Sobre a Aritmética de Curvas Elípticas: O Teorema de Mordell-Weil, a Conjectura de Birch e Swinnerton-Dyer e o Problema dos Números Congruentes

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ✲ ❯❋❇❆

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ✲ ■▼❊

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P●▼❆❚ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦

❙♦❜r❡ ❛ ❆r✐t♠ét✐❝❛ ❞❡ ❈✉r✈❛s ❊❧í♣t✐❝❛s✿ ❖

❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ ❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤

❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❡ ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s

❈♦♥❣r✉❡♥t❡s

❨✉r❡ ❈❛r♥❡✐r♦ ❞❡ ❖❧✐✈❡✐r❛

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❙♦❜r❡ ❛ ❆r✐t♠ét✐❝❛ ❞❡ ❈✉r✈❛s ❊❧í♣t✐❝❛s✿ ❖

❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ ❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤

❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❡ ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s

❈♦♥❣r✉❡♥t❡s

❨✉r❡ ❈❛r♥❡✐r♦ ❞❡ ❖❧✐✈❡✐r❛

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❛✳ ❉r✳ ▼❛♥✉❡❧❛ ❞❛ ❙✐❧✈❛

❙♦✉③❛✳

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❖❧✐✈❡✐r❛✱ ❨✉r❡ ❈❛r♥❡✐r♦ ❞❡✱ ✶✾✾✺

❙♦❜r❡ ❛ ❆r✐t♠ét✐❝❛ ❞❡ ❈✉r✈❛s ❊❧í♣t✐❝❛s✿ ❖ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ ❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❡ ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s✴ ❨✉r❡ ❈❛r♥❡✐r♦ ❞❡ ❖❧✐✈❡✐r❛✳ ✕ ❙❛❧✈❛❞♦r✿ ❯❋❇❆✱ ✷✵✶✽✳

◗✉❛♥t✐❞❛❞❡ ❞❡ ❢♦❧❤❛s ❢✳ ✶✵✷ ✿ ✐❧✳

❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❛✳ ❉r✳ ▼❛♥✉❡❧❛ ❞❛ ❙✐❧✈❛ ❙♦✉③❛✳

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ Pr♦❣r❛♠❛ ❞❡ Pós✲❣r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ✷✵✶✽✳

❘❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s✳

✶✳ ❈✉r✈❛s ❊❧í♣t✐❝❛s✳ ✷✳ ▼♦r❞❡❧❧✲❲❡✐❧✳ ✸✳ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r✳ ✹✳ ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s✳ ■✳ ❙♦✉③❛✱ ▼❛♥✉❡❧❛ ❞❛ ❙✐❧✈❛✳ ■■✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✳ ■■■✳ ❚ít✉❧♦✳

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➚ ❉❡✉s✱ à ♠✐♥❤❛ ❢❛♠✐❧✐❛ ❡ ❛♠✐❣♦s✳

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❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡✱ ❛❣r❛❞❡ç♦ à ❉❡✉s ♣♦r t✉❞♦✱ ♣❡❧♦ ❛♣♦✐♦ q✉❡ ❞❡s❞❡ s❡♠♣r❡ t❡♠ ♠❡ ❝♦♥❝❡❞✐❞♦✱ ♣❡❧❛ ❢♦rç❛ q✉❡ t❡♠ ♠❡ ❞❛❞♦ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❢r❛q✉❡③❛✱ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❡s♣❡r♦✱ ♠♦♠❡♥t♦s ❡st❡s q✉❡ s❡ ♥ã♦ ❢♦ss❡ ♣❡❧❛ ❢é q✉❡ t❡♥❤♦ ❡♠ ❚✐✱ ♥ã♦ t❡r✐❛ s✉♣♦rt❛❞♦✳ ❆♦ ❙❡♥❤♦r ♦ ♠❡✉ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♠✐♥❤❛ ♠ã❡✱ ♠✐♥❤❛s ✐r♠ãs ❡ ✐r♠ã♦s✱ s❡♠♣r❡ ♠❡ ❞❛♥❞♦ ❛♣♦✐♦ ♥❛s ♠✐♥❤❛s t♦♠❛❞❛s ❞❡ ❞❡❝✐sõ❡s✱ ♣r❡♦❝✉♣❛❞♦s ❡♠ s❡♠♣r❡ ♠❡ ❛❥✉❞❛r✳ ❊✉ ❞♦✉ ❣r❛ç❛s ♣❡❧❛ ❢❛♠í❧✐❛ ❛❜❡♥ç♦❛❞❛ q✉❡ t❡♥❤♦ ❡ ❡♠ ❡s♣❡❝✐❛❧ à ♠✐♥❤❛ ♠ã❡✱ q✉❡ s❡♠♣r❡ ❝♦❧♦❝♦✉ ♦s ✜❧❤♦s à ❢r❡♥t❡ ❞❡ ❞✉❛s ♥❡❝❡ss✐❞❛❞❡s✱ ❡ ♥❡❧❛ ♣✉❞❡ ♣❡r❝❡❜❡r ✉♠ ❛♠♦r ❞✐❢❡r❡♥t❡✱ q✉❡ ❞❡✈❡ s❡r ✈❛❧♦r✐③❛❞♦ ❡ ♠✉✐t❛s ❞❛s ✈❡③❡s ♥ã♦ ♣❛r❛♠♦s ♣❛r❛ ♣❡r❝❡❜❡r ♦s ❞❡t❛❧❤❡s ❞❡ss❡ ❝✉✐❞❛❞♦ ❡s♣❡❝✐❛❧ ❡ ❞✐❢❡r❡♥❝✐❛❞♦ q✉❡ ✉♠❛ ♠ã❡ t❡♠ ♣❡❧♦s ✜❧❤♦s✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛ P❛✉❧♦ ❡ ❞♦♥❛ ❱❡r❛✱ ♣❡ss♦❛s s✉♣❡r ❡s♣❡❝✐❛✐s q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ às q✉❛✐s só t❡♥❤♦ ❛ ❛❣r❛❞❡❝❡r✳ ❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ❛ ♣r♦❢❡ss♦r❛ ▼❛♥✉❡❧❛✳ ❆❣r❛❞❡ç♦ ✐♠❡♥s❛♠❡♥t❡ ♣♦r ♣♦❞❡r t❡r ❡st✉❞❛❞♦ ❡ s✐❞♦ ♦r✐❡♥t❛❞♦ ♣❡❧❛ s❡♥❤♦r❛✱ ♣♦r t❡r ♠❡ ❛❥✉❞❛❞♦ ❡♠ ✈ár✐♦s ♠♦♠❡♥t♦s✱ ♣❡❧❛s ❝♦♥✈❡rs❛s s♦❜r❡ ♦ ♠❡✉ ❢✉t✉r♦ ♣r♦✜ss✐♦♥❛❧✱ ♣❡❧❛s ❞✐❝❛s ❡ ❝♦♥s❡❧❤♦s✱ ♣♦r t❡r ♠❡ ♦r✐❡♥t❛❞♦ ❡♠ ♠✐♥❤❛ ♠♦♥♦❣r❛✜❛ ❡ ♥❡st❛ ❞✐ss❡rt❛çã♦ ❡ ♣♦r t❡r ❛❝r❡❞✐❞❛t♦ ❡♠ ♠✐♠ ❝♦♠ r❡s♣❡✐t♦ ❛♦ ❝♦♥t❡ú❞♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❯♠ ♠✉✐t♦ ♦❜r✐❣❛❞♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛✱ à ♣r♦❢❡ss♦r❛ ❈❡❝í❧✐❛ ❙❛❧❣❛❞♦ q✉❡ ♠❡s♠♦ t❡♥❞♦ ♠❡ ❝♦♥❤❡❝✐❞♦ à ♣♦✉❝♦ t❡♠♣♦✱ ❛❝❡✐t♦✉ ♠❡ ♦r✐❡♥t❛r ♥♦ ❢✉t✉r♦ ❞♦✉t♦r❛❞♦ ❡ ❢❛③❡r ♣❛rt❡ ❞❡ss❛ ❜❛♥❝❛✱ t❛♠❜é♠ ♣❡❧❛ ❛❥✉❞❛ q✉❡ ♠❡ ❞❡✉ s♦❜r❡ ❛ t❡♠át✐❝❛ ♣❛r❛ ❡ss❛ ❞✐ss❡rt❛çã♦✳ ❆♦ ♣r♦❢❡ss♦r ▼❛r❝ ❍✐♥❞r②✱ q✉❡ t❛♠❜é♠ ♠❡s♠♦ ♠❡ ❝♦♥❤❡❝❡♥❞♦ à ♣♦✉❝♦ t❡♠♣♦✱ ❛❝❡✐t♦✉ ❢❛③❡r ♣❛rt❡ ❞❡ss❛ ❜❛♥❝❛✳ ➱ ✉♠ ✐♠❡♥s♦ ♣r❛③❡r ❝♦♥t❛r ❝♦♠ ❛ ♣r❡s❡♥ç❛ ❞❡ ✈♦❝ês ♥❛ ❛✈❛❧✐❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳

