Về một lớp đa thức đối xứng

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕

◆●❯❨➍◆ ❚❍➚ ▲❯❾◆

❱➋ ▼❐❚ ▲❰P ✣❆ ❚❍Ù❈ ✣➮■ ❳Ù◆●
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✷✵


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕

◆●❯❨➍◆ ❚❍➚ ▲❯❾◆

❱➋ ▼❐❚ ▲❰P ✣❆ ❚❍Ù❈ ✣➮■ ❳Ù◆●
❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ t♦→♥ sì ❝➜♣
▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✸

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈
❚❙✳ ◆●➷ ❚❍➚ ◆●❖❆◆

❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✷✵

✐

▲í✐ ❝↔♠ ì♥
▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ t❤ü❝ ❤✐➺♥ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ ✣↕✐ ❤å❝
❚❤→✐ ◆❣✉②➯♥ ✈➔ ❤♦➔♥ t ữợ sỹ ữợ ừ ổ
❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ s s tợ ữớ ữợ
ồ ừ ữớ t ự tớ
ữợ ❞➝♥ ✈➔ t➟♥ t➻♥❤ ❣✐↔✐ ✤→♣ ♥❤ú♥❣ t❤➢❝ ♠➢❝ ❝õ❛ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→
tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✳
❚→❝ ❣✐↔ ❝ô♥❣ ✤➣ ❤å❝ t➟♣ ✤÷đ❝ r➜t ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ❝❤✉②➯♥ ♥❣➔♥❤ ❜ê ➼❝❤
❝❤♦ ❝æ♥❣ t→❝ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❜↔♥ t❤➙♥✳ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❝↔♠ ì♥ s➙✉
s➢❝ tỵ✐ ❝→❝ t❤➛② ❣✐→♦✱ ❝ỉ ❣✐→♦ ✤➣ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕② ❧ỵ♣ ❈❛♦ ❤å❝ ❚♦→♥❀
◆❤➔ tr÷í♥❣ ✈➔ ❝→❝ ♣❤á♥❣ ❝❤ù❝ ♥➠♥❣ ❝õ❛ ❚r÷í♥❣❀ ❑❤♦❛ ❚♦→♥ ✕ ❚✐♥✱ tr÷í♥❣
✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ q✉❛♥ t➙♠ ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔
tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳
❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ t➟♣ t❤➸ ❧ỵ♣ ❈❛♦ ❤å❝ ❚♦→♥ ❑✶✷❇ ✤➣
❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥ ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ r➜t ♥❤✐➲✉ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠
❧✉➟♥ ✈➠♥✳
❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣
❣✐ó♣ ✤ï ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ❝❤♦ tæ✐ ❦❤✐ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳
❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✼ ♥➠♠ ✷✵✷✵
❚→❝ ❣✐↔
◆❣✉②➵♥ ❚❤à ▲✉➟♥


✐✐

▼ư❝ ❧ư❝
▲í✐ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶
❈❤÷ì♥❣ ✶✳ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà sỡ ỗ t ✷

✶✳✶✳ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

✶✳✷✳ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà ❤❛✐ ❜✐➳♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✸✳ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà ❜❛ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



ỡ ỗ t ✤❛ t❤ù❝ ❜❛ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✷✳✶✳ ❍å ✤❛ t❤ù❝ {Fm } ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✷✳✷✳ ❍å ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà {Sm } ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



ữỡ ởt ợ tự ố ự ỹ tr ✈➔ ✤❛ t❤ù❝ s❤❛r♣
✤➦❝ ❜✐➺t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶


✐✐✐

▲í✐ ♠ð ✤➛✉
▲✉➟♥ ✈➠♥ ❝â ♠ư❝ ✤➼❝❤ t➻♠ ❤✐➸✉ ✈➲ ❤❛✐ ❜➔✐ t♦→♥ ❝ü❝ trà ✈➲ ✤❛ t❤ù❝ t❤✉➛♥
♥❤➜t✳ ❈→❝ ❜➔✐ t♦→♥ ♥➔② ❝â ❝→❝ ❧í✐ ❣✐↔✐ ✤ì♥ ❣✐↔♥ ❝❤♦ ✤❛ t❤ù❝ ♠ët ❤♦➦❝ ❤❛✐
❜✐➳♥ ✈➔ trð ♥➯♥ ♣❤ù❝ t↕♣ ✈➔ t❤ó ✈à ✈ỵ✐ ✤❛ t❤ù❝ ❜❛ ❤♦➦❝ ♥❤✐➲✉ ❜✐➳♥✳ ❚❛ s➩
t➻♠ ❤✐➸✉ ✈➲ ♠ët ❤å ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ t❤✉➛♥ ♥❤➜t ❜❛ ❜✐➳♥ ♥❤÷ ✈✐➺❝ ❣✐↔✐
q✉②➳t ♥❤ú♥❣ ❜➔✐ t♦→♥ tr➯♥ ✈➔ tr➻♥❤ ❜➔② ❝→❝ t➼♥❤ ❝❤➜t t❤ó ✈à ❦❤→❝ ❝õ❛ ❤å
✤❛ t❤ù❝ ♥➔②✳ ❱➼ ❞ö✿ ❈→❝ ❤➺ sè ❝õ❛ ❝❤ó♥❣ ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❝â t❤➸ ✤÷đ❝ ❜✐➸✉
t❤à ữợ tờ ừ số tự sð ❤ú✉ ♠ët t➼♥❤ ❝❤➜t ❝❤✐❛
❤➳t✳ ❍ì♥ ♥ú❛✱ ❝→❝ ✤❛ tự ữủ t ố ỡ ợ ởt t ❤đ♣ ❝→❝
✤❛ t❤ù❝ ✤÷đ❝ s✐♥❤ r❛ ♥❤÷ ♥❤ú♥❣ ✈➼ ❞ư s❤❛r♣ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲ →♥❤ ①↕
✤❛ t❤ù❝ r✐➯♥❣ ❣✐ú❛ ❝→❝ ❤➻♥❤ ❝➛✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ ♣❤ù❝✳
◆❤✐➲✉ ❦➳t q✉↔ tr t ồ ồ ỵ rst r➡♥❣
♥❤ú♥❣ ✤è✐ t÷đ♥❣ t❤ä❛ ♠➣♥ ♠ët sè ✤✐➲✉ ❦✐➺♥ ❝ü❝ trà ❝â ❝→❝ t➼♥❤ ❝❤➜t r➜t
✤➦❝ ❜✐➺t✳ ❈❤ó♥❣ t❛ ❝â ởt ồ ỡ ừ ỵ ❝❤➥♥❣
❤↕♥✿ ❚r♦♥❣ sè t➜t ❝↔ ❝→❝ ❤➻♥❤ ❝❤ú ♥❤➟t ❝â ❝ị♥❣ ❝❤✉ ✈✐✱ ❤➻♥❤ ❝â ❞✐➺♥ t➼❝❤
❧ỵ♥ ♥❤➜t ❧➔ ♠ët ❤➻♥❤ ✈✉ỉ♥❣✳ ▼ët ✈➼ ❞ư t÷ì♥❣ tü✱ r➡♥❣ ✤ë ❞➔✐ L ❝õ❛ ♠ët
✤÷í♥❣ ❝♦♥❣ ❦➼♥ tr♦♥❣ ♠➦t ♣❤➥♥❣ ✈➔ ❞✐➺♥ t➼❝❤ A ❝õ❛ ♠✐➲♥ ♣❤➥♥❣ ❣✐ỵ✐ ❤↕♥
❜ð✐ L ❧✉ỉ♥ t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ 4πA ≤ L2 ✳ ✣÷í♥❣ ❝♦♥❣ ❝ü❝ trà ♠➔
✤➥♥❣ t❤ù❝ ✤↕t ✤÷đ❝ ✤â ❧➔ ✤÷í♥❣ trá♥✳
✣è✐ ợ t ý tự p ừ p ỵ ❤✐➺✉ ❧➔ R(p)✱ ❧➔ sè ✤ì♥ t❤ù❝
♣❤➙♥ ❜✐➺t ①✉➜t ❤✐➺♥ tr♦♥❣ p ✈ỵ✐ ❤➺ sè ❦❤→❝ ❦❤ỉ♥❣✳ ✣➲ t➔✐ ✤➦t r❛ ♠ö❝ ✤➼❝❤
t➻♠ ❤✐➸✉ ❤❛✐ ❝➙✉ ❤ä✐ s❛✉ ✤➙② ✈➲ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t✿

❈➙✉ ❤ä✐ ✶✳ ❚r♦♥❣ t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ t❤✉➛♥ ♥❤➜t p ❜➟❝ m t❤♦↔

♠➣♥ p = sq ✈ỵ✐ q ❧➔ ✤➛② ✤õ✱ ❤↕♥❣ ❜➨ ♥❤➜t ❝â t❤➸ ❝õ❛ p ❧➔ ❣➻❄ ❈→❝ ✤❛ t❤ù❝
❝â ❤↕♥❣ ❜➨ ♥❤➜t ♥➔② ❧➔ ❝→❝ ✤❛ t❤ù❝ ♥➔♦❄

