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♦→♥ ❝➜♣ ▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ ❚❙✳ ◆●➷ ❚❍➚ ◆●❖❆◆ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✷✵ ✐ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ t❤ü❝ ❤✐➺♥ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ ✣↕✐ t ữợ sỹ ữợ ❝õ❛ ❚❙✳ ◆❣ỉ ❚❤à ◆❣♦❛♥✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ tỵ✐ ữớ ữợ 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❤❛♠ ❣✐❛ ❣✐↔♥❣ ợ trữớ ỏ ❝❤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♥❣ ✶✳ tự ố ự ỹ tr 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 } ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ❈❤÷ì♥❣ ✷✳ ▼ët ❧ỵ♣ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà ✈➔ ✤❛ 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✳ ❱➼ ❞ư✿ ❈→❝ ❤➺ sè ❝õ❛ ❝❤ó♥❣ ❧➔ số õ t ữủ t ữợ tê♥❣ ❝õ❛ ❝→❝ ❤➺ sè ♥❤à t❤ù❝ ✈➔ sð ❤ú✉ ♠ët t➼♥❤ ❝❤➜t ❝❤✐❛ ❤➳t✳ ❍ì♥ ♥ú❛✱ ❝→❝ ✤❛ t❤ù❝ ữủ t ố ỡ ợ ởt t ủ ❝→❝ ✤❛ t❤ù❝ ✤÷đ❝ s✐♥❤ r❛ ♥❤÷ ♥❤ú♥❣ ✈➼ ❞ư s❤❛r♣ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲ →♥❤ ①↕ ✤❛ t❤ù❝ r✐➯♥❣ ❣✐ú❛ ❝→❝ ❤➻♥❤ ❝➛✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ ♣❤ù❝✳ ◆❤✐➲✉ ❦➳t q✉↔ ❤❛② tr♦♥❣ t♦→♥ ❤å❝ ♠✐♥❤ ❤å❛ ❝❤♦ ♥❣✉②➯♥ ỵ rst r ỳ ố tữủ tọ ởt số ✤✐➲✉ ❦✐➺♥ ❝ü❝ trà ❝â ❝→❝ t➼♥❤ ❝❤➜t r➜t ✤➦❝ ❜✐➺t✳ ❈❤ó♥❣ t❛ ❝â ♠ët ✈➔✐ ♠✐♥❤ ❤å❛ ❝ì ❜↔♥ ỵ r số tt ❝→❝ ❤➻♥❤ ❝❤ú ♥❤➟t ❝â ❝ò♥❣ ❝❤✉ ✈✐✱ ❤➻♥❤ ❝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 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 trú ỗ ữỡ ❚r♦♥❣ ❈❤÷ì♥❣ ✶ t❛ tr➻♥❤ ❜➔② ✈➲ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✱ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà ❤❛✐ ❜✐➳♥✱ ✤❛ tự ố ự ỹ tr sỡ ỗ ◆❡✇t♦♥ ❜✐➸✉ ❞✐➵♥ ✤❛ t❤ù❝ ❜❛ ❜✐➳♥ ❝❤♦ 