◆ã♦ ♣♦ss♦ ❞❡✐①❛r ❞❡ ❛❣r❛❞❡❝❡r ❛♦ ♣r♦❢❡ss♦r ❙❛♠✉❡❧ ●♦♠❡s✱ q✉❡ ❛❝r❡❞✐t♦✉ ❡♠ ♠✐♠ ❞❡s❞❡ ❛ ♠✐♥❤❛ ❡♥tr❛❞❛ ♥❛ ❣r❛❞✉❛çã♦✱ ♠❡ ❛❥✉❞♦✉ ♠✉✐t♦ ❡ t❡✈❡ ❣r❛♥❞❡ ✐♥✢✉ê♥❝✐❛ s♦❜r❡ ❛ ❢♦r♠❛çã♦ ❞♦ ♠❡✉ r❛❝✐♦❝í♥✐♦ ♠❛t❡♠át✐❝♦✱ ❡♠ q✉❡ ❣r❛♥❞❡ ♣❛rt❡ ❛♣r❡♥❞✐ ❝♦♠ ❡❧❡✳

❋♦r❛♠ ♠✉✐t♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❇❆ q✉❡ t✐✈❡r❛♠ ♣❛rt✐❝✐♣❛çã♦ ♥❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❡ ♠❡ ❛❥✉❞❛r❛♠ ❡♠ ✈ár✐♦s ♠♦♠❡♥t♦s✱ ❛♦s q✉❛✐s ❞❡✈♦ ❛❣r❛❞❡❝✐♠❡♥t♦✱ ❛♦s ♣r♦❢❡ss♦r❡s ❡ ♣r♦❢❡ss♦r❛s✱ ❈❛r❧♦s ❇❛❤✐❛♥♦✱ ❖s❝❛r✱ ❘✐t❛✱ ❏♦sé ◆❡❧s♦♥✱ ❏♦s❡♣❤✱ P❛✉❧♦ ❱❛r❛♥❞❛s✱ ❱✐t♦r✱ ❏♦✐❧s♦♥✱ ❏✉❛♥ ●♦♥③❛❧❡③✱ ❈❛r♠❡❧❛✱ ❈✐r♦✱ ❚❤✐❛❣♦✱ ❱❛♥❡ss❛✱ ▼❛t❤✐❡✉✱ ❏❡r♦♠❡✱ ❏❛✐♠❡ ❡ ❙❛♠✉❡❧ ❋❡✐t♦s❛✳ ◗✉❡ ❉❡✉s ❛❜❡♥ç♦❡ ❛ ❝❛❞❛ ✉♠ ❞❡ ✈♦❝ês✳

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❝♦♥té♠ ♠✉✐t❛s ♣❡ss♦❛s✱ à t♦❞♦s ✈♦❝ês ✜❝❛♠ ✉♠ ♠✉✐t♦ ♦❜r✐❣❛❞♦ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❡st✉❞♦ ❡ ❞✐✈❡rsã♦✱ ♣❡❧♦s ♠♦♠❡♥t♦s ♥♦ ❘✳❯✳✱ ♣❡❧❛s ❜♦❛s ❝♦♥✈❡rs❛s q✉❡ t✐✈❡ ❝♦♠ ❝❛❞❛ ✉♠ ❞❡ ✈♦❝ês✳ ◆❡st❡ ♠♦♠❡♥t♦ ♥ã♦ ❝✐t❛r❡✐ ♥♦♠❡s✱ ♣♦✐s r❡❛❧♠❡♥t❡ sã♦ ♠✉✐t♦s ❞❡ ✈♦❝ês✱ ♠❛s ❝r❡✐♦ q✉❡ ❝❛❞❛ ✉♠ s❛✐❜❛ ♦ q✉❛♥t♦ s♦✉ ❣r❛t♦✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ❞♦ ◗✉❛rt❡t♦ ❋❛♥tást✐❝♦✳ ❆♦s ❛♠✐❣♦s q✉❡ ❝♦♥❤❡❝✐ ❡ ♣r♦❢❡ss♦r❡s q✉❡ t✐✈❡ ♥♦ ■❋❇❆✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r P❛✉❧♦ ❱✐❝❡♥t❡✱ à ♣r♦❢❡ss♦r❛ ❏✉❛♥✐❝❡ ❡ ❛♦ ♣r♦❢❡ss♦r ❩✐✉❧✱ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳

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❘❡s✉♠♦

❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧ ✭❚✳▼✳❲✮✱ r❡s✉❧t❛❞♦ ❡st❡ q✉❡ ❞✐③ q✉❡ s❡E/K é ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ s♦❜r❡ ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s K✱ ❡♥tã♦ ♦ ❣r✉♣♦E(K)