❈→❝ ✤❛ t❤ù❝ ♥❤÷ ✈➟② ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà✳ ❈➙✉ ❤ä✐ ✶ ❝â ❧✐➯♥ q✉❛♥

✤➳♥ ❝➙✉ ❤ä✐ s❛✉✿


✶

❈➙✉ ❤ä✐ ✷✳ ❚r♦♥❣ sè t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t p ❜➟❝ m t❤♦↔ ♠➣♥

p = sq ✈ỵ✐ q ❧➔ ✤➛② ✤õ✱ ❤↕♥❣ ♥❤ä ♥❤➜t ❝õ❛ p ❝â t❤➸ ❧➔ ❜❛♦ ♥❤✐➯✉❄ ❈→❝ ✤❛
t❤ù❝ ❝â ❤↕♥❣ ❜➨ ♥❤➜t ♥➔② ❧➔ ❝→❝ ✤❛ t❤ù❝ ♥➔♦❄
❈→❝ ✤❛ t❤ù❝ ♥❤÷ ✈➟② ❧➔ ✤❛ t❤ù❝ s❤❛r♣✳
✣è✐ ✈ỵ✐ ✤❛ t❤ù❝ ♠ët ❜✐➳♥✱ ♥❤ú♥❣ ❝➙✉ ❤ä✐ ♥➔② ❦❤ỉ♥❣ t❤ó ✈à ❜ð✐ ✈➻ ♠é✐
✤❛ t❤ù❝ ❦❤→❝ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ❜➟❝ m ❝❤➾ ❧➔ ♠ët ❜ë✐ ❝õ❛ xm ✈➔ ❞♦ ✤â ♥â
❝â ❤↕♥❣ ❜➡♥❣ 1✳ ✣è✐ ✈ỵ✐ ✤❛ t❤ù❝ ❤❛✐ ❜✐➳♥✱ ❝➙✉ tr↔ ❧í✐ ❝ơ♥❣ ❦❤→ ✤ì♥ ❣✐↔♥✳
◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❧✉➟♥ ✈➠♥ ✤✐ s➙✉ ✈➔♦ ✈✐➺❝ t➻♠ ❤✐➸✉ ❝➙✉ tr↔ ❧í✐ ❝❤♦ ❝→❝ ❝➙✉
❤ä✐ tr➯♥ ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜❛ ❜✐➳♥✳
❈➜✉ tró❝ ❝õ❛ ❧✉➟♥ ỗ ữỡ r ữỡ t tr ✈➲
✤❛ t❤ù❝ ✤è✐ ①ù♥❣✱ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà ❤❛✐ ❜✐➳♥✱ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝
trà ❜❛ ❜✐➳♥ ✈➔ sỡ ỗ t tự t❛ ♥❤ú♥❣ ❤➻♥❤
❞✉♥❣ trü❝ q✉❛♥ ✈➲ ✤❛ t❤ù❝✳ ✣➙② ❝ô♥❣ ❧➔ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝➛♥ t❤✐➳t ♣❤ö❝
✈ö ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ❝→❝ ♥ë✐ ❞✉♥❣ t✐➳♣ t❤❡♦ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❈❤÷ì♥❣ ữủ
t ợ t ố q ỳ ọ tr ỵ tt
ự ❚ø ✤â✱ ❤➻♥❤ t❤➔♥❤ ❤å ✤❛ t❤ù❝ ❤❛✐ ❜✐➳♥ ❞✉② ♥❤➜t fm (x, y)
✈ỵ✐ ♥❤ú♥❣ t➼♥❤ ❝❤➜t t❤ó ✈à✳ ❚ø ✤â✱ ❝→❝ ❤å {Fm (x, y, z)}, {Sm (x, y, z)} ữủ
t ợ ố q t t❛ ❧í✐ ❣✐↔✐ ❝õ❛ ❤❛✐ ❝➙✉ ❤ä✐ tr➯♥
tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✳




ữỡ

tự ố ự ỹ tr sỡ ỗ
t
✣❛ t❤ù❝ ✤è✐ ①ù♥❣
❈❤♦ n ∈ N∗✱ tr➯♥ Nn t❛ ởt q tự tỹ ữ s ợ

tỷ tý ỵ ừ Nn (a1 , . . . , an )✱ (b1 , . . . , bn ) t❛ ♥â✐ (a1 , . . . , an ) ≤

(b1 , . . . , bn ) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❤♦➦❝ (a1 , . . . , an ) = (b1 , . . . , bn ) ❤♦➦❝ ∃i ∈
{1, . . . , n} s❛♦ ❝❤♦ a1 = b1 , . . . , ai−1 = bi−1 ✱ ai < bi ✳ ◗✉❛♥ ❤➺ t❤ù tü
♥❤÷ tr➯♥ ❣å✐ ❧➔ q✉❛♥ ❤➺ t❤ù tü tø ✤✐➸♥✳ ✣➙② ❝ô♥❣ ❧➔ ♠ët q✉❛♥ ❤➺ t❤ù
tü t♦➔♥ q ữợ t (a1 , . . . , an ) < (b1 , . . . , bn ) ❝â ♥❣❤➽❛ ❧➔
(a1 , . . . , an ) ≤ (b1 , . . . , bn ) ✈➔ (a1 , . . . , an ) = (b1 , . . . , bn )✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦

A ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ✈➔ ✤❛ t❤ù❝

f (x1 , . . . , xn ) ∈ A[x1 , . . . , xn ]✱ ❣✐↔ sû
m

ci xa1i1 . . . xanin ∈ A[x1 , . . . , xn ],

f (x1 , . . . , xn ) =
i=1

✈ỵ✐ ci ∈ A✱ ci = 0✱ i = 1, . . . , m✱ (ai1 , . . . , ain ) ∈ Nn ✈➔ ♠é✐ ❦❤✐ i =

j t❛ ❝â (ai1 , . . . , ain ) = (aj1 , . . . , ajn )✳ ❑❤✐ ✤â t❛ s➢♣ t➟♣ ❝→❝ ❜ë sè ♠ô
{(ai1 , . . . , ain )|i = 1, . . . , m} t❤❡♦ q✉❛♥ ❤➺ t❤ù tü tø ✤✐➸♥ t❤❡♦ ❝❤✐➲✉ ❣✐↔♠

❞➛♥✳ ❚❤❡♦ t❤ù tü ✤â t❛ ✈✐➳t ❧↕✐ ✤❛ t❤ù❝ f ✱ ❧ó❝ ♥➔② t❛ ♥â✐ ✤❛ t❤ù❝ f ✤÷đ❝
s➢♣ t❤❡♦ ❧è✐ tø ✤✐➸♥✱ ❦❤✐ ✤â tỷ ự ợ ở số ụ ợ t ữủ ❣å✐ ❧➔
❤↕♥❣ tû ❝❛♦ ♥❤➜t ❝õ❛ f ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❈❤♦ A ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ✈➔ f (x1, . . . , xn) ∈
A[x1 , . . . , xn ]✳ ❚❛ ♥â✐ f ❧➔ ♠ët ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝õ❛ n ➞♥ ♥➳✉ f (x1 , . . . , xn ) =


✸

f (xν(1) , . . . , xν(n) ) ✈ỵ✐ ♠å✐ ♣❤➨♣ t❤➳ ν ∈ Sn ✭tr♦♥❣ ✤â f (xν(1) , . . . , xν(n) ✮ ❝â
✤÷đ❝ tø f (x1 , . . . , xn ) ❜➡♥❣ ❝→❝❤ t❤❛② t❤➳ x1 ❜ð✐ xν(1) , . . .✱ t❤❛② t❤➳ xn ❜ð✐

xν(n) ✮✳
❚❛ ♥❤➢❝ ❧↕✐ r➡♥❣✱ tr♦♥❣ ✈➔♥❤ A[x1 , . . . , xn ]✱ t➟♣ t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ ✤è✐
①ù♥❣ ❧➟♣ t❤➔♥❤ ♠ët ✈➔♥❤ ❝♦♥ ❝õ❛ ✈➔♥❤ ✤â✳ ❈→❝ ♣❤➛♥ tû ❝õ❛ A ❝ô♥❣ ❧➔
♥❤ú♥❣ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✳ ◆➳✉ f (x1 , . . . , xn ) ∈
/ A ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ t❤➻

f (x1 , . . . , xn ) ♣❤↔✐ ❝❤ù❛ ❝↔ n ➞♥ ✈➔ ❝â ❝ị♥❣ ❜➟❝ ✤è✐ ✈ỵ✐ ♠é✐ ➞♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ❚r♦♥❣ ✈➔♥❤ ✤❛ t❤ù❝ A[x1, . . . , xn]✱ ①➨t ❝→❝ ✤❛ t❤ù❝ ✤è✐
①ù♥❣ s❛✉ ✤➙② ❣å✐ ❧➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì ❜↔♥ ❝õ❛ n ➞♥ x1 , . . . , xn

