣ ❝❤♦ ❧➔✿ x = 1 ✈➔ x = 4✳
√
❇➻♥❤ ❧✉➟♥✿ ◆❣♦➔✐ r❛✱ t❛ ❝â t❤➸ ❣✐↔✐ t❤❡♦ ❝→❝❤ ✤➦t t = x + 17 − x2✳
✶✳✷✳✼ ❱➼ ❞ö✳ ❚➻♠ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
x3 + y 3 + 1 = 3xy.


✶✾

✣➦t σ1 = x + y, σ2 = xy✳ P❤÷ì♥❣ tr➻♥❤ trð t❤➔♥❤
σ13 − 3σ1 σ2 + 1 = 3σ2 ⇔ (σ1 + 1) σ12 − σ1 + 1 − 3σ2 = 0.
❚r÷í♥❣ ❤đ♣ ✶✿ σ1 + 1 = 0, t❛ ❝â x + y + 1 = 0, ♣❤÷ì♥❣ tr➻♥❤ ❝â ✈æ sè ♥❣❤✐➺♠
♥❣✉②➯♥ x ∈ Z ✈➔ y = −1 − y.
❚r÷í♥❣ ❤đ♣ ✷✿ σ1 2 − σ1 + 1 − 3σ2 = 0. ❚❛ ✈✐➳t ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ð ❞↕♥❣

σ1 2 − σ1 + 1 = 3σ2

✈➔ sû ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✶✳✽ t❛ ❝â

3
σ12 − σ1 + 1 ≤ σ12 ⇔ σ12 − 4σ1 + 4 ≤ 0
4
2
⇔ (σ1 − 2) ≤ 0 ⇔ σ1 = 2 ⇒ σ2 = 1.

❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝â ❤➺

x+y =1
xy = 1.

❍➺ ♥➔② ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❧➔ x = y = 1✳ ◆❤÷ ✈➟②✱ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
✤➣ ❝❤♦ ❧➔
x=1
y=1

✶✳✷✳✽ ❱➼ ❞ö✳

,

x∈Z
y = −1 − x.

❈❤ù♥❣ ♠✐♥❤ r➡♥❣✱ ♥➳✉ ❝→❝ sè u, v, x, y t❤ä❛ ♠➣♥ ❝→❝ ❤➺ t❤ù❝

t❤➻ ✈ỵ✐ ♠å✐ sè tü

u + v = x + y, u2 + v 2 = x2 = y 2
♥❤✐➯♥ n t❛ ❝â un + vn = xn + yn.

✣➦t

σ1 = x + y, σ2 = xy,
α1 = u + v, α2 = uv.

❑❤✐ ✤â
x+y =u+v
x 2 + y 2 = u2 + v 2

⇔

x+y =u+v
⇔
(x + y)2 − 2xy = (u + v)2 − 2uv

σ1 = α 1
2 = 2 .

ỵ ỡ t ♠é✐ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ✤➲✉ ❜✐➸✉ ❞✐➵♥ ❞✉② ♥❤➜t q✉❛ ✤❛ t❤ù❝
❝õ❛ ❝→❝ ❜✐➳♥ ❧➔ ❝→❝ ✤❛ t❤ù❝ ✤â✐ ①ù♥❣ ❝ì sð✳ ●✐↔ sû xn +yn = ϕ(α1, α2)✳ t❤➳ t❤➻ t❛
❝â un + vn = ϕ(α1, α2)✳ ❉♦ α1 = σ1, α2 = σ2✱ ♥➯♥ t❛ ❝â ϕ(σ1, σ2) = ϕ(α1, α2)✳
❚ø ✤â s✉② r❛ xn + yn = un + vn..✳


✷✵

✶✳✷✳✷ ❱➲ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐

◆❤✐➲✉ ❜➔✐ t♦→♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ ✤÷đ❝ ❣✐↔✐ ♠ët ❝→❝❤ ❞➵ ❞➔♥❣ ♥❤í →♣
❞ư♥❣ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✳ ✣➸ ♠✐♥❤ ❤å❛✱ ①➨t ♠ët sè ✈➼ ❞ö s❛✉✳
✶✳✷✳✾ ❱➼ ❞ö ✭❬✷❪✮✳ ●✐↔ sû x1 , x2 ❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐
ax2 + bx + c = 0, (a = 0).