❞♦s s❡✉s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳ ❆ss✐♠ E(K) ∼= E(K)tor ⊕Zr, ♣❛r❛

❛❧❣✉♠ r ≥ 0✱ ❡♠ q✉❡ ❡ss❡ ✐♥✈❛r✐❛♥t❡ r é ❝❤❛♠❛❞♦ ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡ E✳ P♦r ✜♠✱ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❞♦✐s ❢❛♠♦s♦s ♣r♦❜❧❡♠❛s ❛r✐t♠ét✐❝♦s✱ s❡♥❞♦ ❡❧❡s ❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❡ ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s ❡ s❡rá ❞✐s❝✉t✐❞❛ ❛ r❡❧❛çã♦ ❡♥tr❡ ❡ss❡s ❞♦✐s ♣r♦❜❧❡♠❛s✳

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❆❜str❛❝t

❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ st✉❞② t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✱ ♠♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ❞❡♠♦♥str❛t✐♦♥ ♦❢ t❤❡ ▼♦r❞❡❧❧✲❲❡✐❧ ❚❤❡♦r❡♠ ✭▼✳❲✳❚✮ ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞✱ ✇❤✐❝❤ r❡s✉❧ts t❤❛t ✐❢ E/K ✐s ❛ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ♦♥ ❛ ♥✉♠❜❡rs ✜❡❧❞ K✱ t❤❡♥ t❤❡ E(K) ❣r♦✉♣ ♦❢ ✐ts K✲

r❛t✐♦♥❛❧ ♣♦✐♥ts ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤✉s E(K)∼= E(K)tor ⊕Zr, ❢♦r s♦♠❡ r ≥ 0✱ ✇❤❡r❡

t❤❛t ✐♥✈❛r✐❛♥t r ✐s ❝❛❧❧❡❞ t❤❡ ❛❧❣❡❜r❛✐❝ r❛♥❦✳ ❋✐♥❛❧❧②✱ t✇♦ ❢❛♠♦✉s ❛r✐t❤♠❡t✐❝❛❧ ♣r♦❜❧❡♠s ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞✱ t❤❡ ❇✐r❝❤ ❛♥❞ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❈♦♥❥❡❝t✉r❡ ❛♥❞ t❤❡ ❈♦♥❣r✉❡♥t ◆✉♠❜❡rs Pr♦❜❧❡♠✱ ❛♥❞ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠s ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞✳

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❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ Pr❡❧✐♠✐♥❛r❡s ✼

✶✳✶ ❱❛r✐❡❞❛❞❡s ❆✜♥s ❡ Pr♦❥❡t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✶✳✶ ❱❛r✐❡❞❛❞❡s ❆✜♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✶✳✷ ❱❛r✐❡❞❛❞❡s Pr♦❥❡t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷ ❈✉r✈❛s ❆❧❣é❜r✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷✳✶ ▼❛♣❛s ❡♥tr❡ ❈✉r✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷✳✷ ❉✐✈✐s♦r❡s ❞❡ ✉♠❛ ❝✉r✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✷✳✸ ❉✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✷✳✹ ❖ ❚❡♦r❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲❘♦❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✷ ❈✉r✈❛s ❊❧í♣t✐❝❛s ✸✵

✷✳✶ ▲❡✐ ❞❡ ❣r✉♣♦ ♣❛r❛ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✷ ❘❡❞✉çã♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✸ ❖ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧ ✹✺

✸✳✶ ❆ ✈❡rsã♦ ❢r❛❝❛ ❞♦ ❚❡♦r❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✶✳✶ ❖ ❡♠♣❛r❡❧❤❛♠❡♥t♦ ❞❡ ❑✉♠♠❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✶✳✷ ❆ ❞❡♠♦♥str❛çã♦ ❞❛ ✈❡rsã♦ ❢r❛❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✷ ❖ ❚❡♦r❡♠❛ ❞❛ ❉❡s❝✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✸ ❆❧t✉r❛s ♥♦ ❊s♣❛ç♦ Pr♦❥❡t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✹ ❆❧t✉r❛s ❡♠ ❈✉r✈❛s ❊❧í♣t✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺

✹ ❆ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ✭❇❙❉✮ ❡ ♦ Pr♦❜❧❡♠❛ ❞♦s

◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s ✭P◆❈✮ ✼✸

✹✳✶ ❆ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✹✳✷ ❖ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶

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■♥tr♦❞✉çã♦

❖ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❞❡st❡ tr❛❜❛❧❤♦ sã♦ ❛s ❝❤❛♠❛❞❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s✳ ❙✉❛s ❛♣❛✲ r✐çõ❡s ✐♥✐❝✐❛✐s r❡♠♦♥t❛♠ à ●ré❝✐❛ ❆♥t✐❣❛✱ ♥♦ ❧✐✈r♦ ❆r✐t❤♠❡t✐❝❛ ❞❡ ❉✐♦♣❤❛♥t✉s✳ ▼✉✐t♦s ♠❛t❡♠át✐❝♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✱ ❛ ❡①❡♠♣❧♦ ❞❡ P♦✐♥❝❛ré✱ ❇✐r❝❤✱ ❙✇✐♥♥❡rt♦♥✲❉②❡r✱ ▲❡♥str❛ ❞❡♥tr❡ ♦✉tr♦s s❡ ❞❡♣❛r❛r❛♠ ❝♦♠ ♣r♦❜❧❡♠❛s q✉❡ ♦s ❧❡✈❛r❛♠ ❛ ❡st✉❞❛r ❛ ♥❛t✉r❡③❛ ❞❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s✳ ❆t✉❛❧♠❡♥t❡✱ s❛❜❡✲s❡ ❜❛st❛♥t❡ ❛ s❡✉ r❡s♣❡✐t♦ ❡ ❞❡ s✉❛ ✉t✐❧✐❞❛❞❡ ❡♠ ❛♣❧✐❝❛çõ❡s q✉❡ ✈ã♦ ❞❡s❞❡ ❛ ▼❛t❡♠át✐❝❛ ♣✉r❛ à ❋ís✐❝❛ ❡ ❈r✐♣t♦❣r❛✜❛✱ ❡♠❜♦r❛ ♠✉✐t♦s s❡❥❛♠ t❛♠❜é♠ ♦s ♣r♦❜❧❡♠❛s ❡ ❝♦♥❥❡❝t✉r❛s q✉❡ ❛s ❡♥✈♦❧✈❡♠✳ ❆❧❣✉♥s ❞❡st❡s ♣r♦❜❧❡♠❛s s❡rã♦ ❛❜♦r❞❛❞♦s ❛q✉✐✳