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 ❜✐➳♥ ♣❤ù❝✳ ❚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✳ ✷ ❈❤÷ì♥❣ ✶ tự ố ự ỹ tr sỡ ỗ t ✶✳✶✳ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❈❤♦ n ∈ N∗✱ tr➯♥ Nn t❛ ✤à♥❤ ♥❣❤➽❛ ♠ët q✉❛♥ ❤➺ t❤ù tü ♥❤÷ s ợ tỷ tý ỵ 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 ✱ < bi ✳ ◗✉❛♥ ❤➺ t❤ù tü ♥❤÷ tr➯♥ ❣å✐ ❧➔ q✉❛♥ ❤➺ t❤ù tü tø ✤✐➸♥✳ ✣➙② ❝ô♥❣ ❧➔ ♠ët q✉❛♥ ❤➺ t❤ù tü t♦➔♥ ♣❤➛♥✳ ❚❛ q✉② ÷ỵ❝ ✈✐➳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 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 B ✳ ❉♦ ✤â t❛ ❝â α(A, B, 0) = γ(A − 1, B, 0) + γ(A, B − 1, 0) = (−1)B + (−1)B−1 = ❚❛ ✈ø❛ ①➨t ①♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ❝â ➼t ♥❤➜t ♠ët tr♦♥❣ sè A, B ❤♦➦❝ C ❜➡♥❣ ❦❤ỉ♥❣✳ ❚r÷í♥❣ ❤đ♣ ✸✳❚✐➳♣ t❤❡♦ t❛ ①➨t A > B ≥ C > 0✳ ❚❛ ✈✐➳t (A, B, C) = (2r + − j, k, j − k) ✈ỵ✐ ❝→❝ sè tü ♥❤✐➯♥ j ✈➔ k t❤ä❛ ♠➣♥ j > k ✳ ❚❛ ❝â α(2r+1 − j, k, j − k) =γ(2r − j, k, j − k) + γ(2r − (j − 1), k − 1, (j − 1) − (k − 1)) + γ(2r − (j − 1), k, (j − 1) − k) =(−1)j j k + (−1)j−1 j−1 j−1 + (−1)j−1 k−1 k = ✭✷✳✷✸✮ ❚r÷í♥❣ ❤đ♣ ✹✳ ❚❛ ①➨t tr÷í♥❣ ❤đ♣ t✐➳♣ t❤❡♦ ❧➔ A = B > C > 0✳ ❚❛ ✈✐➳t (A, B, C) = (2r + − j, 2r + − j, 2j − 2r − 1) ✈ỵ✐ ♠ët sè j ✳ ❑❤✐ ✤â t❛ ❝â α(2r+1 − j, 2r + − j, 2j − 2r − 1) =γ(2r − j, 2r − (j − 1), 2(j − 1) − 2r + 1) + γ(2r − (j − 1), 2r − j, 2(j − 1) − 2r + 1) + γ(2r − (j − 1), 2r + − j, 2(j − 1) − 2r) =(−1)j−1 =(−1)j−1 j−1 2r − j j−1 2r − j + (−1)j−1 + =(−1)j−1 K2r+1,2r+1−j j−1 2r − j j 2r + − j + (−1)j−1 j−1 2r + − j ✷✻ ❚r÷í♥❣ ❤đ♣ ✺✳ ❚r÷í♥❣ ❤đ♣ ❝✉è✐ ❝ị♥❣✱ A = B = C = 2r 3+ t❤➻ t❛ ❝â 2r + = 6L + ✈➔ ❦❤✐ ✤â A = B = C = 2L + 1✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝â α(2L + 1, 2L + 1,2L + 1) =γ(2L, 2L + 1, 2L + 1) + γ(2L + 1, 2L, 2L + 1) + γ(2L + 1, 2L + 1, 2L) =3γ(2L + 1, 2L + 1, 2L) =3γ(2r − (4L + 1), 2L + 1, 2L) =3(−1)4L+1 4L + 2L + =(−1)4L+1 4L + 4L + 4L + + + 2L + 2L + 2L =(−1)4L+1 4L + + 2L + 4L + 2L + =(−1)4L+1 K6L+3,2L+1 ❱➻ ✈➟② ♥➳✉ r ≡ 0, mod t❤➻ t❛ ❝â 4r+1 s.q2r+1 =x2r+1 + y 2r+1 + z 2r+1 + (−1)j−1 K2r+1,2r+1−j j=r+1 (xy)2r+1−j z 2(j−r)−1 + (xz)2r+1−j y 2(j−r)−1 + x2(j−r)−1 (yz)2r+1−j r−2 =x2r+1 + y 2r+1 + z 2r+1 + r−J 2J+1 (xy) z (−1)r+J K2r+1,r−J J=0 r−J 2J+1 + (xz) y + x2J+1 (yz)r−J ❱➻ 2r + = 6L + tø ❜✐➸✉ t❤ù❝ ❝✉è✐ ❝ị♥❣ t❛ ♥❤➟♥ ✤÷đ❝ ❤➺ sè ❝õ❛ ❤↕♥❣ tû ù♥❣ ✈ỵ✐ ✤ì♥ t❤ù❝ x2L+1 y 2L+1 z 2L+1 (1)4L+1 K6L+3,2L+1 trũ ợ tỷ tữỡ ự Sm ố ợ trữớ ủ m t ✈✐➳t m = 2r ✈➔ ✤➦t q2r (x, y, z) = γ(a, b, c)xa y b z c , ✷✼ tr♦♥❣ ✤â γ(2r − − j, k, j − k) = (−1)j−1 j k , ✈ỵ✐ 2r − − j ≥ k ≥ j − k ≥ 0✳ ❚❛ ❝â γ(σ(2r − − j, k, j − k)) ❝â ❣✐→ trà ❦❤ỉ♥❣ ✤ê✐ ✈ỵ✐ ❜➜t ❦ý ❤♦→♥ ✈à σ ❝õ❛ (2r − − j, k, j − k)✳ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü tr➯♥ t❛ ❝ơ♥❣ ❝â s.q2r = S2r ứ ự ỵ t❛ t❤➜② ❦❤✐ m ≡ 1, mod 6✱ t❤➻ R(Sm ) = m+5 ✳ ❉♦ ✤â✱ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✳✶✵ ❝↔ Fm ✈➔ Sm ✤➲✉ ❧➔ ❝→❝ ✤❛ t❤ù❝ sr õ m ự sỹ tỗ t ❝õ❛ ✤❛ t❤ù❝ s❤❛r♣ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❞✉② ♥❤➜t ✈ỵ✐ ♠é✐ m ❧➫✳ ❉✬❆♥❣❡❧♦ ✈➔ ▲❡❜❧ ✤÷❛ r❛ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❝ö t❤➸ ✤➸ ①➙② ❞ü♥❣ ❝→❝ ✈➼ ❞ö ✈➲ ✤❛ t❤ù❝ s❤❛r♣ tø fm ❜➡♥❣ ❝→❝❤ t❤❛② ❝→❝ ❜✐➸✉ tự tr fm tự ỗ ữ t ♠♦❞✉❧♦(x + y − 1)✳ Ð ✤➙② ♠ët ❦ÿ t❤✉➟t t÷ì♥❣ tü ✤÷đ❝ sû ❞ư♥❣ ✤➸ ✤✐ tø Fm ✤➳♥ Sm ✳ ▼ët ❝❤➻❛ ❦❤â❛ q✉❛♥ trå♥❣ ♠➔ ❝❤ó♥❣ t❛ s➩ sû ❞ö♥❣ ❧➦♣ ❧↕✐ ð ✤➙② ❧➔ Fm (x, y, z) ≡ mod(x + y + z)✳ ❚❛ ✤➦t m r= ❦❤✐ ✤â r m (−1)k Km,k xm−2k y k z k Gm (x, y, z) := x + k=1 ≡ (−1)m [y m + z m ] mod(x + y + z) ✭✷✳✷✹✮ ▼➺♥❤ ✤➲ ✷✳✷✳✸ s❛✉ ✤➙② ♠ỉ t↔ tø♥❣ tr÷í♥❣ ❤đ♣ ✤➸ ❝â t❤➸ t ữủ Sm tứ Fm rữợ t t ❤å❛ q✉→ tr➻♥❤ tr➯♥ ❜➡♥❣ ♠ët ✈➼ ❞ö✳ ❱➼ ❞ö ✷✳✷✳✷✳ ❱ỵ✐ m = 13✱ t❛ ❝â F13 =x13 + y 13 + z 13 − 13x11 yz + 65x9 y z − 156x7 y z + 182x5 y z − 91x3 y z + 13xy z t❛ t❤➜② 156x7 y z = 65x7 y z + 91x7 y z ❚❤❛② ✈➔♦ ✈➔ s➢♣ ①➳♣ ❧↕✐✱ t❛ ✤÷đ❝ F13 =x13 + y 13 + z 13 − 13x11 yz + 65x9 y z − 65x7 y z + 13xy z ✷✽ − 91x7 y z + 182x5 y z − 91x3 y z =x13 + y 13 + z 13 + 13xyz x5 −x5 + 5x3 yz − 5xy z + y z − 91x3 y z x2 x2 − 2yz + y z ❚❛ ✤➦t −x5 + 5x3 yz − 5xy z = −G5 ✈➔ x2 − 2yz = G2 ✳ ❑➳t ❤đ♣ ✈ỵ✐ ✷✳✷✹ t❛ ❝â F13 =x13 + y 13 + z 13 + 13xyz x5 (−G5 ) + y z − 91x3 y z x2 (G2 ) + y z ≡x13 + y 13 + z 13 + 13xyz x5 y + z + y z − 91x3 y z x2 y + z + y z mod(x + y + z) =S13 ▼➺♥❤ ✤➲ ✷✳✷✳✸✳ ❈❤♦ Gm = Fm +(−y)m +(−z)m ✈ỵ✐ m ≥ ✈➔ r = ❑❤✐ ✤â t❛ ❝â m ✳ ✶✳ ◆➳✉ m = 6L + ❤♦➦❝ m = 6L − t❤➻ Fm =xm + y m + z m L−1 (−1)r−j Km,r−j (xyz)2j+1 (−x)r−3j−1 Gr−3j−1 + (yz)r−3j−1 , + j=0 ✭✷✳✷✺✮ ♥➳✉ m = 6L + t❤➻ Fm = ❱➳ ♣❤↔✐ ❝õ❛ ✭✷✳✷✺✮ + (−1)r−L Km,r−L (xyz)2L+1 ✭✷✳✷✻✮ ❚❛ ♥❤➟♥ ✤÷đ❝ Sm ❜➡♥❣ ❝→❝❤ t❤❛② (−1)r−3j−1 Gr−3j−1 ❜ð✐ ❜✐➸✉ t❤ù❝ y r−3j−1 + z r−3j−1 ✱ ❝❤ó♥❣ ỗ ữ ợ t (x + y + z)✳ ✷✳ ◆➳✉ m = 6L − ❤♦➦❝ m = 6L − 4✱ Fm = − xm − y m − z m L−1 (−1)r−j Km,r−j (xyz)2j (−x)r−3j Gr−3j + (yz)r−3j , + j=0 ✭✷✳✷✼✮ ✷✾ ❚r♦♥❣ ❦❤✐ ♥➳✉ m = 6L Fm = ❱➳ ♣❤↔✐ ❝õ❛ ✭✷✳✷✼✮ + (−1)r−L Km,r−L (xyz)2L ✭✷✳✷✽✮ ❚❛ ♥❤➟♥ ✤÷đ❝ −Sm ❜➡♥❣ ❝→❝❤ t❤❛② (−1)r−3j Gr−3j ❜➡♥❣ y r−3j + z r−3j ú ỗ ữ ợ t x + y + z ✳ ❚❛ s➩ sû ❞ö♥❣ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ fm ✤÷đ❝ ✤÷❛ r❛ tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✶✳✼ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ♥➔②✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ❝❤♦ tr÷í♥❣ ❤đ♣ m ❧➫✱ tr÷í♥❣ ❤đ♣ m ❝❤➤♥ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü✳ ❚❛ ✤➦t Pm ❧➔ ✈➳ ♣❤↔✐ ❝õ❛ ❝æ♥❣ t❤ù❝ ✭✷✳✷✺✮ ❤♦➦❝ ✭✷✳✷✻✮✳ ❘ã r➔♥❣✱ ❦❤✐ t❤❛② (−1)r−3j−1 Gr−3j−1 ❜ð✐ ❜✐➸✉ t❤ù❝ y r−3j−1 +z r−3j−1 t❛ ♥❤➟♥ ✤÷đ❝ ❝ỉ♥❣ t❤ù❝ Sm ✳ ◆❤÷ ✈➟② sỷ ỗ ữ tự t s r Pm ≡ Sm mod(x + y + z) ▼➦t ❦❤→❝ Sm = (x + y + z).qm ✭♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ỵ t s r Pm tr ❦❤✐ x + y + z = 0✳ ❱➻ ✈➟②✱ ♥➳✉ t❛ t❤❛② z ❜ð✐ −1 ✈➔ ❝ë♥❣ t❤➯♠ ❤➡♥❣ sè 1✱ t❛ ♥❤➟♥ ✤÷đ❝ ♠ët ✤❛ t❤ù❝ ❤❛✐ ❜✐➳♥ pm (x, y) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ tø ✭✶✮ ✤➳♥ ✭✹✮ tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✶✳✼✳ ❚❤➟t ✈➟② pm (x, y) ♥❤➟♥ ❣✐→ trà tr➯♥ ✤÷í♥❣ t❤➥♥❣ x + y = ✭❞♦ Pm ♥❤➟♥ ❣✐→ trà ❦❤✐ x + y + z = 0✮❀ pm (0, 0) = 0❀ pm (ηx, η y) = pm (x, y) ✈ỵ✐ η ❧➔ ♠ët ❝➠♥ ♥❣✉②➯♥ t❤õ② ❜➟❝ m ❝õ❛ ✤ì♥ ✈à❀ deg pm = m✳ ❚❤❡♦ ♠➺♥❤ ✤➲ ✷✳✶✳✼ ✈➔ ✷✳✶✳✾ t❛ s✉② r❛ pm (x, y) ≡ fm (x, y)✳ ✣➳♥ ✤➙②✱ t❛ ❧↕✐ trø pm (x, y) ✤✐ 1❀ t❤✉➛♥ ♥❤➜t ❤â❛ ✈➔ t❤❛② t❤➳ z ❜➡♥❣ −z t❛ ♥❤➟♥ ❧↕✐ ✤÷đ❝ Pm s✉② r❛ Pm = Fm ✳ ❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✷✳✷✳✹✳ Sm(x, y, z) = xm + ym + zm tr♦♥❣ Zm[x, y, z] ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ m ❧➔ sè ♥❣✉②➯♥ tè✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ ▼➺♥❤ ✤➲ ✷✳✷✳✸ t❛ ❝â ❝→❝ ❤➺ sè ❝õ❛ Sm ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❝→❝ ❤➺ sè ❝õ❛ Fm t q ỵ t❛ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✸✵ ❑➳t ❧✉➟♥ ❝õ❛ ❧✉➟♥ ✈➠♥ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❧ỵ♣ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ t❤✉➛♥ ♥❤➜t t❤ỉ♥❣ q✉❛ ✈✐➺❝ tr↔ ❧í✐ ❤❛✐ ❝➙✉ ❤ä✐ tê♥❣ q✉→t tr♦♥❣ ♥❤ú♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✳ ▲✉➟♥ ✈➠♥ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉ ✤➙②✿ ✶✳ ❱ỵ✐ ♠é✐ m ≥ 1, tỗ t t ởt