σ1 =

xi
1≤i≤n

σ2 =

xi xj
1≤i
...
xi1 xi2 · · · xik

σk =
1≤i1 <···
...
σn =x1 · · · xn .
❈→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì ❜↔♥ ✤â♥❣ ♠ët ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ỵ
tt tự ố ự sỷ g(x1 , . . . , xn ) ❧➔ ✤❛ t❤ù❝ ✤è✐
①ù♥❣ ❝õ❛ A[x1 , . . . , xn ]✱ ❦❤✐ ✤â ✤❛ t❤ù❝ ♥❤➟♥ ✤÷đ❝ tø g(x1 , . . . , xn ) ❜➡♥❣
❝→❝❤ t❤❛② x1 ❜ð✐ σ1 ✳✳✳✱ t❤❛② xn ❜ð✐ σn ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ g(σ1 , . . . , σn ) ✈➔ ❣å✐
❧➔ ✤❛ t❤ù❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì ❜↔♥✳ ❱➔ ✤❛ t❤ù❝ g(σ1 , . . . , σn ) ụ
tự ố ự

ỵ sỷ f (x1, . . . , xn) ∈ A[x1, . . . , xn] ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣
❦❤→❝ 0✳ ❑❤✐ õ tỗ t t tự h(x1 , . . . , xn ) ∈ A[x1 , . . . , xn ] s❛♦

❝❤♦ f (x1 , . . . , xn ) = h(σ1 , . . . , σn )✳

✶✳✷✳ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà ❤❛✐ ❜✐➳♥
❈❤♦ s ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì ❜↔♥ ❜➟❝ ♥❤➜t n ❜✐➳♥✳ ❈❤➥♥❣ ❤↕♥ ✈ỵ✐ n = 2✱

s(x, y) = x + y; ❦❤✐ n = 3 t❛ ❝â s(x, y, z) = x + y + z ✳ ❈❤♦ k ♠ët sè

ổ t ỵ Vk ổ ✈❡❝tì ❝→❝ ✤❛ t❤ù❝ t❤✉➛♥
♥❤➜t ❜➟❝ k ✱ n ❜✐➳♥ ✈ỵ✐ ❤➺ sè ♣❤ù❝✱ ❝ị♥❣ ✈ỵ✐ ✤❛ t❤ù❝ 0✳ ❚❛ ❝➛♥ ♠ët sè ❦❤→✐
♥✐➺♠✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ▼ët ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t q ∈ Vk ✤÷đ❝ ❣å✐ ❧➔ ✤➛② ✤õ
♥➳✉ ♠å✐ ✤ì♥ t❤ù❝ ❜➟❝ k ✱ n ❜✐➳♥ ✤➲✉ ①✉➜t ❤✐➺♥ tr♦♥❣ q ✈ỵ✐ ❤➺ sè ❦❤→❝ 0✳
❈❤➥♥❣ ❤↕♥✱ ✈ỵ✐ n = 2 t❤➻ q1 (x, y) = x3 − x2 y + xy 2 − y 3 ❧➔ ✤❛ t❤ù❝ ❜➟❝
✸ ✤➛② ✤õ✱ ♥❣÷đ❝ ❧↕✐ q2 (x, y) = x3 + y 3 ❧➔ ❦❤æ♥❣ ✤➛② ✤õ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷✳ ❈❤♦ p ❧➔ ✤❛ t❤ù❝ n ❜✐➳♥✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❤↕♥❣ ừ p ỵ
R(p) số ỡ tự ①✉➜t ❤✐➺♥ tr♦♥❣ p ✈ỵ✐ ❤➺ sè ❦❤→❝ ❦❤ỉ♥❣✳
✣è✐ ✈ỵ✐ tr÷í♥❣ ❤đ♣ ✤❛ t❤ù❝ ❤❛✐ ❜✐➳♥✱ ❝→❝ ❈➙✉ ❤ä✐ ✶ ✈➔ ❈➙✉ ❤ä✐ ✷ t❛ ✤➲✉
❝â ❝➙✉ tr↔ ❧í✐ ❦❤→ rã r➔♥❣✳ ❚❛ ❝â ❦➳t q✉↔ s❛✉✳

▼➺♥❤ ✤➲ ✶✳✷✳✸✳ ❈❤♦ m ≥ 1, ❦❤✐ ✤â t❛ ❝â
p(x, y) = xm + (−1)m−1 y m ,
❧➔ ♠ët ✤❛ t❤ù❝ s❤❛r♣ ❤❛✐ ❜✐➳♥✳ ❍ì♥ ♥ú❛✱ ♥➳✉ m ❧➫ t❤➻ p ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣
❝ü❝ trà✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â ♥❤➟♥ ①➨t ♠å✐ ✤ì♥ t❤ù❝ ❤❛✐ ❜✐➳♥ ❜➟❝ ♠ ✤➲✉ ❦❤ỉ♥❣
❝â t❤÷ì♥❣ ✤➛② ✤õ✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû t❛ ❝â ✤ì♥ t❤ù❝ xa y b ♠➔ ❝â t❤÷ì♥❣

xa y b
= q(x, y) ❧➔ ♠ët ✤❛ t❤ù❝ ✤➛② ✤õ✳ ❑❤✐ ✤â
x+y

xa y b = (x + y)q(x, y)
❚❤❛② y = 0 ✈➔♦ ✤➥♥❣ t❤ù❝ tr➯♥✱ t❛ s✉② r❛ 0 = xq(x, 0)✳ ❉♦ ✤â q(x, 0) = 0
ổ ỵ q(x, y) ❧➔ ✤❛ t❤ù❝ ✤➛② ✤õ ♥➯♥ ♥â ❝❤ù❛ ✤ì♥ t❤ù❝ ❞↕♥❣

xr ✱ ✈➔ ❦❤✐ ✤â ❤✐➸♥ ♥❤✐➸♥ q(x, 0) = cxr = 0 ✭✈ỵ✐ c ♥➔♦ ✤â✮✳
❚❛ ①➨t ✤❛ t❤ù❝ ❤❛✐ ❤↕♥❣ tû p(x, y) = xm + (−1)m−1 y m . ❚❛ ❝â ♣❤➙♥ t➼❝❤

p(x, y) = (x + y)(xm−1 − xm−2 y + · · · + (−1)m−1 y m−1 )
❚❛ ✤➦t

q(x, y) = xm−1 − xm−2 y + · · · + (−1)m−1 y m−1 ,


✺

❦❤✐ ✤â✱ rã r➔♥❣ q ❧➔ ✤❛ t❤ù❝ t❤÷ì♥❣ ✤➛② ✤õ ❝õ❛ p(x, y)✳ ◆❤÷ ✈➟② p(x, y) ❧➔
✤❛ t❤ù❝ ❝â ❤↕♥❣ ❜➨ ♥❤➜t t❤ä❛ ♠➣♥ p = sq ✈ỵ✐ q ❧➔ ✤➛② ✤õ✳ ❱➟② p ❧➔ ♠ët
✤❛ t❤ù❝ s❤❛r♣ ❤❛✐ ❜✐➳♥✳
❑❤✐ m ❧➫ t❛ ❝â

p(x, y) = xm + y m
❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ♥➯♥ p ❧➔ ✤❛ tự ố ự ỹ tr

ú ỵ m = 2r ❧➔ ♠ët sè ❝❤➤♥ t❛ ❝â
✭✐✮ ✣❛ t❤ù❝

p(x, y) = xm + (−1)m−1 y m
❦❤æ♥❣ ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✳ ❚❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ❦❤ỉ♥❣ ❝â
✤❛ t❤ù❝ ✤è✐ ①ù♥❣ t❤✉➛♥ ♥❤➜t ❤❛✐ ❜✐➳♥ ✭✈ỵ✐ ❤➺ sè tr➯♥ C✮ ❜➟❝ ❝❤➤♥ ❝â
❤❛✐ ❤↕♥❣ tû ❝❤✐❛ ❤➳t ❝❤♦ x + y. ❚❤➟t ✈➟② ❣✐↔ sû ♥❣÷đ❝ ❧↕✐✱ t❛ ❝â ✤❛
t❤ù❝ p(x, y) = cxa y b + cxb y a ✤è✐ ①ù♥❣ t❤✉➛♥ ♥❤➜t ❤❛✐ ❜✐➳♥ ❜➟❝ a + b
❝❤➤♥ ❝â ❤❛✐ ❤↕♥❣ tû s❛♦ ❝❤♦ p(x, y) = (x + y)q(x, y) ✈ỵ✐ q(x, y) ❧➔ ✤❛
t❤ù❝ ♥➔♦ ✤â✳ ❱ỵ✐ y = −x✱ t❛ ❝â p(x, −x) = 0q(x, −x) = 0✳ ▼➦t ❦❤→❝✱

✈➻ a + b ❝❤➤♥ ♥➯♥ a ✈➔ b ❝â ❝ò♥❣ t➼♥❤ ❝❤➤♥ ❧➫✳ ❉♦ ✈➟② (−1)a = (−1)b ✳
❚❛ s✉② r❛

0 = p(x, −x) = cxa (−x)b + cxb (−x)a = 2(1)a cxa+b .
ởt ổ ỵ
tự

p(x, y) = x2r + 2(−1)r−1 xr y r + y 2r
❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❜➟❝ m = 2r ❝â ❤↕♥❣ ❜➡♥❣ ❜❛ t❤ä❛ ♠➣♥
r−1
r

r−1 r

p(x, y) = (x + y)(x + (−1)

