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  1. ❈♦♠♣✉t✐♥❣ ❝r✐t✐❝❛❧ ♣♦✐♥ts ✉s✐♥❣ ●rö❜♥❡r ❇❛s❡s✿ ❈♦♠♣❧❡①✐t② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ❏❡❛♥✲❈❤❛r❧❡s ❋❛✉❣èr❡ ▼♦❤❛❜ ❙❛❢❡② ❊❧ ❉✐♥ P✐❡rr❡✲❏❡❛♥ ❙♣❛❡♥❧❡❤❛✉❡r ❯P▼❈ ✕ ❈◆❘❙ ✕ ■◆❘■❆ P❛r✐s ✲ ❘♦❝q✉❡♥❝♦✉rt ▲■P✻ ✕ ❙❆▲❙❆ t❡❛♠ ❙é♠✐♥❛✐r❡ ❆❧❣♦r✐t❤♠s ❏❛♥✉❛r② ✸✶✱ ✷✵✶✶ ✶✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  2. ❈r✐t✐❝❛❧ ♣♦✐♥ts ▲❡t ❢ ✶ ❢ ♣ ① ✶ ① ♥ ❜❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❉ ✭ ♣ ♥ ✶✮ ❛♥❞ ▼ ❜❡ t❤❡ ♣ ♥ ✶ ♠❛tr✐① ❢ ✶ ❢ ✶ ① ✷ ① ♥ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✳ ❢ ♣ ❢ ♣ ① ✷ ① ♥ ♥ s✉❝❤ t❤❛t ❢ ✶ ① ❈♦♠♣✉t❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ① ❢ ♣ ① ✵ ❛♥❞ ❘❛♥❦ ▼ ♣ ✳ ❑ ✵ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t② ✳ P♦❧②♥♦♠✐❛❧ s②st❡♠ s♦❧✈✐♥❣ ✳ Pr♦❜❧❡♠ st❛t❡♠❡♥t ( ① − ✷ ) ✷ + ( ② − ✷ ) ✷ + ③ ✷ − ✶ = ✵ . � � ✷ ( ① − ✷ ) ✷ ( ② − ✷ ) ✷ ③ ▼ = ✶ ✵ ✵ ✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  3. ❈r✐t✐❝❛❧ ♣♦✐♥ts ▲❡t ❢ ✶ ❢ ♣ ① ✶ ① ♥ ❜❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❉ ✭ ♣ ♥ ✶✮ ❛♥❞ ▼ ❜❡ t❤❡ ♣ ♥ ✶ ♠❛tr✐① ❢ ✶ ❢ ✶ ① ✷ ① ♥ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✳ ❢ ♣ ❢ ♣ ① ✷ ① ♥ ♥ s✉❝❤ t❤❛t ❢ ✶ ① ❈♦♠♣✉t❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ① ❢ ♣ ① ✵ ❛♥❞ ❘❛♥❦ ▼ ♣ ✳ ❑ ✵ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t② ✳ P♦❧②♥♦♠✐❛❧ s②st❡♠ s♦❧✈✐♥❣ ✳ Pr♦❜❧❡♠ st❛t❡♠❡♥t ( ① − ✷ ) ✷ + ( ② − ✷ ) ✷ + ③ ✷ − ✶ = ✵ . � � � � ✷ ( ② − ✷ ) ▼ = ✷ ③ = ✵ ✵ ✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  4. ❑ ✵ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t② ✳ P♦❧②♥♦♠✐❛❧ s②st❡♠ s♦❧✈✐♥❣ ✳ Pr♦❜❧❡♠ st❛t❡♠❡♥t ( ① − ✷ ) ✷ + ( ② − ✷ ) ✷ + ③ ✷ − ✶ = ✵ . � � � � ✷ ( ② − ✷ ) ▼ = ✷ ③ = ✵ ✵ ❈r✐t✐❝❛❧ ♣♦✐♥ts ▲❡t ❢ ✶ , . . . , ❢ ♣ ∈ Q [ ① ✶ , . . . , ① ♥ ] ❜❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❉ ✭ ♣ ≤ ♥ − ✶✮ ❛♥❞ ▼ ❜❡ t❤❡ ♣ × ( ♥ − ✶ ) ♠❛tr✐①  ∂ ❢ ✶ ∂ ❢ ✶  . . . ∂ ① ✷ ∂ ① ♥ ✳ ✳ ✳✳✳  ✳ ✳   . ✳ ✳    ∂ ❢ ♣ ∂ ❢ ♣ . . . ∂ ① ✷ ∂ ① ♥ ❈♦♠♣✉t❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ① ∈ R ♥ s✉❝❤ t❤❛t ❢ ✶ ( ① ) = · · · = ❢ ♣ ( ① ) = ✵ ❛♥❞ ❘❛♥❦ ( ▼ ) < ♣ ✳ ✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  5. Pr♦❜❧❡♠ st❛t❡♠❡♥t ( ① − ✷ ) ✷ + ( ② − ✷ ) ✷ + ③ ✷ − ✶ = ✵ . � � � � ✷ ( ② − ✷ ) ▼ = ✷ ③ = ✵ ✵ ❈r✐t✐❝❛❧ ♣♦✐♥ts ▲❡t ❢ ✶ , . . . , ❢ ♣ ∈ Q [ ① ✶ , . . . , ① ♥ ] ❜❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❉ ✭ ♣ ≤ ♥ − ✶✮ ❛♥❞ ▼ ❜❡ t❤❡ ♣ × ( ♥ − ✶ ) ♠❛tr✐①  ∂ ❢ ✶ ∂ ❢ ✶  . . . ∂ ① ✷ ∂ ① ♥ ✳ ✳ ✳✳✳  ✳ ✳   . ✳ ✳    ∂ ❢ ♣ ∂ ❢ ♣ . . . ∂ ① ✷ ∂ ① ♥ ❈♦♠♣✉t❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ① ∈ R ♥ s✉❝❤ t❤❛t ❢ ✶ ( ① ) = · · · = ❢ ♣ ( ① ) = ✵ ❛♥❞ ❘❛♥❦ ( ▼ ) < ♣ ✳ ❑ ✵ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t② ✳ � P♦❧②♥♦♠✐❛❧ s②st❡♠ s♦❧✈✐♥❣ ✳ ✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  6. ❖t❤❡r ❛♣♣❧✐❝❛t✐♦♥s✿ ❙❛♠♣❧✐♥❣ ♣♦✐♥ts ❈♦♥♥❡❝t✐✈✐t② q✉❡r✐❡s ◗✉❛♥t✐✜❡r ❡❧✐♠✐♥❛t✐♦♥ ✳ ✳ ✳ ❈❛♥♥②✱ ●r✐❣♦r✐❡✈✱ ❱♦r♦❜❥♦✈✱ ❇❛s✉✱ P♦❧❧❛❝❦✱ ❘♦②✱ ❇❛♥❦✱ ●✐✉st✐✱ ❍❡✐♥t③✱ P❛r❞♦✱ ❘♦✉✐❧❧✐❡r✱ ❙❛❢❡②✱ ❙❝❤♦st✱ ❍♦♥❣✱ ✳ ✳ ✳ ▼♦t✐✈❛t✐♦♥s P♦❧②♥♦♠✐❛❧ ▼✐♥✐♠✐③❛t✐♦♥  ( ① ✶ ,..., ① ♥ ) ∈ R ♥ ① ✶ ♠✐♥  s✳t✳ ❢ ✶ ( ① ✶ , . . . , ① ♥ ) = · · · = ❢ ♣ ( ① ✶ , . . . , ① ♥ ) = ✵ .  ❆ss✉♠♣t✐♦♥ ✿ t❤❡ ♠✐♥✐♠✉♠ ✐s r❡❛❝❤❡❞ ♦♥ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ❡①❛❝t s♦❧✉t✐♦♥✴❝❡rt✐✜❡❞ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✳ ✸✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  7. ❖t❤❡r ❛♣♣❧✐❝❛t✐♦♥s✿ ❙❛♠♣❧✐♥❣ ♣♦✐♥ts ❈♦♥♥❡❝t✐✈✐t② q✉❡r✐❡s ◗✉❛♥t✐✜❡r ❡❧✐♠✐♥❛t✐♦♥ ✳ ✳ ✳ ❈❛♥♥②✱ ●r✐❣♦r✐❡✈✱ ❱♦r♦❜❥♦✈✱ ❇❛s✉✱ P♦❧❧❛❝❦✱ ❘♦②✱ ❇❛♥❦✱ ●✐✉st✐✱ ❍❡✐♥t③✱ P❛r❞♦✱ ❘♦✉✐❧❧✐❡r✱ ❙❛❢❡②✱ ❙❝❤♦st✱ ❍♦♥❣✱ ✳ ✳ ✳ ▼♦t✐✈❛t✐♦♥s P♦❧②♥♦♠✐❛❧ ▼✐♥✐♠✐③❛t✐♦♥  ( ① ✶ ,..., ① ♥ ) ∈ R ♥ ℓ ( ① ✶ , . . . , ① ♥ ) ♠✐♥  s✳t✳ ❢ ✶ ( ① ✶ , . . . , ① ♥ ) = · · · = ❢ ♣ ( ① ✶ , . . . , ① ♥ ) = ✵ .  ❆ss✉♠♣t✐♦♥ ✿ t❤❡ ♠✐♥✐♠✉♠ ✐s r❡❛❝❤❡❞ ♦♥ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ❡①❛❝t s♦❧✉t✐♦♥✴❝❡rt✐✜❡❞ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✳ ✸✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  8. ▼♦t✐✈❛t✐♦♥s P♦❧②♥♦♠✐❛❧ ▼✐♥✐♠✐③❛t✐♦♥  ( ① ✶ ,..., ① ♥ ) ∈ R ♥ ℓ ( ① ✶ , . . . , ① ♥ ) ♠✐♥  s✳t✳ ❢ ✶ ( ① ✶ , . . . , ① ♥ ) = · · · = ❢ ♣ ( ① ✶ , . . . , ① ♥ ) = ✵ .  ❆ss✉♠♣t✐♦♥ ✿ t❤❡ ♠✐♥✐♠✉♠ ✐s r❡❛❝❤❡❞ ♦♥ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ❡①❛❝t s♦❧✉t✐♦♥✴❝❡rt✐✜❡❞ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✳ ❖t❤❡r ❛♣♣❧✐❝❛t✐♦♥s✿ ❙❛♠♣❧✐♥❣ ♣♦✐♥ts ❈♦♥♥❡❝t✐✈✐t② q✉❡r✐❡s ◗✉❛♥t✐✜❡r ❡❧✐♠✐♥❛t✐♦♥ ✳ ✳ ✳ ❈❛♥♥②✱ ●r✐❣♦r✐❡✈✱ ❱♦r♦❜❥♦✈✱ ❇❛s✉✱ P♦❧❧❛❝❦✱ ❘♦②✱ ❇❛♥❦✱ ●✐✉st✐✱ ❍❡✐♥t③✱ P❛r❞♦✱ ❘♦✉✐❧❧✐❡r✱ ❙❛❢❡②✱ ❙❝❤♦st✱ ❍♦♥❣✱ ✳ ✳ ✳ ✸✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

  9. ●❡♦♠❡tr✐❝ ❛❧❣♦r✐t❤♠s ✿ ❇❛s✉✱ P♦❧❧❛❝❦✱ ❘♦②✱ ❇❛♥❦✱ ●✐✉st✐✱ ❍❡✐♥t③✱ ▼✬❇❛❦♦♣✱ P❛r❞♦✱ ❙❝❤♦st✱ ❙❛❢❡②✱✳✳✳ ●rö❜♥❡r ❜❛s✐s ✇❡❧❧✲s✉✐t❡❞ ❢♦r ❞❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s ✭❋❛✉❣èr❡✱ ❙❛❢❡②✱ ❙✳ ■❙❙❆❈ ✷✵✶✵✮✳ ❊①♣❡r✐♠❡♥t❛❧❧②✱ ❝♦♠♣✉t✐♥❣ ❝r✐t✐❝❛❧ ♣♦✐♥ts ✇✐t❤ ●❇ ❛❧❣♦r✐t❤♠s ✭❘❆●❧✐❜✱ ❋●❜ ✐♥ ▼❛♣❧❡✮ ❝❛♥ s♦❧✈❡ ❧❛r❣❡ ♣r♦❜❧❡♠s✳ ●♦❛❧ ✶ ❆r✐t❤♠❡t✐❝ ❝♦♠♣❧❡①✐t② ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❝r✐t✐❝❛❧ ♣♦✐♥ts ✳ ●♦❛❧ ✷ ❯s❡ t❤❡ str✉❝t✉r❡ ♦❢ ❋ ▼❛①▼✐♥♦rs ▼ t♦ s♣❡❡❞✲✉♣ t❤❡ ❝♦♠♣✉t❛t✐♦♥s✳ ▼♦t✐✈❛t✐♦♥s ✭■■✮ � ♥ − ✶ ❉ ♣ ( ❉ − ✶ ) ♥ − ♣ ✳ � ●❡♥❡r✐❝❛❧❧②✱ ♥❜✳ ❝r✐t✳ ♣♦✐♥ts✿ ♣ − ✶ ✹✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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