tự f = fm (x, y) ❜➟❝ m s❛♦ ❝❤♦ f (0, 0) = 0, f (x, y) = ❦❤✐ x + y = ✈➔ f ❧➔ Γ(m, 2)✲ ❜➜t ❜✐➳♥✳ ✣❛ t❤ù❝ ♥➔② ❝â ♥❤ú♥❣ ❞↕♥❣ ❜✐➸✉ ❞✐➵♥ ❦❤→❝ ♥❤❛✉ r➜t t❤ó ✈à ✭t❤➸ ❤✐➺♥ tr♦♥❣ ❝→❝ ▼➺♥❤ ✤➲ ✷✳✶✳✼✱ ▼➺♥❤ ✤➲ ✷✳✶✳✽ ✈➔ ▼➺♥❤ ✤➲ ✷✳✶✳✾✮✳ ✷✳ ❳➙② ❞ü♥❣ ❝æ♥❣ t❤ù❝ ❝õ❛ ❤å Fm ✭m ≥ 1✮ ♥❤ú♥❣ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜❛ ❜✐➳♥ ❝❤✐❛ ❤➳t ❝❤♦ x + y + z ợ tữỡ ởt tự ✤➛② ✤õ✳ ✣➦❝ ❜✐➺t ❦❤✐ m ❧➫ Fm ❧➔ s❤❛r♣ ✭▼➺♥❤ ✤➲ ✷✳✶✳✶✵✮✳ ✸✳ ❳➙② ❞ü♥❣ ❝æ♥❣ t❤ù❝ ❝õ❛ ❤å Sm ✭m ≥ 1✮ ♥❤ú♥❣ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ü❝ trà ❜❛ ❜✐➳♥✱ ✤➦❝ ❜✐➺t ❦❤✐ m ❧➔ sè tü ♥❤✐➯♥ ❝❤✐❛ ❝❤♦ ❞÷ ❤♦➦❝ t❤➻ Sm ụ tự sr ỵ ố ❧✐➯♥ ❤➺ ❣✐ú❛ Sm ✈➔ Fm ✤÷đ❝ ①➙② ❞ü♥❣ t❤ỉ♥❣ q✉❛ t➼♥❤ ❝❤➜t ✤➦❝ ❜✐➺t ❝õ❛ ✤❛ t❤ù❝ fm tr♦♥❣ ♣❤➛♥ ✭✶✳✮ ✭▼➺♥❤ ✤➲ ✷✳✷✳✸✮✳ ✸✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ▲➯ ❚❤à ❚❤❛♥❤ ◆❤➔♥✱ ✭✷✵✶✺✮✱ ●✐→♦ tr➻♥❤ ỵ tt tự t ố ❣✐❛ ❍➔ ◆ë✐✳ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ❇r♦♦❦s ❏✳✱ P♦❧②♥♦♠✐❛❧s✧✱ ✭✷✵✶✾✮✱ ❆♠❡r✳ ✧❆♥ ■♥t❡r❡st✐♥❣ ❋❛♠✐❧② ♦❢ ▼♦♥t❤❧②✱ ✶✷✻✿✻✱ ✺✷✼✲✺✹✵✱ ▼❛t❤✳ ❙②♠♠❡tr✐❝ ❉❖■✿ ✶✵✳✶✵✽✵✴✵✵✵✷✾✽✾✵✳✷✵✶✾✳✶✺✽✹✺✶✹ ✳ ❬✸❪ ❉✬❆♥❣❡❧♦✱ ❏✳✱ ✭✷✵✵✹✮✱ ✧◆✉♠❜❡r✲t❤❡♦r❡t✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❝❡rt❛✐♥ ❈❘ ♠❛♣♣✐♥❣s✧✱ ❏✳ ●❡♦♠✳ ❆♥❛❧✳ ✶✹✭✷✮✿ ✷✶✺✲✷✷✾✳ ❬✹❪ ❉✬❆♥❣❡❧♦✱ ❏✳ P✳✱ ✭✶✾✾✸✮✱ ❙❡✈❡r❛❧ ❈♦♠♣❧❡① ❱❛r✐❛❜❧❡s ❛♥❞ t❤❡ ●❡♦♠❡tr② ♦❢ ❘❡❛❧ ❍②♣❡rs✉r❢❛❝❡s✳ ❇♦❝❛ ❘❛t♦♥✱ ❋▲✿ ❈❘❈ Pr❡ss✳ ❬✺❪ ❉✬❆♥❣❡❧♦✱ ❏✳✱ ❑♦s✱ ❙✳✱ ❘✐❡❤❧✱ ❊✳✱ ✭✷✵✵✸✮✱ ✧❆ s❤❛r♣ ❜♦✉♥❞ ❢♦r t❤❡ ❞❡❣r❡❡ ♦❢ ♣r♦♣❡r ♠♦♥♦♠✐❛❧ ♠❛♣♣✐♥❣s ❜❡t✇❡❡♥ ❜❛❧❧s✧✱ ❏✳ ●❡♦♠✳ ❆♥❛❧✳ ✶✸ ✭✹✮✿ ✺✽✶✲✺✾✸✳ ❬✻❪ ❉✬❆♥❣❡❧♦✱ ❏✳✱ ▲❡❜❧✱ ❏✳✱ ✭✷✵✵✾✮✱ ✧❈♦♠♣❧❡①✐t② r❡s✉❧ts ❢♦r ❈❘ ♠❛♣♣✐♥❣s ❜❡t✇❡❡♥ s♣❤❡r❡s✧✱ ■♥t ❏✳ ▼❛t❤✳ ✷✾✭✷✮✿ ✶✹✾✲✶✻✻✳ ❬✼❪ ▲❡❜❧✱ ❏✳✱ P❡t❡rs✱ ❍✳✱ ✭✷✵✶✶✮✱ ✧P♦❧②♥♦♠✐❛❧s ❝♦♥st❛♥t ♦♥ ❛ ❤②♣❡r♣❧❛♥❡ ❛♥❞ ❈❘ ♠❛♣s ♦❢ ❤②♣❡rq✉❛❞r✐❝✧✱ ▼♦s❝✳ ▼❛t❤✳ ❏✳ ✶✶✭✷✮✿ ✷✽✺✲✸✶✺✳