(−1)j xr−1−j y j ),

y )(
j=0

tr♦♥❣ ✤â
r−1
r

q(x, y) = (x + (−1)

r−1 r

(−1)j xr−1−j y j ),

y )(
j=0

❧➔ ✤➛② ✤õ ♥➯♥ p ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà ❤❛✐ ❜✐➳♥ ❜➟❝ ❝❤➤♥ ✈➔ ❝â
❤↕♥❣ ❜➡♥❣ ❜❛✳


✻

✶✳✸✳ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà ❜❛ ❜✐➳♥
✣➲ t➔✐ ✤➦t r❛ ♠ư❝ ✤➼❝❤ t➻♠ ❤✐➸✉ ❝➙✉ tr↔ ❧í✐ ❝❤♦ ❝→❝ ❝➙✉ ❤ä✐ ✶✱ ❝➙✉ ❤ä✐ ✷
tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜❛ ❜✐➳♥✿

❈➙✉ ❤ä✐ ✸✳ ❚r♦♥❣ t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ ❜❛ ❜✐➳♥ ✤è✐ ①ù♥❣ t❤✉➛♥ ♥❤➜t p ❜➟❝ m
❝â ♠ët t❤÷ì♥❣ ✤➛② ✤õ✱ ❤↕♥❣ ❜➨ ♥❤➜t ❝â t❤➸ ❝õ❛ p ❜❛♦ ♥❤✐➯✉❄ ❈→❝ ✤❛ t❤ù❝
❝â ❤↕♥❣ ❜➨ ♥❤➜t ♥➔② ❧➔ ❝→❝ ✤❛ t❤ù❝ ♥➔♦❄

❈➙✉ ❤ä✐ ✹✳ ❚r♦♥❣ sè t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ ❜❛ ❜✐➳♥ t❤✉➛♥ ♥❤➜t p ❜➟❝ m ❝â

♠ët t❤÷ì♥❣ ✤➛② ✤õ✱ ❤↕♥❣ ❜➨ ♥❤➜t ❝â t❤➸ ❝õ❛ p ❧➔ ❜❛♦ ♥❤✐➯✉❄ ❈→❝ ✤❛ t❤ù❝
❝â ❤↕♥❣ ❜➨ ♥❤➜t ♥➔② ❧➔ ❝→❝ ✤❛ t❤ù❝ ♥➔♦❄
✣➲ t➔✐ t➟♣ tr✉♥❣ ✈➔♦ ✈✐➺❝ t➻♠ ❤✐➸✉ ♠ët ❤å {Fm } ❝→❝ ✤❛ t❤ù❝ s❤❛r♣ ✈➔
❤å {Sm } ❝→❝ ✤❛ t❤ù❝ ố ự ỹ tr ợ ộ m ữỡ r ❝❤÷ì♥❣

2✱ ❝❤ó♥❣ t❛ s➩ t➻♠ ❤✐➸✉ ❦ÿ ❝ỉ♥❣ t❤ù❝ tê♥❣ q✉→t ❝õ❛ Sm ✳ ▼ët sè ✤❛ t❤ù❝
❜➟❝ ❧➫ ✤➛✉ t✐➯♥ ❝õ❛ ❤å {Sm } ❧➔
S1 = x + y + z
S3 = x3 + y 3 + z 3 − 3xyz
S5 = x5 + y 5 + z 5 + 5xyz(xy + zx + yz)

S7 = x7 + y 7 + z 7 − 7xyz(x2 y 2 + x2 z 2 + y 2 z 2 )
S9 = x9 + y 9 + z 9 + 9xyz(x3 y 3 + x3 z 3 + y 3 z 3 ) − 30x3 y 3 z 3
S11 = x11 + y 11 + z 11 − 11xyz(x4 y 4 + x4 z 4 + y 4 z 4 )
+ 55x3 y 3 z 3 (xy + xz + yz)
S13 = x13 + y 13 + z 13 + 13xyz(x5 y 5 + x5 z 5 + y 5 z 5 )
− 91x3 y 3 z 3 (x2 y 2 + x2 z 2 + y 2 z 2 ),
✈➔ ♠ët sè ✤❛ t❤ù❝ ❜➟❝ ❝❤➤♥ ✤➛✉ t✐➯♥ ❧➔

S2 = x2 + y 2 + z 2 + 2(xy + xz + yz)
S4 = x4 + y 4 + z 4 − 2(x2 y 2 + x2 z 2 + y 2 z 2 )
S6 = x6 + y 6 + z 6 + 2(x3 y 3 + x3 z 3 + y 3 z 3 ) − 9x2 y 2 z 2
S8 = x8 + y 8 + z 8 − 2(x4 y 4 + x4 z 4 + y 4 z 4 ) + 16x2 y 2 z 2 (xy + xz + yz).




ỡ ỗ t tự
❝â t❤➸ ❜✐➸✉ ❞✐➵♥ ❝→❝ ✤❛ t❤ù❝ ❜❛ ❜✐➳♥ ♠ët trỹ q sỡ ỗ
t ỡ ỗ t ♠ët ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜❛ ❜✐➳♥ g(x, y, z)
ỗ t tr õ ộ tữỡ ự ợ ởt ✤ì♥ t❤ù❝ ❝â ❤➺ sè ❦❤→❝ ❦❤ỉ♥❣
①✉➜t ❤✐➺♥ tr♦♥❣ g ✳ ▼ët ❝↕♥❤ ✤÷đ❝ ♥è✐ ❣✐ú❛ ❤❛✐ ✤➾♥❤ t÷ì♥❣ ù♥❣ ✈ỵ✐ ❤❛✐ ✤ì♥
t❤ù❝ m1 ✈➔ m2 ♥➳✉ ❝â ❤❛✐ ✤ì♥ t❤ù❝ ❦❤→❝ ♥❤❛✉ λ1 ✈➔ λ2 ❝ò♥❣ ❝â ❜➟❝ ♠ët
s❛♦ ❝❤♦ λ1 m1 = λ2 m2 ✳ ❚❛ ♠✐♥❤ ❤å❛ sỡ ỗ t tự t t
g(x, y, z) = x3 + 3x2 y − 2x2 z + xyz z 3



ỡ ỗ t ❝õ❛ g(x, y, z)

ỡ ỗ t ừ g(x) ỗ t ỗ ự ợ tỷ
ổ t ữ tr



ỡ ỗ t rút ồ ừ g(x, y, z)

õ ởt sỡ ỗ t ồ ỡ t ộ
ừ ỗ t ợ số ừ tỷ tữỡ ự t sỡ ỗ tr ởt
t ựt t ❤✐➺♥ t➜t ❝↔ ❝→❝ ✤ì♥ t❤ù❝ ❝â t❤➸ ❝â ❝õ❛ tự t
t ỗ ự ợ tỷ õ số ổ
sỡ ỗ s tr tữỡ ự ợ ❧ơ② t❤ø❛ ❝❛♦ ❤ì♥ ❝õ❛

z ✱ ❝→❝ ❧ơ② t❤ø❛ ❝❛♦ ❤ì♥ ❝õ❛ x ð ❜➯♥ tr→✐ ✈➔ ❝→❝ ❧ơ② t❤ø❛ ❝❛♦ ❤ì♥ ❝õ❛ y ð
❜➯♥ ♣❤↔✐✳ ❍➻♥❤ ✶✳✷ ❧➔ sì ỗ t ừ g t t ủ q ữợ

ỡ ỗ t ừ tự ố ự ỹ tr t t

sỡ ỗ tữ ✈➻ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❧➔ ❝→❝
✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝â ❤↕♥❣ ♥❤ä ♥❤➜t t x + y + z ợ tữỡ ❧➔ ✤❛
t❤ù❝ ✤➛② ✤õ✳ ❍➻♥❤ ✶✳✸✱ ✶✳✹ ✈➔ ✶✳✺ ❜✐➸✉ t sỡ ỗ t ừ S3 , S5 S7 ✳






ỡ ỗ t ừ S3

ỡ ỗ ◆❡✇t♦♥ ❝õ❛ s5






ỡ ỗ t ừ S7




ữỡ

ởt ợ tự ố ự ỹ tr
t❤ù❝ s❤❛r♣ ✤➦❝ ❜✐➺t
✷✳✶✳ ❍å ✤❛ t❤ù❝ {Fm} ✈➔ ❝→❝ t➼♥❤ ❝❤➜t
❈➙✉ ❤ä✐ ✶ ✈➔ ❈➙✉ ❤ä✐ ✷ ❞÷í♥❣ ♥❤÷ ❧➔ t❤✉➛♥ tó② ✤↕✐ sè ♥➯♥ t❤➟t ❜➜t ♥❣í
❧➔ ❤å ❝→❝ ✤❛ t❤ù❝ s❤❛r♣ {Fm } ✤➣ ✤÷đ❝ ♣❤→t ♠✐♥❤ tr t ở
ự ỵ tt ừ ❜✐➳♥ ♣❤ù❝✳ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ t❛ ♠æ t↔ ♠è✐
q✉❛♥ ❤➺ tr➯♥ ✈➔ ♣❤→t ❜✐➸✉ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ✤❛ t❤ù❝ s❤❛r♣✳ ❏✳ ❉✬❆♥❣❡❧♦
✤÷đ❝ ❣❤✐ ♥❤➟♥ ❝❤➼♥❤ ❧➔ ♥❣÷í✐ ✤➣ ♣❤→t ❤✐➺♥ r❛ ❝→❝ ✤❛ t❤ù❝ ✤â ✈➔ t➼♥❤ ❝❤➜t
❝õ❛ ❝❤ó♥❣✳