❆s ❝✉r✈❛s ❡❧í♣t✐❝❛s sã♦ ♦❜❥❡t♦s ❞❡ ♥❛t✉r❡③❛ ❣❡♦♠étr✐❝❛✱ q✉❡ ❛❞♠✐t❡♠ ✉♠❛ ❡s✲ tr✉t✉r❛ ❛❧❣é❜r✐❝❛ ❞❡ ❣r✉♣♦ ❞❡✜♥✐❞❛ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✳ ❊st❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❣❡ô♠❡tr♦✲ ❛❧❣é❜r✐❝♦ ❢❛③ ❞❡st❡s ✉♠ ❞♦s ♦❜❥❡t♦s ♠❛✐s ❢❛s❝✐♥❛♥t❡s ❞❛ ♠❛t❡♠át✐❝❛✱ ❝♦♠ ✈❛r✐❛❞❛s ❛♣❧✐✲ ❝❛çõ❡s✳ ❆q✉✐K ❞❡♥♦t❛rá ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s ❡ K ✉♠ ❢❡❝❤♦ ❛❧❣é❜r✐❝♦✳

❯♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ é ✉♠❛ ❝✉r✈❛ ♥♦ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦ P2✱ ❝♦♠ ✉♠ ♣♦♥t♦ ❜❛s❡ ❡s♣❡✲

❝í✜❝❛❞♦✱ s❛t✐s❢❛③❡♥❞♦ ✉♠❛ ❡q✉❛çã♦ ❤♦♠♦❣ê♥❡❛ ❞❛ ❢♦r♠❛

Y2Z +a1XY Z+a3Y Z2 =X3+a2X2Z +a4XZ2+a6Z3,

♦♥❞❡ O = [0,1,0] é ♦ ♣♦♥t♦ ❜❛s❡ ❡ a1, . . . , a6 ∈ K✳ ❆ ❡q✉❛çã♦ ❛❝✐♠❛ é ❝❤❛♠❛❞❛ ❡q✉❛çã♦

❞❡ ❲❡✐❡rstr❛ss✱ ❡ ♣♦❞❡ s❡r ❝♦❧♦❝❛❞❛ ❡♠ ❝♦♦r❞❡♥❛❞❛s ♥ã♦ ❤♦♠♦❣ê♥❡❛s ❢❛③❡♥❞♦ x = X

Z ❡ y= Y

Z✱ ♦❜t❡♥❞♦

E :y2+a1xy+a3y=x3+a2x2+a4x+a6.

▲❡♠❜r❛♥❞♦ q✉❡ ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ♣♦♥t♦ ♥♦ ✐♥✜♥✐t♦✱ ❞❡♥♦t❛❞♦ ♣♦r O✳ ❙❡ a1, . . . ,

a6 ∈K✱ ❞✐③❡♠♦s q✉❡ E ❡stá ❞❡✜♥✐❞❛ s♦❜r❡K✱ ❡ ❡s❝r❡✈❡♠♦s E/K✳

P♦❞❡♠♦s ❞❡✜♥✐r t❛♠❜é♠ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ❝♦♠♦ s❡♥❞♦ ✉♠ ♣❛r (E,O)✱ ♦♥❞❡

E é ✉♠❛ ❝✉r✈❛ ♣r♦❥❡t✐✈❛ ❞❡ ❣ê♥❡r♦ ✶✱ ❡ O ✉♠ ♣♦♥t♦ ❞❡ E✳ ❆♠❜❛s ❛s ❞❡✜♥✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❝✉r✈❛ ❞❛❞❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ❞❡✜♥✐çã♦✱ ❡❧❛ s❛t✐s❢❛③ ❛ s❡❣✉♥❞❛ ❞❡✜♥✐çã♦✱ ❡ q✉❡ s❡ ❝♦♥s✐❞❡r❛r♠♦s ✉♠❛ ❝✉r✈❛ ♣❡❧❛ s❡❣✉♥❞❛ ❞❡✜♥✐çã♦✱ ❡①✐st✐rá ✉♠ ♠♦❞❡❧♦ ❡♠P2 ✐s♦♠♦r❢♦ à ❝✉r✈❛ ❡♠ q✉❡stã♦✳

❖ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❞❡ E✱ ❞❡♥♦t❛❞♦ ♣♦r E(K)✱ é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦

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❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s s♦❜r❡ ❛ ❝✉r✈❛ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s ❡♠K✱ ✐st♦ é✱

E(K) ={(x, y)K2 :y2+a1xy+a3y =x3+a2x2+a4x+a6} ∪ {O},

♦♥❞❡a1, . . . , a6 ∈K✳

❊st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❡st✉❞❛r ❛ ❣❡♦♠❡tr✐❛ ❞❡ss❛s ❝✉r✈❛s ♣❛r❛ ❡♥tã♦✱ ❜✉s❝❛r ❡♥t❡♥❞❡r ❛ s✉❛ ❛r✐t♠ét✐❝❛ ♥♦ s❡♥t✐❞♦ ❞❡ ❜✉s❝❛r ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛ ❡str✉t✉r❛ ❞♦s s❡✉s ♣♦♥t♦sK✲r❛❝✐♦♥❛✐s✳

❍❡♥r② P♦✐♥❝❛ré ❝♦♥❥❡❝t✉r♦✉ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s r❛❝✐♦♥❛✐s s❡r✐❛ ✉♠ ❣r✉♣♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❡ ▼♦r❞❡❧❧ ♣r♦✈♦✉ ❡ss❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❝✉r✈❛s ❡❧í♣t✐❝❛s r❛❝✐♦♥❛✐s ❡♠ ✶✾✷✷✱ ❡ ❡♠ ✶✾✷✽✱ ❲❡✐❧ ❡st❡♥❞❡✉ ❡ss❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❝✉r✈❛s ❡❧í♣t✐❝❛s s♦❜r❡ ❝♦r♣♦s ❞❡ ♥ú♠❡r♦s✱ t❡♥❞♦ ❤♦❥❡ ♦ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❙❡E/K é ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ s♦❜r❡ ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s✱ ❡♥tã♦E(K) é ✉♠ ❣r✉♣♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ❝♦♠

E(K)∼=E(K)tor ⊕Zr,

♣❛r❛ ❛❧❣✉♠ r ≥ 0✳ ❊ss❡ r ❝❤❛♠❛r❡♠♦s ❞❡ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡ E✱ ❡ E(K)tor ♦ ❣r✉♣♦ ❞❡

t♦rçã♦✱ q✉❡ é ♦ s✉❜❣r✉♣♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❞❡ ♦r❞❡♠ ✜♥✐t❛✳ ❖ ❈❛♣ít✉❧♦ ✸ s❡rá ❞❡❞✐❝❛❞♦ à ❞❡♠♦♥str❛çã♦ ❞❡ss❡ r❡s✉❧t❛❞♦✳