❱ỵ✐ n ❧➔ sè ♥❣✉②➯♥✱ ✤➦t Sn = x1n + x2n.
❛✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
aSn+2 + bSn+1 + cSn = 0.

❜✮ ⑩♣ ❞ư♥❣✿ ❑❤ỉ♥❣ ❦❤❛✐ tr✐➸♥✱ ❤➣② t➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝
A= 1+

❛✮ ❚❛ ❝â
❉♦ ✤â

√

2

5

+ 1−

√

5

2 .

n
n
xn+2
+ xn+2
= (xn+1
+ xn+1
1
2
1
2 )(x1 + x2 ) − (x1 + x2 )x1 x2 .

Sn+2 = Sn+1 (x1 + x2 ) − Sn x1 x2 .

❚r♦♥❣ ❜✐➸✉ t❤ù❝ tr➯♥ t❤❛② x1 + x2 = − ab ✈➔ x1x2 = ac t❛ ✤÷đ❝
c
b
Sn+2 = − Sn+1 − Sn ,
a
a

❤❛②
❜✮ ✣➦t

aSn+2 + bSn+1 + cSn = 0.
√
x1 = 1 + 2, x2 = 1 − 2 t❤➻ x1 , x2 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
√
√
x − (1 + 2) x − (1 − 2) = 0 ⇔ x2 − 2x 1 = 0.

ợ ữỡ tr tr tự tr ❝➙✉ ❛✮ trð t❤➔♥❤

Sn+2 = 2Sn+1 + Sn (Sn = x1 n + x2 n ).

❚r♦♥❣ ✤➥♥❣ t❤ù❝ tr➯♥ ❧➛♥ ❧÷đt ❝❤♦ n = 0, 1, 2, 3, 4 t❛ t➼♥❤ ✤÷đ❝
S2 = 2S1 + S0 = 2.2 + 2 = 6, S3 = 2S2 + S1 = 2.6 + 2 = 14,

✭✶✳✶✵✮


✷✶

❱➟②

S4 = 2S3 + S2 = 2.14 + 6 = 34, S5 = 2S4 + S3 = 2.34 + 14 = 82.

A = (1 +

✶✳✷✳✶✵ ❱➼ ❞ö

0, (a = 0)✳

√

2)5 + (1 −

√

2)5 = 82.

✭❬✷❪✮✳ ●✐↔ sû x1, x2 ❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ax2 +bx+c =

❚❤➔♥❤ ❧➟♣ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ ❝â ❝→❝ ♥❣❤✐➺♠ ❧➔ x12 ✈➔ x22✳
✣➦t σ1 = x1 + x2, σ2 = x1x2, s2 = x12 + x22, s = y1 + y2, p = y1y2✳ ❚❛ ❝â
s2 = σ1 2 − 2σ2 ✳ ❚❤❡♦ ✣à♥❤ ❧➼ ❱✐❡t❡ t❛ ❝â
c
b
σ 1 = x 1 + x 2 = − , σ2 = x 1 x 2 = .
a
a

❉♦ ✤â
2

s = s2 = x 1 + x 2

2

b
= −
a

2

c
c
− 2 , p = σ 2 2 = x1 2 x2 2 =
a

a

2

.

❱➟② ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ ❝➛♥ ❧➟♣ s➩ ❧➔

y 2 − sy + p = 0.

❤❛②

a2 y 2 − (b2 − 2ac)y + c2 = 0.

✶✳✷✳✶✶ ❱➼ ❞ư✳

❚❤➔♥❤ ❧➟♣ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ z 2 + pz + q = 0 ❝â ❝→❝ ♥❣❤✐➺♠
z1 = x1 6 − 2x2 2 , z2 = x2 6 − 2x1 2

tr♦♥❣ ✤â x1, x2 ❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✿ x2 − x − 3 = 0.
❚❤❡♦ ✤à♥❤ ❧➼ ❱✐❡t❡ t❛ ❝â
▼➦t ❦❤→❝✱ t❛ ❝â

σ1 = x1 + x2 = 1, σ2 = x1 x2 = −3.

−p = z1 + z2 = (x1 6 − 2x2 2 ) + (x2 6 − 2x1 2 )
= (x1 6 + x2 6 ) − 2(x1 2 + x2 2 )
= s6 − 2s2
= (σ1 6 − 6σ1 4 σ2 + 9σ1 2 σ2 2 − 2σ2 3 ) − 2(σ1 2 − 2σ2 ) = 140.