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❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❈♦♥❣r❡s♦ ❞❡ ❧❛ ❘❡❛❧ ❙♦❝✐❡❞❛❞ ▼❛t❡♠át✐❝❛ ❊s♣❛ñ♦❧❛ ✕ ➪✈✐❧❛ ✷✵✶✶ ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s

❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s K ✐s ❛♥ ❛❧❣❡❜r❛✐❝❛❧❧② ❝❧♦s❡❞ ✜❡❧❞ φ : P ✶ K → P ✷ K ♣♦❧②♥♦♠✐❛❧ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ φ ( t ) = ( ✉ ✶ ( t ) : ✉ ✷ ( t ) : ✉ ✸ ( t )) ❤♦♠♦❣❡♥❡♦✉s ✇✐t❤ ❣❝❞ ( ✉ ✐ ( t )) = ✶ ❈ := ■♠ ( φ ) ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s

❚❤❡ ✭❝♦♠♣✉t❛t✐♦♥❛❧✮ ■♠♣❧✐❝✐t✐③❛t✐♦♥ Pr♦❜❧❡♠ ●✐✈❡♥ ( ✉ ✶ ( t ) : ✉ ✷ ( t ) : ✉ ✸ ( t )) ✱ ❝♦♠♣✉t❡ ❊ ( ❳ ✶ , ❳ ✷ , ❳ ✸ ) ∈ K [ ❳ ✶ , ❳ ✷ , ❳ ✸ ] s✉❝❤ t❤❛t ❈ = { ❊ ( ❳ ✶ , ❳ ✷ , ❳ ✸ ) = ✵ } ✉ ( t ) = ( ✷ t ✶ t ✷ , t ✷ ✶ − t ✷ ✷ , t ✷ ✶ + t ✷ ❊ ( ❳ ✶ , ❳ ✷ , ❳ ✸ ) = ❳ ✷ ✶ + ❳ ✷ ✷ − ❳ ✷ ✷ ) → ✸ ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s

❚❤❡ ✭❝♦♠♣✉t❛t✐♦♥❛❧✮ ■♥✈❡rs✐♦♥ Pr♦❜❧❡♠ ●✐✈❡♥ ( ✉ ✶ ( t ) : ✉ ✷ ( t ) : ✉ ✸ ( t )) ✱ ❝♦♠♣✉t❡ ❋ ✶ ( ❳ ✶ , ❳ ✷ , ❳ ✸ ) , ❋ ✷ ( ❳ ✶ , ❳ ✷ , ❳ ✸ ) ∈ K [ ❳ ✶ , ❳ ✷ , ❳ ✸ ] s✉❝❤ t❤❛t ( ❋ ✶ : ❋ ✷ ) ✉ ( t ) ��� P ✶ K ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ P ✶ ❈ → ❈ K ✉ ( t ) = ( ✷ t ✶ t ✷ , t ✷ ✶ − t ✷ ✷ , t ✷ ✶ + t ✷ ✷ ) → ( ❋ ✶ , ❋ ✷ ) = ( ❳ ✷ + ❳ ✸ , ❳ ✶ ) ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s

✭♠❡t❛✮✲❋❛❝t ■❢ ❙✐♥❣ ( ❈ ) ✐s ✏s♠❛❧❧✑✱ t❤❡♥ ❜♦t❤ t❤❡ ✐♠♣❧✐❝✐t✐③❛t✐♦♥✴✐♥✈❡rs✐♦♥ ♣r♦❜❧❡♠ ❣❡ts ❡❛s✐❡r ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s

❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡s ❜② ❧✐♥❡s ❈ ❤❛s ♦♥❡ s✐♥❣✉❧❛r ♣♦✐♥t ♦❢ ♠✉❧t✐♣❧✐❝✐t② ❡q✉❛❧ t♦ ❞❡❣ ( ❈ ) − ✶ ❇② ❇é③♦✉t✬s t❤❡♦r❡♠✱ ❛♥② ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ s✐♥❣✉❧❛r ♣♦✐♥t ✐♥t❡rs❡❝ts ❈ ✐♥ ❛♥♦t❤❡r s✐♥❣❧❡ ♣♦✐♥t ❚❤❡ ♣❡♥❝✐❧ ♦❢ ❧✐♥❡s ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ s✐♥❣✉❧❛r ♣♦✐♥t ♣r♦❞✉❝❡s ❛ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ ✉ ( t ) ♦❢ t❤❡ ❝✉r✈❡ ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s

❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡s ❜② ❧✐♥❡s ❆ss✉♠❡ ♣ = ( ✵ : ✵ : ✶ ) ✐s t❤❡ s✐♥❣✉❧❛r ♣♦✐♥t ♦❢ ❈ ❋♦r ❛♥② ❛ ( t ) , ❜ ( t ) ♦❢ ❞❡❣r❡❡s ❞ − ✶ , ❞ r❡s♣❡❝t✐✈❡❧② s✉❝❤ t❤❛t ❣❝❞ ( ❛ , ❜ ) = ✶✱ ✇❡ ❤❛✈❡ ✉ ( t ) = ( t ✶ ❛ ( t ) , t ✷ ❛ ( t ) , ❜ ( t )) ❊ ( ❳ ) = ❜ ( ❳ ✶ , ❳ ✷ ) − ❳ ✸ ❛ ( ❳ ✶ , ❳ ✷ ) ( ❋ ✶ , ❋ ✷ ) = ( ❳ ✶ , ❳ ✷ ) ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s ❆♥② ❝✉r✈❡ ♣❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❧✐♥❡s ♣❛ss✐♥❣ t❤r♦✉❣ ♣ ✐s ♦♥❡ ♦❢ t❤❡ ❛❜♦✈❡

❚❤❡ ●❡♦♠❡tr② ♦❢ ❈ ❛r♦✉♥❞ ✐ts s✐♥❣✉❧❛r ♣♦✐♥t ❲r✐t❡ τ � ❛ ( ❳ ✶ , ❳ ✷ ) = ❝ ( ❞ ❥ ❳ ✷ − ❡ ❥ ❳ ✶ ) ν ❥ ❥ = ✶ ❝ ∈ K \ { ✵ } , ( ❞ ❥ : ❡ ❥ ) � = ( ❞ ❦ : ❡ ❦ ) ✐❢ ❥ � = ❦ , ν ❥ ∈ N ❚❤❡r❡ ❛r❡ τ ❞✐✛❡r❡♥t ❜r❛♥❝❤❡s ♦❢ ❈ ♣❛ss✐♥❣ t❤r♦✉❣❤ ♣ ❚❤❡ t❛♥❣❡♥t t♦ t❤❡ ❜r❛♥❝❤ γ ❥ ( t ) ❛t t ❥ ✐s t❤❡ ❧✐♥❡ ❞ ❥ ❳ ✷ − ❡ ❥ ❳ ✶ = ✵ ❉✐✛❡r❡♥t ❜r❛♥❝❤❡s ❤❛✈❡ ❞✐✛❡r❡♥t t❛♥❣❡♥ts ✭♥♦ t❛❝♥♦❞❡s✮ ❚❤❡ ♦r❞❡r ♦❢ ❝♦♥t❛❝t ♦❢ ❈ ✇✐t❤ ❞ ❥ ❳ ✷ − ❡ ❥ ❳ ✶ = ✵ ❛t ♣ ✐s ❡q✉❛❧ t♦ ν ❥ − ✶ ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s

❇♦♥✉s ❚r❛❝❦ ❆ s❡t ♦❢ ♠✐♥✐♠❛❧ ❣❡♥❡r❛t♦rs ♦❢ t❤❡ ❘❡❡s ❆❧❣❡❜r❛ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ ❣✐✈❡♥ ❜② ✉ ( t ) ✐s ✈❡r② ❡❛s② t♦ ❣❡t ✐♥ t❡r♠s ♦❢ ❛ ( t ) , ❜ ( t ) ❈♦①✱ ❍♦❢❢♠❛♥✱ ❲❛♥❣ ✷✵✵✽ ❇✉sé ✷✵✵✾ ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s

❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s ✭ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❚❡r❡s❛ ❈♦rt❛❞❡❧❧❛s✮ ❆ ❝✉r✈❡ ✐s ♣❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❧✐♥❡s ✐✛ t❤❡r❡ ❡①✐st ❛ ❜✐r❛t✐♦♥❛❧ ( ❋ ✶ : ❋ ✷ ) ♠♦r♣❤✐s♠ ❈ ��� P ✶ K s✉❝❤ t❤❛t ❞❡❣ ( ❋ ✶ , ❋ ✷ ) = ✶ ❆ ❝✉r✈❡ ✐s ♣❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❝♦♥✐❝s ✐✛ t❤❡r❡ ❡①✐st ❛ ❜✐r❛t✐♦♥❛❧ ( ❋ ✶ : ❋ ✷ ) ♠♦r♣❤✐s♠ ❈ ��� P ✶ K s✉❝❤ t❤❛t ❞❡❣ ( ❋ ✶ , ❋ ✷ ) = ✷ ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s

P❡♥❝✐❧s ♦❢ ❈♦♥✐❝s ❋ ✶ ( ❳ ) , ❋ ✷ ( ❳ ) ∈ K [ ❳ ✶ , ❳ ✷ , ❳ ✸ ] ❤♦♠♦❣❡♥❡♦✉s ♦❢ ❞❡❣r❡❡ ✷ ❈ ✐s ♣❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ( ❋ ✶ , ❋ ✷ ) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s②st❡♠  ✵ t ✶ ❋ ✷ ( ❳ ) − t ✷ ❋ ✶ ( ❳ ) =  ✵ ❊ ( ❳ ) =  ❤❛s ✷ ❞ − ✶ s♦❧✉t✐♦♥s ✐♥ P ✷ K ❛♥❞ ♦♥❡ ✭ ✉ ( t ) ✮ ✐♥ P ✷ K ( t ) ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s

❙✐♥❣✉❧❛r✐t✐❡s ❙✐♥❣ ( ❈ ) ⊂ ❱ ( ❋ ✶ , ❋ ✷ ) ❈✉r✈❡s ♣❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❝♦♥✐❝s ❤❛✈❡ ❛t ♠♦st ✹ s✐♥❣✉❧❛r ♣♦✐♥ts ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s

▲✐♥❡s ✈s ❈♦♥✐❝s ❆♥② ❝✉❜✐❝ ✐s ♣❛r❛♠❡t❡r✐③❛❜❧❡ ❜♦t❤ ❜② ❧✐♥❡s ❛♥❞ ❝♦♥✐❝s✦ ( ✸ t ✶ t ✷ ✷ , ✸ t ✷ ✶ t ✷ , t ✸ ✶ + t ✷ ✉ ( t ) = ✷ ) � ( ❳ ✷ , ❳ ✶ ) ( ❋ ✶ ( ❳ ) , ❋ ✷ ( ❳ )) = ( ❳ ✷ ✶ , ❳ ✷ ✷ − ✸ ❳ ✷ ❳ ✸ ) ❳ ✸ ✶ + ❳ ✸ ✷ − ✸ ❳ ✶ ❳ ✷ ❳ ✸ ❊ ( ❳ ) = ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s

▲✐♥❡s ✈s ❈♦♥✐❝s ✭❈♦rt❛❞❡❧❧❛s ✲ ❉✬✮ ■❢ ❞❡❣ ( ❈ ) > ✸ , t❤❡♥ t❤❡ ❝✉r✈❡ ❝❛♥♥♦t ❜❡ ♣❛r❛♠❡t❡r✐③❡❞ ❜② ❜♦t❤ ❧✐♥❡s ❛♥❞ ❝♦♥✐❝s✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✐❢ ❈ ✐s ♣❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❢♦r♠s ♦❢ ❞❡❣r❡❡ ❞ ❛♥❞ ❞ ′ ✱ t❤❡♥ ❞ + ❞ ′ ≥ ❞❡❣ ( ❈ ) ●❡♥❡r✐❝❛❧❧②✱ ❈ ✐s ♣❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❢♦r♠s ♦❢ ❞❡❣r❡❡ ❞❡❣ ( ❈ ) − ✷ ❈❛r❧♦s ❉✬❆♥❞r❡❛ ❘❛t✐♦♥❛❧ P❧❛♥❡ ❈✉r✈❡s P❛r❛♠❡t❡r✐③❛❜❧❡ ❜② ❈♦♥✐❝s