●r❛♥❞❡ ♣❛rt❡ ❞❛ ♣❡sq✉✐s❛ ❡♠ ❣❡♦♠❡tr✐❛ ❛r✐t♠ét✐❝❛✱ ár❡❛ q✉❡ ✉s❛ ❞❡ ♠ét♦❞♦s ❞❛ ❣❡♦♠❡tr✐❛ ❛❧❣é❜r✐❝❛ ♣❛r❛ ♦❜t❡r r❡s✉❧t❛❞♦s ❛r✐t♠ét✐❝♦s✱ é ♦ ❡st✉❞♦ ❞❡ ♣♦♥t♦s r❛❝✐♦♥❛✐s ❡♠ ✈❛r✐❡❞❛❞❡s✳ ❆ss✐♠✱ é ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ s❛❜❡r ♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ♦ s✉❜❣r✉♣♦ ❞❡ t♦rçã♦ ❡ ♦ ♣♦st♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳ ❙♦❜r❡ ♦ s✉❜❣r✉♣♦ ❞❡ t♦rçã♦ ✭♣♦♥t♦s ❞❡ ♦r❞❡♠ ✜♥✐t❛✮✱ ▼❛③✉r ♣r♦✈♦✉ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❝✉r✈❛s ❡❧í♣t✐❝❛s r❛❝✐♦♥❛✐s✱ q✉❡ ❞❡s❝r❡✈❡ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♦ s✉❜❣r✉♣♦ ❞❡ t♦rçã♦✳

❙❡❥❛ E/Q ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳ ❊♥tã♦ E(Q)tor é ✐s♦♠♦r❢♦ ❛ ✉♠ ❞♦s s❡❣✉✐♥t❡s

❣r✉♣♦s

Z/nZ, ❝♦♠ 1≤n ≤10 ♦✉ n = 12;

Z/2ZZ/2nZ, ❝♦♠ 1n 4.

❈❛❧❝✉❧❛r ♦ ♣♦st♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧✐♣t✐❝❛ ❡♠ ❣❡r❛❧ é ✉♠ ♣r♦❜❧❡♠❛ ❞✐❢í❝✐❧✱ ❡ s♦❜r❡ ✐ss♦✱ ❡♠ ✶✾✻✺✱ ❇✳ ❏✳ ❇✐r❝❤ ❡ ❙✐r ❍✳ P✳ ❋✳ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❝♦♥❥❡❝t✉r❛r❛♠ q✉❡ ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ r❛❝✐♦♥❛❧ s❡r✐❛ ✐❣✉❛❧ ❛ ✉♠ ♦✉tr♦ ✐♥✈❛r✐❛♥t❡✱ ❛❣♦r❛ ❛♥❛❧ít✐❝♦✱ r❡❧❛❝✐♦♥❛❞♦ ❛ ▲✲sér✐❡ ❞❛ ❝✉r✈❛✱ q✉❡ ❞❡✜♥✐r❡♠♦s ❧♦❣♦ ♠❛✐s✳

❙❡❥❛ E : y2 = x3 +ax+b✱ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ r❛❝✐♦♥❛❧ ♥❛ ❢♦r♠❛ ❞❡ ❲❡✐❡rstr❛ss

r❡❞✉③✐❞❛✱ ❝♦♠ a, b ∈ Z ❡ ∆ s❡✉ ❞✐s❝r✐♠✐♥❛♥t❡✳ ❉❛❞♦ p ♣r✐♠♦✱ s❡❥❛ Z → Fp✱ z 7→ z✱ ❛

r❡❞✉çã♦ ♠ó❞✉❧♦ p❡ ❝♦♥s✐❞❡r❡ ❛ ❝✉r✈❛

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ap =p+ 1−#Ep(Fp),

♦♥❞❡Ep(Fp)é ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦sFp✲r❛❝✐♦♥❛✐s ❞❛ ❝✉r✈❛Ep✱ ❝❤❛♠❛❞❛ ❛ r❡❞✉çã♦ ♠ó❞✉❧♦

p ❞❡ E✳ ❉✐③❡♠♦s q✉❡ p é ✉♠ ♣r✐♠♦ ❞❡ ❜♦❛ r❡❞✉çã♦ s❡ ❛ ❝✉r✈❛ Ep é ♥ã♦ s✐♥❣✉❧❛r✱ ❝❛s♦

❝♦♥trár✐♦ ❞✐③❡♠♦s q✉❡ p é ❞❡ ♠á r❡❞✉çã♦✳ ❚❡♠♦s q✉❡ ♦s ♣r✐♠♦s ❞❡ ❜♦❛ r❡❞✉çã♦ sã♦ ❡①❛t❛♠❡♥t❡ ♦s ♣r✐♠♦s q✉❡ ♥ã♦ ❞✐✈✐❞❡♠ ♦ ❞✐s❝r✐♠✐♥❛♥t❡∆✳

❙❡♥❞♦ E/Q ✉♠❛ ❝✉r✈❛ r❛❝✐♦♥❛❧ ❝♦♠ ❡q✉❛çã♦ E :y2 =x3+ax+b✱ ❝♦♠ a, bZ✳

❙✉❛ L✲sér✐❡ é ❞❡✜♥✐❞❛ ♣❡❧♦ ♣r♦❞✉t♦ ❞❡ ❊✉❧❡r

L(E, s) = Y

p |∆ 1 1−app−s

Y

p ∤ ∆

1

1−app−s+p1−2s

,

❝♦♠ sC✳

❚❡♠♦s q✉❡ ♦ ♣r♦❞✉t♦ ❛❝✐♠❛ ❞❡✜♥❡ ✉♠❛ sér✐❡ ❞❡ ❉✐r✐❝❤❧❡t

L(E, s) =

X

n=1

bn

ns,

❡ ♥❡ss❡ ❝❛s♦✱ bp =ap✱ ♣❛r❛ t♦❞♦ p ♣r✐♠♦✳ ➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ❛ sér✐❡ ❛❝✐♠❛ ❝♦♥✈❡r❣❡

♣❛r❛ t♦❞♦ ♣♦♥t♦s❞♦ ♣❧❛♥♦ ❝♦♠Re(s)> 3

2✳ ●r❛ç❛s ❛♦ ❚❡♦r❡♠❛ ❞❛ ▼♦❞✉❧❛r✐❞❛❞❡ ♣r♦✈❛❞♦

♣♦r ❲✐❧❡s ❡♠ ❬❲✐❧❡s✷❪✱ é ♣♦ssí✈❡❧ ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❝♦♥t✐♥✉❛çã♦ ❛♥❛❧ít✐❝❛ ❞❡L(E, s)

❛ t♦❞♦ ♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✳ ❆ss✐♠ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ ❡①♣❛♥sã♦ ❞❡L(E, s) ❡♠ t♦r♥♦ ❞❡

s= 1✳ ❚❡♠♦s ❡♥tã♦ ❛ ✈❡rsã♦ ❢r❛❝❛ ❞❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r✳