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳ ▼ët →♥❤ ①↕ ❧✐➯♥ tö❝ f : X → Y

❣✐ú❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥

tỉ♣ỉ ✤÷đ❝ ❣å✐ ❧➔ r✐➯♥❣ ♥➳✉ t↕♦ ↔♥❤ ❝õ❛ ♠ët t➟♣ ❝♦♠♣❛❝t tr♦♥❣ Y ❧➔ ♠ët
t➟♣ ❝♦♠♣❛❝t tr♦♥❣ X ✱ tù❝ ❧➔ ✈ỵ✐ ♠é✐ t➟♣ ❝♦♠♣❛❝t K tr♦♥❣ Y t❛ ❝â f −1 (K)
❧➔ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ X ✳
▼ët ✈➜♥ ✤➲ q✉❛♥ trå♥❣ ❧➔ t➻♠ ❤✐➸✉ ✈➲ →♥❤ ①↕ r✐➯♥❣ ❝❤➾♥❤ ❤➻♥❤ ❣✐ú❛ ❝→❝
❤➻♥❤ ❝➛✉ tr♦♥❣ ♥❤ú♥❣ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ ♣❤ù❝ ❝â sè ❝❤✐➲✉ ❦❤→❝ ♥❤❛✉✳ Ð ✤➙②
t❛ ❝❤➾ t➟♣ tr✉♥❣ ✈➔♦ ①➨t ❝→❝ →♥❤ ①↕ ✤❛ t❤ù❝✳ ❚❛ ❦➼ ❤✐➺✉ Cn ❧➔ ❦❤æ♥❣ ❣✐❛♥
❊✉❝❧✐❞ ♣❤ù❝ n ❝❤✐➲✉✳ ❱ỵ✐ z = (z1 , z2 , . . . , zn ) ∈ Cn ✱ t❛ ✤➦t z

2

n

=

|zj |2

j=1
n

ừ z ỵ Bn ❧➔ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à ♠ð tr♦♥❣ C ✱ ✤â
❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû z ∈ Cn ❝â z < 1✳
❈❤♦ p : Cn → CN ❧➔ ♠ët →♥❤ ①↕ ✤❛ t❤ù❝ ❝❤➾♥❤ ❤➻♥❤✱ ❦❤✐ ✤â p : Bn → BN
❧➔ →♥❤ ①↕ r✐➯♥❣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ p ❜✐➳♥ ♠➦t ❝➛✉ ✤ì♥ ✈à tr♦♥❣ Cn t❤➔♥❤ ♠➦t


✶✷

❝➛✉ ✤ì♥ ✈à tr♦♥❣ CN ✱ tù❝ ❧➔ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
n

N

p(z)

2

|pk (z)| = 1 ♥➳✉ z
2

=

2

✭✷✳✶✮

|zj |2 = 1.

=
j=1

k=1

❱➼ ❞ư ✷✳✶✳✷✳ ❳➨t →♥❤ ①↕ tø C2 ✈➔♦ C3 ✤÷đ❝ ❝❤♦ ❜ð✐
√
(z, w) → (z 3 , 3zw, w3 ).

◆➳✉ |z|2 + |w|2 = 1✱ t❤➻

√

|z 3 |2 + | 3zw|2 + |w3 |2 = |z 3 |2 + 3|zw|2 |z|2 + |w|2 + |w3 |2
= |z|6 + 3|z|4 |w|2 + 3|z|2 |w|4 + |w|6
= |z|2 + |w|2

3

= 1,

✣➙② ❧➔ →♥❤ ①↕ ✤❛ t❤ù❝ r✐➯♥❣ tø B2 ✈➔♦ B3 ✳
❚❛ ①➨t tr÷í♥❣ ❤đ♣ n = 2✳ ✣ë ♣❤ù❝ t↕♣ ❝õ❛ →♥❤ ①↕ ✤❛ t❤ù❝ p : B2 → BN
♣❤ö t❤✉ë❝ ✈➔♦ sè ❝❤✐➲✉ N ❀ ♠ët ❣✐↔ t❤✉②➳t ❝õ❛ ❉✬❆♥❣❡❧♦ ✭tr♦♥❣ ❬✹❪✮ ♥â✐
r➡♥❣ ❜➟❝ m ❝õ❛ ♠ët →♥❤ ①↕ r✐➯♥❣ ♥❤÷ ✈➟② ❧✉ỉ♥ t❤ä❛ ♠➣♥
✭✷✳✷✮

m ≤ 2N − 3.

✣➳♥ ♥➠♠ ✷✵✵✸✱ tr♦♥❣ ❜➔✐ ❜→♦ ❬✺❪ ❝→❝ t→❝ ❣✐↔ ❉✬❆♥❣❡❧♦✱ ❑♦s ✈➔ ❘✐❡❤❧ ✤➣
❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② tr♦♥❣ tr÷í♥❣ ❤đ♣ t➜t ❝↔ ❝→❝ t❤➔♥❤
♣❤➛♥ ❝õ❛ p ❧➔ ❝→❝ ỡ tự ỗ tớ ụ ữ r ởt ồ ỳ →♥❤ ①↕
✤ì♥ t❤ù❝ ❝❤♦ tr÷í♥❣ ❤đ♣ m = 2N − 3 ợ ộ m ữỡ
ừ s❤❛r♣ ❝õ❛ ❤å ❧➔
(m−1)/2

p(z, w)

2

2 m

= |z|

2 m

+ |w|

Km,k |z|2

+

m−2k

|w|2

k

, ✭✷✳✸✮

k=1

tr♦♥❣ ✤â ❝→❝ ❤➺ sè Km,k ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ Km,0 = 1 ✈➔

Km,k =

m−k
k

+

m−1−k
k−1

✈ỵ✐

1≤k≤

m
.
2

✭✷✳✹✮

❚❤❛② |z|2 , |w|2 ❜➡♥❣ (x, y) t ữủ ởt tự tỹ ợ ❝→❝
❤➺ sè ❦❤ỉ♥❣ ➙♠ ♥❤➟♥ ❣✐→ trà 1 tr➯♥ ✤÷í♥❣ t❤➥♥❣ x + y = 1 ✈➔ ❝â ❤↕♥❣

N=

m+3
✳ ◆❤÷ ✈➟② t❛ ❝â ♠ët s♦♥❣ →♥❤ ❣✐ú❛ ❝→❝ →♥❤ ①↕ ✤ì♥ t❤ù❝ r✐➯♥❣
2


✶✸

❜✐➳♥ B2 ✈➔♦ BN ✈➔ ❧ỵ♣ P ❝→❝ ✤❛ t❤ù❝ ❤❛✐ ❜✐➳♥ ❝â ❤➺ sè ❦❤æ♥❣ ➙♠ ♥❤➟♥
❣✐→ trà 1 tr➯♥ ✤÷í♥❣ t❤➥♥❣ x + y = 1✳
❍❛✐ ❦➳t q✉↔ s❛✉ ✤➣ ✤÷đ❝ ❝→❝ t→❝ ❣✐↔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t tr♦♥❣ ❬✺❪ q✉❛
✤â ❝ô♥❣ ❝❤♦ t❛ t❤➜② ♠è✐ ❧✐➯♥ ❤➺ ❝❤➦t ❝❤➩ ❣✐ú❛ ❜➔✐ t♦→♥ ✈➲ ✤❛ t❤ù❝ ✈➔ ❜➔✐
t♦→♥ ỹ tr tr ự

ỵ sỷ p ❧➔ ♠ët ✤❛ t❤ù❝ ❤❛✐ ❜✐➳♥ t❤ü❝ x, y t❤ä❛ ♠➣♥

✭✶✮ p(x, y) = 1 tr➯♥ ✤÷í♥❣ t❤➥♥❣ x + y = 1,
✭✷✮ ▼é✐ ❤➺ sè ❝õ❛ p(x, y) ✤➲✉ ❦❤ỉ♥❣ ➙♠✳

●å✐ N ❧➔ sè ✤ì♥ t❤ù❝ ♣❤➙♥ ❜✐➺t tr♦♥❣ p ✈➔ d ❧➔ ❜➟❝ ❝õ❛ p✳ ❑❤✐ ✤â d ≤

2N − 3✳ ❱➔ ❦➳t q✉↔ ✤↕t ✤÷đ❝ ❝ü❝ tr tự ợ ộ N 2 ổ tỗ t↕✐
✤❛ t❤ù❝ t❤ä❛ ♠➣♥ ✭✶✮ ✈➔ ✭✷✮ ❝â ❜➟❝ ❧➔ 2N 3.
ứ ỵ tr t ụ t❤✉ ✤÷đ❝ ❤➺ q✉↔ s❛✉✿