❆ ❡①♣❛♥sã♦ ❞❡ ❚❛②❧♦r ❞❡ L(E, s) ❡♠ t♦r♥♦ ❞❡ s= 1 t❡♠ ❛ ❢♦r♠❛

L(E, s) =cr(s−1)r+cr+1(s−1)r+1+ t❡r♠♦s ❞❡ ❣r❛✉s ♠❛✐♦r❡s,

❡♠ q✉❡ cr6= 0 ❡ r é ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡ E✳

❉❡s❞❡ ❛ s✉❛ ❢♦r♠✉❧❛çã♦✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣❛r❝✐❛✐s ❢♦r❛♠ ❡♥❝♦♥tr❛❞♦s✱ ♠❛s ❛té ❤♦❥❡ ♦ ♣r♦❜❧❡♠❛ ❝♦♥t✐♥✉❛ ❡♠ ❛❜❡rt♦✱ s❡♥❞♦ q✉❡ ❡♠ ✷✵✵✵ ♦ ❈❧❛② ▼❛t❤❡♠❛t✐❝s ■♥st✐t✉t❡ ♦ ❧✐st♦✉ ❝♦♠♦ ✉♠ ❞♦s ♣r♦❜❧❡♠❛s ❞♦ ♠✐❧ê♥✐♦✳

◆♦ ❈❛♣ít✉❧♦ ✹✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❝♦♥❥❡❝t✉r❛✱ q✉❡ s❡♥❞♦ ✈❡r❞❛❞❡✐r❛ t❡rá ❝♦♠♦ ❝♦♥✲ s❡q✉ê♥❝✐❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r s❡ ✉♠ ❞❛❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦n é ♦✉ ♥ã♦ ✉♠ ♥ú♠❡r♦ ❝♦♥❣r✉❡♥t❡✳ P♦r ✉♠ ♥ú♠❡r♦ ❝♦♥❣r✉❡♥t❡✱ ❡♥t❡♥❞❡♠♦s ✉♠ r❛❝✐♦♥❛❧ q✉❡ é ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❝♦♠ ❧❛❞♦s r❛❝✐♦♥❛✐s✳ ❆té ♦ ♠♦♠❡♥t♦ ♥ã♦ t❡♠♦s ❝♦♥❤❡✲ ❝✐♠❡♥t♦ ❞❡ ✉♠ t❛❧ ❛❧❣♦r✐t♠♦✱ ❡ ❡ss❡ é ♦ ❝❤❛♠❛❞♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s✱ ❞♦ q✉❛❧ ❢❛❧❛r❡♠♦s ♥❡ss❡ tr❛❜❛❧❤♦✳

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❉❛❞♦ n>1✱ ❞❡✜♥✐♠♦s ❛ ❝✉r✈❛ ❡❧í♣t✐❝❛

En:y2 =x3−n2x.

❙♦❜r❡ En t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

n é ❝♦♥❣r✉❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ En t❡♠ ✐♥✜♥✐t♦s ♣♦♥t♦s r❛❝✐♦♥❛✐s✳

P❡❧❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✱ n é ✉♠ ♥ú♠❡r♦ ❝♦♥❣r✉❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡En é ♣♦s✐t✐✈♦✳

❙♦❜r❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡❝✐❞✐r s❡ ✉♠ ❞❛❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n é ❝♦♥❣r✉❡♥t❡✱ ❚✉♥♥❡❧❧ ❡♠ ❬❚✉♥♥❡❧❧❪✱ ♣r♦✈♦✉ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s✳ ❉❡✜♥❛

An= #{(x, y, z)∈Z3 :n= 2x2+y2 + 8z2},

Bn = #{(x, y, z)∈Z3 :n = 2x2+y2+ 32z2},

Cn= #{(x, y, z)∈Z3 :n= 8x2 + 2y2+ 16z2},

Dn= #{(x, y, z)∈Z3 :n= 8x2 + 2y2+ 64z2}.

❙❡ n é í♠♣❛r ❡ ✉♠ ♥ú♠❡r♦ ❝♦♥❣r✉❡♥t❡✱ ❡♥tã♦ An= 2Bn✳ ❙❡ n é ♣❛r ❡ ❝♦♥❣r✉❡♥t❡✱ ❡♥tã♦

Cn = 2Dn✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ❛ ✈❡rsã♦ ❢r❛❝❛ ❞❡ ❇❙❉ ✈❛❧❡ ♣❛r❛ En✱ ❡♥tã♦ s❡ n é í♠♣❛r ❡

An= 2Bn ♦✉ s❡ n é ♣❛r ❡ Cn= 2Dn✱ ❡♥tã♦ n é ❝♦♥❣r✉❡♥t❡✳

◆♦ ❈❛♣ít✉❧♦ ✶✱ ✐♥tr♦❞✉③✐r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s s♦❜r❡ ✈❛r✐❡❞❛❞❡s ❛❧❣é❜r✐❝❛s ❛✜♥s ❡ ♣r♦❥❡t✐✈❛s✱ ❝♦♥❝❡✐t♦s ❝♦♠♦ s✉❛✈✐❞❛❞❡ ❡ ❞✐♠❡♥sã♦ s❡rã♦ ❞❡✜♥✐❞♦s✳ ❆✐♥❞❛ ♥❡ss❡ ❝❛♣ít✉❧♦✱ ❞❛r❡♠♦s ❛t❡♥çã♦ às ✈❛r✐❡❞❛❞❛❞❡s ❛❧❣é❜r✐❝❛s ❞❡ ❞✐♠❡♥sã♦ ✶✱ ❛s ❝❤❛♠❛❞❛s ❝✉r✈❛s ❛❧❣é❜r✐❝❛s✳ ❘❡s✉❧t❛❞♦s ❝♦♠♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❇é③♦✉t ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲❘♦❝❤ s❡rã♦ ❡♥✉♥❝✐❛❞♦s✳

◆♦ ❈❛♣ít✉❧♦ ✷✱ ❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s s❡rã♦ ✐♥tr♦❞✉③✐❞❛s✳ ▼♦str❛r❡♠♦s ❝♦♠♦ é ✐♥tr♦✲ ❞✉③✐❞❛ à ❡str✉t✉r❛ ❞❡ ❣r✉♣♦ ❡♠ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✱ ❡ ❡♥❝♦♥tr❛r❡♠♦s ❡①♣r❡ssõ❡s ❛❧❣é❜r✐❝❛s ♣❛r❛ ❡ss❛ ♦♣❡r❛çã♦ ❞❡ ❣r✉♣♦ q✉❡ s❡rá ❞❡✜♥✐❞❛ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✳ ❚❛♠❜é♠ ❡st❛r❡♠♦s ✐♥✲ t❡r❡ss❛❞♦s ❡♠ ❡st✉❞❛r ❛ ❝✉r✈❛ q✉❛♥❞♦ ✈✐st❛ s♦❜r❡ ♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞♦ ❝♦r♣♦K ♣♦r ✉♠❛ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✱ ❛ ✜♠ ❞❡ ❞❡✜♥✐r ❛ r❡❞✉çã♦ ❞❛ ❝✉r✈❛ ❝♦♠ r❡❧❛çã♦ ❛ ✈❛❧♦r✐③❛çã♦✳

(15)

(16)

❈❛♣ít✉❧♦ ✶

Pr❡❧✐♠✐♥❛r❡s

❊st❡ t❡①t♦ tr❛t❛rá ❞❛s ❝❤❛♠❛❞❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s✱ ❡ ♣❛r❛ ✐ss♦ ♥❡❝❡ss✐t❛♠♦s ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ❆❧❣é❜r✐❝❛✳ ◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ✐♥tr♦❞✉③✐❞♦s ♥♦çõ❡s ❜ás✐❝❛s ❞❛ ❧✐♥❣✉❛❣❡♠ ❞❛ ●❡♦♠❡tr✐❛ ❆❧❣é❜r✐❝❛ q✉❡ s❡rã♦ ♦ s✉♣♦rt❡ ✐♥✐❝✐❛❧ ♣❛r❛ ♦s ❞❡♠❛✐s ❝❛♣ít✉❧♦s✳ P❛r❛ t❛❧ ❛❜♦r❞❛❣❡♠ ❛ss✉♠✐r❡♠♦s q✉❡ ♦ ❧❡✐t♦r ❡st❡❥❛ ❢❛♠✐❧✐❛r✐③❛❞♦ ❝♦♠ ❛❧❣✉♠❛s ❢❡rr❛♠❡♥t❛s ❞❛ ➪❧❣❡❜r❛ ❈♦♠✉t❛t✐✈❛ ❡ ❚❡♦r✐❛ ❞❡ ●❛❧♦✐s✳ ■♥❞✐❝❛♠♦s ❛s r❡❢❡rê♥❝✐❛s ❬❆t✐②❛❤❪ ❡ ❬▼♦r❛♥❞✐❪✳ ❯s❛r❡♠♦s|X| ♣❛r❛ ❞❡♥♦t❛r ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞♦ ❝♦♥❥✉♥t♦ X✳

❆ ♣r✐♥❝✐♣❛❧ r❡❢❡rê♥❝✐❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ ❝❛♣ít✉❧♦ é ♦ ❧✐✈r♦ ✭❬❙✐❧✈❡r✲ ♠❛♥❪✮✳ ❉✉r❛♥t❡ ♦ t❡①t♦ K ❞❡♥♦t❛rá ✉♠ ❝♦r♣♦ ♣❡r❢❡✐t♦✱ ❡ K ♦ s❡✉ ❢❡❝❤♦ ❛❧❣é❜r✐❝♦ ❡ G(K/K) ♦ ❣r✉♣♦ ❞❡ ●❛❧♦✐s ❞❡ss❛ ❡①t❡♥sã♦✳

✶✳✶ ❱❛r✐❡❞❛❞❡s ❆✜♥s ❡ Pr♦❥❡t✐✈❛s

◆❡st❛ s❡çã♦ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ❡①♣♦s✐çã♦ s♦❜r❡ ✈❛r✐❡❞❛❞❡s ❛❧❣é❜r✐❝❛s✱ ❞❡✜♥✐♥❞♦ ❝♦♥❝❡✐t♦s ❝♦♠♦ s✉❛✈✐❞❛❞❡✱ ❞✐♠❡♥sã♦✱ ❛♥❡❧ ❧♦❝❛❧ ❞❡♥tr❡ ♦✉tr♦s✳ ❆❧❣✉♥s r❡s✉❧t❛❞♦s s❡rã♦ ♣r♦✈❛❞♦s✱ ❡ ♦✉tr♦s ❡st❛rã♦ ❛❝♦♠♣❛♥❤❛❞♦s ❞❡ r❡❢❡rê♥❝✐❛s ♣❛r❛ s✉❛ ❞❡♠♦♥str❛çã♦✳

✶✳✶✳✶ ❱❛r✐❡❞❛❞❡s ❆✜♥s

❉❡✜♥✐çã♦ ✶✳✶✳ ❖n✲❡s♣❛ç♦ ❛✜♠ ✭s♦❜r❡ K✮ é ♦ ❝♦♥❥✉♥t♦ ❞❛s n✲✉♣❧❛s ❝♦♠ ❡♥tr❛❞❛s ❡♠ K

An=An(K) = {(x1, . . . , xn) : xi ∈K}.

❙✐♠✐❧❛r♠❡♥t❡✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❞❡ An é ♦ ❝♦♥❥✉♥t♦

An(K) ={(x1, . . . , xn) : xi ∈K}.

➱ ✐♠❡❞✐❛t♦ ✈❡r✐✜❝❛r q✉❡ ❣r✉♣♦ ❞❡ ●❛❧♦✐s G(K/K) ❛❣❡ s♦❜r❡ An✱ ❞❛ s❡❣✉✐♥t❡

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❢♦r♠❛✿ P❛r❛τ G(K/K)❡ P An

Pτ = (τ(x1), . . . , τ(xn)).

❆ss✐♠✱ An(K) ♣♦❞❡ s❡r ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r

An(K) ={P An: Pτ =P, τ G(K/K)}.

❙❡❥❛ K[X] =K[x1, . . . , xn] ♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s ❡♠ n ✈❛r✐á✈❡✐s ❡ s❡❥❛ I ⊆K[X]

✉♠ ✐❞❡❛❧✳ P❛r❛ ❝❛❞❛I ❛ss♦❝✐❛♠♦s ✉♠ s✉❜❝♦♥❥✉♥t♦ V(I) ❞❡An V(I) = {P ∈An : f(P) = 0, ∀f ∈I}. ❉✐ss♦✱ s❡❣✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦✳

❉❡✜♥✐çã♦ ✶✳✷✳ ❯♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ ✭❛✜♠✮ é q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❞❛ ❢♦r♠❛ V(I)✳ ❙❡

V é ✉♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦✱ ♦ ✐❞❡❛❧ ❞❡ V é ❞❡✜♥✐❞♦ ❝♦♠♦

I(V) = {f ∈K[X] : f(P) = 0, ∀P ∈V}.

❯♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ é ❞❡✜♥✐❞♦ s♦❜r❡ K s❡ s❡✉ ✐❞❡❛❧ I(V) ♣♦❞❡ s❡r ❣❡r❛❞♦ ♣♦r ♣♦❧✐♥ô✲

♠✐♦s ❡♠K[X]✱ ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r V /K✳

❙❡ V é ❞❡✜♥✐❞♦ s♦❜r❡ K✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❞❡ V é ♦ ❝♦♥❥✉♥t♦

V(K) =V An(K).