❍➺ q✉↔ ✷✳✶✳✹✳ ●✐↔ sû f ❧➔ ♠ët →♥❤ ①↕ ✤ì♥ t❤ù❝ ❝❤➾♥❤ ❤➻♥❤ r✐➯♥❣ tø ❤➻♥❤
❝➛✉ ✤ì♥ ✈à tr♦♥❣ C2 ✈➔♦ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à tr♦♥❣ CN ✳ ❑❤✐ ✤â ❜➟❝ ❝õ❛ ❢ ❦❤ỉ♥❣
✈÷đt q✉→ 2N − 3✳ ❱➔ ❦➳t q✉↔ ❝ô♥❣ ✤↕t ❝ü❝ trà✳
❚❤❛② ✤ê✐ ❝ỉ♥❣ t❤ù❝ ✭✷✳✸✮ ♠ët ❝❤ót t❛ ✤à♥❤ ♥❣❤➽❛ ✤÷đ❝ ởt ồ {fm }
ỳ tự ỗ ✈➔ ❜➟❝ ❧➫✱ ♠➦❝ ❞ò ❝❤➾ ❝â ♥❤ú♥❣ ✤❛ t❤ù❝
❜➟❝ ❧➫ ♠ỵ✐ t❤✉ë❝ ❧ỵ♣ P ✿
m
2

fm (x, y) = xm (y)m +

Km,k xm2k y k .



k=1

ữ ỵ r ộ fm ❧➔ ❜➜t ❜✐➳♥ q✉❛ →♥❤ ①↕ (x, y) → (ηx, η 2 y) tr♦♥❣ ✤â η
❧➔ ♠ët ❝➠♥ ♥❣✉②➯♥ t❤õ② ❜➟❝ m ❝õ❛ 1✳ ▼ö❝ ✤➼❝❤ ❝õ❛ t❛ ❧➔ sû ❞ö♥❣ ❤å {fm }
✤➸ ①→❝ ✤à♥❤ ❤å {Fm } ♥❤ú♥❣ ✤❛ t❤ù❝ s❤❛r♣✳
▲❡❜❧ ✈➔ P❡t❡rs tr♦♥❣ ❜➔✐ ❜→♦ ❬✼❪ ✤➣ ❝â ♠ët sè ❦➳t q✉↔ q✉❛♥ trå♥❣ tr♦♥❣

✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ →♥❤ ①↕ ✤ì♥ t❤ù❝ r✐➯♥❣ ❣✐ú❛ ❝→❝ ❤➻♥❤ ồ r
r n = 2 ữợ ❧÷đ♥❣ ✈➲ ❜➟❝ ❝õ❛ ❝ỉ♥❣ t❤ù❝ ✭✷✳✷✮ t❤ä❛ ♠➣♥ ❝❤♦ ♠ët
❧ỵ♣ ❝→❝ ✤❛ t❤ù❝ ❝❤ù❛ ❧ỵ♣ P ✳ ❱ỵ✐ ♠é✐ ✤❛ t❤ù❝ p ∈ P ✱ t❛ t❤✉➛♥ ♥❤➜t ❤â❛

p(x, y) − 1 ❜ð✐ ❜✐➳♥ z ✈➔ s❛✉ ✤â t❤❛② z ❜➡♥❣ −z t❛ ♥❤➟♥ ✤÷đ❝ ♠ët ✤❛ t❤ù❝
t❤✉➛♥ ♥❤➜t P (x, y, z) ❜➟❝ m t❤ä❛ ♠➣♥ P (x, y, z) = q(x, y, z)(x + y + z)
✈ỵ✐ ✤❛ t❤ù❝ t❤÷ì♥❣ q ❧➔ t❤✉➛♥ ♥❤➜t ❝â ❜➟❝ m − 1✳ ✣✐➲✉ ❦✐➺♥ p ❝â ❤➺ sè




ổ ởt õ ỷ ỵ ợ ❝→❝ ❞➜✉ ❤✐➺✉ ❝õ❛ ❝→❝ ❤➺ sè ❝õ❛

m+5
❝â t❤➸ ✤÷đ❝
2
❝❤ù♥❣ ♠✐♥❤ ✈ỵ✐ ❣✐↔ t❤✉②➳t ②➳✉ ❤ì♥✱ r➡♥❣ P s✐♥❤ r❛ tø ♠ët ♣❤➛♥ tû ❝õ❛ P ✳
●✐↔ t❤✉②➳t ✤➛✉ t✐➯♥ ❧➔ P ❦❤ỉ♥❣ ❝â ♥❤➙♥ tû ❝❤✉♥❣ ❧➔ ✤ì♥ t❤ù❝ ữỡ
tt tự sỡ ỗ t ừ q ồ ự ữủ ỵ
s

P Ptrs r r ữợ ữủ R(P )

ỵ Ptr P (x, y, z) ❧➔ ♠ët ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t
❜➟❝ m s❛♦ ❝❤♦ P (x, y, z) = q(x, y, z)(x + y + z) ✈ỵ✐ q ❧➔ ♠ët ✤❛ t❤ù❝ t❤✉➛♥

♥❤➜t ❜➟❝ m − 1✳ ●✐↔ sû ❝→❝ ❤↕♥❣ tû ❝õ❛ P ❦❤ỉ♥❣ ❝â ❝❤✉♥❣ ♥❤➙♥ tû ✤ì♥
t❤ù❝ ❜➟❝ ❞÷ì♥❣ ✈➔ sì ỗ t ừ q ỗ t tổ ✤â t❛ ❝â
m+5
R(P ) ≥

.
2

❍➺ q✉↔ ✷✳✶✳✻✳ ◆➳✉ P (x, y, z) ❧➔ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ m ❝â t❤÷ì♥❣ ✤➛②
m+5
.
2
❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t P (x, y, z) ❜➟❝ m ❜➜t ❦ý s❛♦ ❝❤♦

✤õ t❤➻ R(P ) ≥

P (x, y, z) = (x + y + z)q(x, y, z)
✈ỵ✐ q ❧➔ ✤➛② ✤õ✳ ❑❤✐ ✤â ❝→❝ ❤↕♥❣ tû ❝õ❛ P ❦❤ỉ♥❣ ❝â ❝❤✉♥❣ ♥❤➙♥ tû ✤ì♥
t❤ù❝ ❜➟❝ ❞÷ì♥❣✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû ❝→❝ ❤↕♥❣ tû ❝õ❛ P ❝â ❝❤✉♥❣ ♥❤➙♥ tû ✤ì♥
t❤ù❝ ❜➟❝ ❞÷ì♥❣ xa y b z c ✳ ❚❛ s✉② r❛ P ❝❤✐❛ ❤➳t ❝❤♦ xa y b z c . ❑❤æ♥❣ ♠➜t t➼♥❤
tê♥❣ q✉→t✱ t❛ ❝â t❤➸ ❣✐↔ sû a ≥ 1. ❑❤✐ ✤â P ❝❤✐❛ ❤➳t ❝❤♦ x. ❉♦ ✤â

0 = P (0, y, z) = (y + z)q(0, y, z).
❚❛ s✉② r❛ q(0, y, z) = 0. ◆❤÷♥❣ ✤✐➲✉ ♥➔② ❧➔ ổ ỵ q ừ q ự
tû ❞↕♥❣ cy r z s . ✈➔ ❞♦ ✈➟② q(0, y, z) = cy r z s = 0✳
▼➦t q ừ sỡ ỗ t ❝õ❛ q ❧➔ ❧✐➯♥ t❤ỉ♥❣✳ ⑩♣ ❞ư♥❣

m+5
.
2
❇➡♥❣ ❝→❝❤ t❤✉➛♥ ♥❤➜t ❤â❛ ✤❛ t❤ù❝ fm (x, y) − 1 ❜ð✐ ❜✐➳♥ z ✈➔ s❛✉ ✤â
t❤❛② z ❜➡♥❣ −z ✱ t❛ ♥❤➟♥ ữủ ồ tự {Fm } ố ợ ú t
tự tr t =
ỵ t s✉② r❛ R(P ) ≥

m
2

(−1)k Km,k xm−2k y k z k . ✭✷✳✻✮

Fm (x, y, z) = xm − (−y)m − (−z)m +
k=1




ợ ộ tr m ữỡ t õ fm ❧➔ ✤❛ t❤ù❝ ❞✉② ♥❤➜t t❤ä❛ ♠➣♥
❜è♥ ✤✐➲✉ ❦✐➺♥ tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✶✳✼ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❬✸❪✳ ❚❛ ❣å✐ η ❧➔
♠ët ❝➠♥ ♥❣✉②➯♥ t❤õ② ❜➟❝ m ❝õ❛ 1✱ ✤➦t
m−1

(1 − η j x − η 2j y).

fm,2 (x, y) = 1



j=0

ợ m trữợ f = fm,2 ❧➔ ✤❛ t❤ù❝ ❞✉② ♥❤➜t t❤ä❛ ♠➣♥
❜è♥ ✤✐➲✉ ❦✐➺♥✿

✭✶✮ f (0, 0) = 0;
✭✷✮ f (x, y) = 1 ❦❤✐ x + y = 1;
✭✸✮ f ❝â ❜➟❝ m;

✭✹✮ f ηx, η 2 y = f (x, y) ✈ỵ✐ η ❧➔ ♠ët ❝➠♥ ♥❣✉②➯♥ t❤õ② ❜➟❝ m ❝õ❛ ✤ì♥ ✈à

✭f t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ♥➔② t❛ ❣å✐ f ❧➔ Γ(m, 2) ✲ ❜➜t ❜✐➳♥✮✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â t❤➸ ❦✐➸♠ tr❛ trü❝ t✐➳♣ fm,2 t❤♦↔ ♠➣♥ ❜è♥ t➼♥❤ ❝❤➜t
tr➯♥✳ ❇➙② ❣✐í t❛ ❣✐↔ sû ❝â ✤❛ t❤ù❝ f (x, y) t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ❝→❝ t➼♥❤ ❝❤➜t
✭✶✮✲✭✹✮✳ ❚❛ ✈✐➳t f (x, y) =

crs xr y s ✭crs ❦❤→❝ ❦❤æ♥❣✮✳ ❑❤✐ ✤â f ηx, η 2 y =

crs xr y s η r+2s ❚❤❡♦ t➼♥❤ ❝❤➜t ✭✹✮✱ f (x, y) = f ηx, η 2 y ✱ ✈➔ s♦ s→♥❤ ❤➺ sè
❤❛✐ ✤❛ t❤ù❝ ♥➔②✱ t❛ s✉② r❛ crs = crs η r+2s ✳ ❉♦ ✈➟② η r+2s = 1 ✈➔ ❞♦ ✤â r + 2s
❝❤✐❛ ❤➳t ❝❤♦ m✳
❚❤❡♦ t➼♥❤ ❝❤➜t ✭✶✮✱ f ❦❤æ♥❣ ❝â ❤↕♥❣ tû ❤➡♥❣✳ ❚❤❡♦ t➼♥❤ ❝❤➜t ✭✸✮✱ t❛ ❝❤➾
❝➛♥ ①➨t r + 2s = km ợ r + s m ữ t s r❛ ❝→❝ ✤ì♥ t❤ù❝ ①✉➜t
❤✐➺♥ tr♦♥❣ f ❧➔ xkm−2s y s ✈ỵ✐ 1 ≤ k ≤ 2 ✈➔

mk
m(k − 1)
≤s≤
.
2−1
2
❇➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤♦ t❛ ❦❤➥♥❣ ✤à♥❤✿ ♥➳✉ 0 ≤ s ≤ m ✈➔ s ❧➔ sè ♠ô ❝õ❛

y tr♦♥❣ ❝→❝ ✤ì♥ t❤ù❝ ❜➜t ❜✐➳♥ t❤➻ ♥â ①✉➜t ❤✐➺♥ ♥❤✐➲✉ ♥❤➜t ♠ët ❧➛♥✳
❚❛ ❝â t❤➸ ✈✐➳t
f (x, y) =

crs xr y s

tr♦♥❣ ✤â r + 2s = km ✈ỵ✐ sè ♥❣✉②➯♥ ❦ t❤ä❛ ♠➣♥ 1 ≤ k ≤ 2 ✈➔ r + s ≤ m✳
❚❛ t❤✉➛♥ ♥❤➜t ❤â❛ ✤❛ t❤ù❝ f ❜➡♥❣ ❝→❝❤ t❤❛② t❤➳ ♠é✐ ✤ì♥ t❤ù❝ crs xr y s ❜ð✐

crs xr y s (x + y)m−r−s ✳ ❑❤✐ ✤â✱ t❛ s➩ ♥❤➟♥ ✤÷đ❝ ✤❛ t❤ù❝ g t❤ä❛ ♠➣♥ ✭✶✮✱ ✭✷✮


✶✻

✈➔ ✭✸✮✱ ✈➔ g ❝ô♥❣ ❧➔ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ m✳ ❚➼♥❤ ❝❤➜t t❤✉➛♥ ♥❤➜t ✈➔
t➼♥❤ ❝❤➜t ✭✷✮ s✉② r❛ g(x, y) = (x + y)m ✳ ❚❤➟t ✈➟②✱ ♥➳✉ x + y = λ✱ ✈ỵ✐ λ ❧➔

x y
+ = 1 ✈➔ t❛ ❝â
λ λ
x y
(✈➻ g t❤✉➛♥ ♥❤➜t)
g(x, y) = λm g( , )
λ λ
= λm
(✈➻ g t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ✭✷✮)

♠ët sè ♣❤ù❝ ❦❤→❝ ✵✱ t❤➻

= (x + y)m .
❉♦ ✤â g(x + y) = (x + y)m ♥❤÷ ❧➔ ✤❛ t❤ù❝✳ ◆❤÷ ✈➟②✱ t❛ ❝â

(x + y)m = g(x, y) =

crs xr y s (x + y)m−r−s .

✭✷✳✽✮

❚❛ ❝â ✤❛ t❤ù❝ xr y s (x + y)m−r−s ❝â ❜➟❝ t❤❡♦ ❜✐➳♥ x ❧✉æ♥ ❦❤→❝ ♥❤❛✉ ✈➻
♠é✐ s ❝❤➾ ①✉➜t ❤✐➺♥ ✤ó♥❣ ♠ët ❧➛♥ ✈➻ ✈➟② ❝→❝ ✤❛ t❤ù❝ ♥➔② ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥
t➼♥❤✳ ❉♦ ✈➟②✱ tø ❝æ♥❣ t❤ù❝ ✭✷✳✽✮ t❛ s✉② r❛ ❝→❝ ❤➺ sè crs ✤÷đ❝ ①→❝ ✤à♥❤ ❞✉②
♥❤➜t✳ ❱➻ ✈➟② f ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❜ð✐ ❜è♥ t➼♥❤ ❝❤➜t tr➯♥✳
❈→❝ ❦➳t q✉↔ t✐➳♣ t❤❡♦ s➩ ❞➛♥ ❝❤♦ t❛ t❤➜② fm,2 ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝
✭✷✳✼✮ trị♥❣ ✈ỵ✐ ✤❛ t❤ù❝ fm ❜ð✐ ❝ỉ♥❣ t❤ù❝ ✭✷✳✺✮✱ ✈➔ ♥❣♦➔✐ r❛ ♥â ❝á♥ ❝â ♠ët
sè ❜✐➸✉ ❞✐➵♥ ✈æ ❝ị♥❣ ❜➜t ♥❣í ♠➔ ♥❤➻♥ ❜➲ ♥❣♦➔✐ ❝❤ó♥❣ r➜t ➼t sü ❧✐➯♥ q✉❛♥✳
❚❛ ✤➦t✿

A1 = x; A2 = x2 + 2y

✭✷✳✾✮

An+2 = xAn+1 + yAn

✭✷✳✶✵✮

❑❤✐ ✤â t❛ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ n ≥ 1 ✤÷đ❝ r➡♥❣ degAn = n
ỗ tớ ợ ồ tở C t õ

Am (x, λ2 y) = λm Am (x, y).
❝ô♥❣ ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ m✳ ❚❤➟t ✈➟② t❛ ❝â

Am+2 (λx, λ2 y) = (λx)Am+1 (λx, λ2 y) + (λ2 y)Am (λx, λ2 y)
= (λx).λm+1 Am+1 (x, y) + λ2 y.λm Am (x, y)
= λm+2 (xAm+1 (x, y) + yAm (x, y))
= λm+2 Am+2 (x, y).

✭✷✳✶✶✮


✶✼

✣➦❝ ❜✐➺t ❦❤✐ λ ❧➔ ❝➠♥ ♥❣✉②➯♥ t❤õ② ❜➟❝ m ❝õ❛ ✤ì♥ ✈à tø ❝ỉ♥❣ t❤ù❝ ✭✷✳✶✶✮
t❛ ❝â

Am (λx, λ2 y) = Am (x, y).

✭✷✳✶✷✮

❚❛ ✤➦t hm = Am + (−1)m+1 y m . ❑❤✐ ✤â t❛ ❝â hm = fm,2 .

▼➺♥❤ ✤➲ ✷✳✶✳✽✳ ❱ỵ✐ ♠å✐ m ≥ 1 t❛ ❝â fm,2 = Am + (−1)m+1ym := hm.
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ❜➡♥❣ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ hm t❤ä❛ ♠➣♥
❝→❝ t➼♥❤ ❝❤➜t ✭✶✮ ✤➳♥ ✭✹✮ tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✶✳✼
❚❛ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ hm t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t ✭✶✮ ✈➔ ✭✸✮✳ ❚➼♥❤ ❝❤➜t
✭✹✮ ✤÷đ❝ s✉② r❛ tø ❝ỉ♥❣ t❤ù❝ ✭✷✳✶✷✮✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ hm t❤ä❛ ♠➣♥ t➼♥❤
❝❤➜t ✭✷✮ ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ m✿
✭✰✮ ❱ỵ✐ m = 1 t❛ ❝â h1 = x + y.
✭✰✮ ❱ỵ✐ m = 2 t❛ ❝â h2 = A2 − y 2 = x2 + 2y − y 2 = 1 ❦❤✐ x + y = 1✳
✭✰✮ ❱ỵ✐ m ≥ 2 t❛ ❝â

hm+1 (x + y) = Am+1 (x, y) + (−1)m+2 y m+1
= xAm (x, y) + yAm−1 (x, y) + (−1)m y m+1
= x(Am (x, y) + (−1)m+1 y m ) − x(−1)m+1 y m + y(Am−1 (x, y)
+ (−1)m y m−1 ) − y(−1)m y m−1
= x(hm ) + y(hm−1 ) + (−1)m xy m − (−1)m y m + (−1)m y m+1
= x(hm ) + y(hm−1 ) + (−1)m y m (x + y) − (−1)m y m

❚r➯♥ ✤÷í♥❣ t❤➥♥❣ x + y = 1 t❛ ❝â hm = hm−1 = 1 ✭t❤❡♦ ❣✐↔ t❤✐➳t q✉②
♥↕♣✮ ♥➯♥

hm+1 (x + y) = x + y + 0 = 1.
❱➟② hm t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ✭✷✮✳ ⑩♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✷✳✶✳✼ t❛ s✉② r❛

fm,2 = Am + (−1)m+1 y m .
❚✐➳♣ t❤❡♦✱ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ Am ✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ fm (x, y) trị♥❣
✈ỵ✐ fm,2 ✳ ❉♦ ✤â t❛ s➩ ❝â ❦➳t ❧✉➟♥ fm ❧➔ ✤❛ t❤ù❝ ❞✉② ♥❤➜t t❤ä❛ ♠➣♥ ❜è♥
t➼♥❤ ❝❤➜t tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✶✳✼✳ ❚❛ ✤➦t✿
m
2

Km,k xm−2k y k ,

Bm = xm +
k=1

✭✷✳✶✸✮


✶✽

Km,k =

m−k
k

+

m−1−k
k−1

✈ỵ✐

1≤k≤

m
.
2

▼➺♥❤ ✤➲ ✷✳✶✳✾✳ ❱ỵ✐ ❦➼ ❤✐➺✉ tr➯♥✱ Bm t❤ä❛ ♠➣♥ ❝→❝ ❝æ♥❣ t❤ù❝ ✭✷✳✾✮ ✈➔
✭✷✳✶✵✮✳ ❉♦ ✈➟② Bm = Am ✈➔ t❛ ❝â fm (x, y) = fm,2 (x, y).

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â B1 = x = A1 ✈➔ B2 = x2 + 2y = A2 ✳ ▼➦t ❦❤→❝✱ ✈ỵ✐

m ≥ 1, t❛ ❝â
m+1
2

Km+1,k xm+1−2k y k )

xBm+1 + yBm = x(xm+1 +
k=1

m
2

+ y(xm +

Km,k xm−2k y k )
k=1

m+1
2

= xm+2 + xm y +

Km+1,k xm+2−2k y k
k=1
m
2

Km,k xm−2k y k+1

+
k=1

=x

m+2

m+2−2

+x

y(1 + Km+1,1 ) + B.

Ð ✤➙②✱
♥➳✉ m ❧➫ t❤➻ B =

m+1
2

(Km+1,k + Km,k−1 )xm+2−2k y k ✳

k=2
m
2

♥➳✉ m ❝❤➤♥ t❤➻ B =

k=2

(Km+1,k + Km,k−1 )xm+2−2k y k + Km, m2 y

m+2
2

▼➦t ❦❤→❝✱ t❛ ❧➛♥ ❧÷đt t➼♥❤ ❝→❝ ❤➺ tû ❝õ❛ ✤❛ t❤ù❝ tr➯♥
✭✰✮ Km+1,k + Km,k−1

=

m+1−k
k

+

m−k

k−1

=

m+1−k
k

+

m+1−k
k−1

+

=

m+2−k
k

+

m+1−k
k−1

= Km+2,k.

+

m−k+1
m−k

+
k−1
k−2
m−k
k−1

+

m−k
k−2

✳


✶✾

✭✰✮ m = 2n✭❝❤➤♥✮✿ Km, m2 =

m
m−1
+
1
0

✭✰✮ 1 + Km+1,1 = 1 +

=

m+1
m

+
1
0
❚ø ✤â t❛ ❝â✿

n
n−1
+
n
n−1

= Km+2, m+2
2 .

=m+2

= Km+2,1.

m+2
2

Km+2,k xm+2−2k y k + xm+2 = Bm+2 .

xBm+1 + yBm =
k=1

❚ù❝ ❧➔ Bm+2 = xBm+1 + yBm ❤❛② Bm = Am ✈ỵ✐ ♠å✐ m. ⑩♣ ❞ư♥❣ ▼➺♥❤
✤➲ ✷✳✶✳✽ t❛ s✉② r❛
m
2

Km,k xm−2k y k = fm,2 (x, y).

fm (x, y) = xm − (−y)m +
k=1

◆❤÷ tr➯♥ t❛ ✤➣ ✤➲ ❝➟♣✱ ✤❛ t❤ù❝ Fm ❝â ✤÷đ❝ ❜➡♥❣ ❝→❝❤ t❤✉➛♥ ♥❤➜t ❤â❛
✤❛ tự fm (x, y) 1 rỗ t z −z ✳
m
2

Fm (x, y, z) = xm − (−y)m − (−z)m +

(−1)k Km,k xm−2k y k z k .
k=1

❚❛ ❝â ❦➳t q✉↔ s❛✉ ✤➙② ✈➲ Fm ✳

▼➺♥❤ ✤➲ ✷✳✶✳✶✵✳ ❚❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤➙②✿
✭✐✮ Fm (x, y, z) ❝❤✐❛ ❤➳t ❝❤♦ x + y + z ✈➔ t❤÷ì♥❣ ❧➔ ♠ët ✤❛ t❤ù❝ ✤➛② ✤õ✳

m+5
✳ ❱➻ ✈➟② Fm ❧➔ ✤❛ t❤ù❝ s❤❛r♣✳
2
❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❚❛ ❝❤ù♥❣ ♠✐♥❤ Fm = Qm .(x + y + z) ✈ỵ✐
✭✐✐✮ ◆➳✉ m ❧➫✱ R(Fm ) =

min{m−1−j,j−1}

m−1

Qm =

(−1)

j

j=1

k=0
m
2

(−1)j

+
j=0

m−1−j
k

xm−1−k−j z j y k + y j z k

m−1−j
xm−1−2j y j z j ,
j

✭✷✳✶✹✮

✷✵

❚❤➟t ✈➟②✱ t❛ ✤➦t Qm =

γ(a, b, c)xa y b z c ❦❤✐ ✤â
a + min{b, c}
a + b + c = m − 1 ✈➔ γ(a, b, c) = (−1)max{b,c}
.
min{b, c}
✣➸ ❝❤ù♥❣ ♠✐♥❤ Fm = Qm .(x + y + z) t❛ s➩ ❦✐➸♠ tr❛ ❝→❝ ❤➺ sè ❝õ❛ ❤❛✐ ✤❛
t❤ù❝✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ❝❤ó♥❣ ❜➡♥❣ ♥❤❛✉✳
✭✰✮ ❉➵ ❦✐➸♠ tr❛ ❝❤♦ ♥❤ú♥❣ ❤➺ sè ❝õ❛ xm , y m , z m . ●å✐ α(A, B, C) ❧➔ ❤➺ sè

xA y B z C tr♦♥❣ Qm .(x + y + z)✳ ❑❤✐ ✤â s✉② r❛
α(A, B, C) = γ(A − 1, B, C) + γ(A, B − 1, C) + γ(A, B, C − 1).
❚r÷í♥❣ ❤đ♣ B > C :

α(A, B, C) = (−1)B

A−1+C
C

+ (−1)B−1

A+C
C
+ (−1)B

= (−1)B

A−1+C
C

+

A+C −1
C −1

A−1+C
C −1
+ (−1)B−1

= (−1)B

A+C
C

A+C
C

+ (−1)B−1

A+C
C

= 0.

❚r÷í♥❣ ❤đ♣ B < C, ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝â α(A, B, C) = 0.
❚r÷í♥❣ ❤ñ♣ B = C, t❛ ❝â α(A, B, C) ❧➔ ❤➺ sè ❝õ❛ xm−2B y B z B tr♦♥❣ ✤❛
t❤ù❝ Qm .(x + y + z). ợ ữ ỵ A + 2B = m ✈➔ A + B = m − B t❛ ❝â

α(A, B, C) = γ(A − 1, B, B) + γ(A, B − 1, B) + γ(A, B, B − 1)
= (−1)B

A−1+B
B

+ (−1)B

A+B−1
B−1
+ (−1)B

= (−1)B

A+B
B

= (−1)B Km,B .

+ (−1)B

A+B−1
B−1

A+B−1
B−1