◆❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ ③❡r♦s ❡♠ ❝♦♠✉♠ ❞❡ ♣♦❧✐♥ô♠✐♦s ❡♠ ✉♠ ✐❞❡❛❧✱ ♠❛s ♣♦❞❡rí❛♠♦s t❡r t♦♠❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ S q✉❛❧q✉❡r ❞❡ ♣♦❧✐♥ô♠✐♦s ❡ t♦♠❛❞♦V(S) ❝♦♠♦ ♥❛ ❞❡✜♥✐çã♦ ❞❡ V(I)✳ ❚❡♠♦s q✉❡ V(S) =V(hSi)✱ ♦♥❞❡

hSi é ♦ ✐❞❡❛❧ ❣❡r❛❞♦ ♣♦r S✳

❙❛❜❡♠♦s q✉❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❇❛s❡ ❞❡ ❍✐❧❜❡rt✱ t♦❞♦s ♦s ✐❞❡❛✐s ❞❡ K[X] ❡ K[X]

sã♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦s✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ t♦❞♦ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ é ❛ ✐♥t❡rs❡çã♦ ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❝♦♥❥✉♥t♦s ❛❧❣é❜r✐❝♦s ❣❡r❛❞♦s ♣♦r ❝♦♥❥✉♥t♦s ✉♥✐tár✐♦s ❞❡ ❝❡rt♦s ♣♦❧✐♥ô♠✐♦s✳

❯♠ ♦✉tr♦ r❡s✉❧t❛❞♦ ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛ é ♦ ❚❡♦r❡♠❛ ❞♦s ❩❡r♦s ❞❡ ❍✐❧❜❡rt✳

❚❡♦r❡♠❛ ✶✳✸ ✭◆✉❧❧st❡❧❧❡♥ss❛t③✮✳ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦✳ ❙❡ I é ✉♠ ✐❞❡❛❧ ❞❡ K[x1, . . . , xn]✱ ❡♥tã♦

I(V(I)) =√I,

♦♥❞❡√I❞❡♥♦t❛ ♦ r❛❞✐❝❛❧ ❞❡I✱ ♦✉ s❡❥❛✱√I ={f K[x1, . . . , xn] :fm ∈I, ♣❛r❛ ❛❧❣✉♠m>

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❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✶✱ ❚❡♦r❡♠❛ ✶✳✸❆✱ ❬❍❛rts❤♦r♥❡❪✮✳

❉❡✜♥✐çã♦ ✶✳✹✳ ❯♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ ❛✜♠ V é ❝❤❛♠❛❞❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ✭❛✜♠✮ s❡ I(V) é ✉♠ ✐❞❡❛❧ ♣r✐♠♦ ❡♠ K[X]✳

❊①❡♠♣❧♦ ✶✳✺✳

◆❛ ✜❣✉r❛ t❡♠♦s ❞♦✐s ❡①❡♠♣❧♦s ❞❡ ❝♦♥❥✉♥t♦s ❛❧❣é❜r✐❝♦s✱ ♦♥❞❡ ♠♦str❛♠♦s ❛♣❡♥❛s ♦s ♣♦♥t♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s r❡❛✐s✳ ❊♠ a) t❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ ✈❛r✐❡❞❛❞❡✱ ❥á ♦ ❝♦♥❥✉♥t♦

❛❧❣é❜r✐❝♦ ❡♠ b) ♥ã♦ é ✉♠❛ ✈❛r✐❡❞❛❞❡✳

❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ I(V(x2y)) =phx2yi=hx2yié ♣r✐♠♦✱ ❡♥tã♦V(x2y)é

✉♠❛ ✈❛r✐❡❞❛❞❡✳ ❆❣♦r❛✱ s❡♥❞♦f(x, y) =x−y❡g(x, y) =x+y✱ t❡♠♦s q✉❡f(1,−1) = 26= 0

❡g(1,1) = 26= 0✳ ▼❛s ❝♦♠♦(1,−1),(1,1)∈V(x2y2)❡♥tã♦xy, x+y6∈I(V(x2y2))

❆ss✐♠✱ I(V(x2y2)) ♥ã♦ é ♣r✐♠♦✱ ✈✐st♦ q✉❡ (xy)·(x+y) =x2y2 I(V(x2y2))

❙❡❥❛ V /K ✉♠❛ ✈❛r✐❡❞❛❞❡✱ ✐st♦ é✱ V é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡✜♥✐❞❛ s♦❜r❡ K✳ ❊♥tã♦✱ ♦ ❛♥❡❧ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛✜♥s ❞❡ V /K é ❞❡✜♥✐❞♦ ♣♦r

K[V] = K[X]

I(V /K).

❖ ❛♥❡❧ K[V] é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡✳ ❙❡✉ ❝♦r♣♦ q✉♦❝✐❡♥t❡ ✭❝♦r♣♦ ❞❡ ❢r❛çõ❡s✮ é

❞❡♥♦t❛❞♦ ♣♦r K(V) ❡ é ❝❤❛♠❛❞♦ ♦ ❝♦r♣♦ ❞❡ ❢✉♥çõ❡s ❞❡ V /K✳ ❙✐♠✐❧❛r♠❡♥t❡✱ K[V] ❡

K(V) sã♦ ❞❡✜♥✐❞♦s s✉❜st✐t✉✐♥❞♦ K ♣♦r K✳

❉❡✜♥✐♠♦s ❛❣♦r❛ ✉♠❛ ♥♦çã♦ ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛✱ q✉❡ é ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞✐♠❡♥sã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡✳

❉❡✜♥✐çã♦ ✶✳✻ ✭❉✐♠❡♥sã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡✮✳ ❙❡❥❛ V ✉♠❛ ✈❛r✐❡❞❛❞❡✳ ❆ ❞✐♠❡♥sã♦ ❞❡ V✱ ❞❡♥♦t❛❞❛ ♣♦r dim(V)✱ é ♦ ❣r❛✉ ❞❡ tr❛♥s❝❡♥❞ê♥❝✐❛ ❞❡ K(V) s♦❜r❡ K✳

❊①❡♠♣❧♦ ✶✳✼✳

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✶✵

• ❙❡ V =V(f)✱ ♣❛r❛ ❛❧❣✉♠ f K[X]K✱ ❡♥tã♦ dim(V) =n1✳ ❉✐③❡♠♦s q✉❡ V é ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡✳

◗✉❛♥❞♦ ❡st✉❞❛♠♦s ♦❜❥❡t♦s ❣❡♦♠étr✐❝♦s✱ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ s❛❜❡r s❡ ♣❛r❡❝❡ r❛③♦❛✈❡❧♠❡♥t❡ ✏s✉❛✈❡✑✳ ❆ ❞❡✜♥✐çã♦ q✉❡ s❡❣✉❡✱ ❢♦r♠❛❧✐③❛ ❡ss❛ ✐❞é✐❛ ❡♠ t❡r♠♦s ❞♦ ❝r✐tér✐♦ ❏❛❝♦❜✐❛♥♦ ✉s✉❛❧ ♣❛r❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❡s♣❛ç♦ t❛♥❣❡♥t❡✳

❉❡✜♥✐çã♦ ✶✳✽✳ ❙❡❥❛ V ✉♠❛ ✈❛r✐❡❞❛❞❡✱ P ∈ V✱ ❡ f1, . . . , fm ∈ K[X] ✉♠ ❝♦♥❥✉♥t♦ ❞❡

❣❡r❛❞♦r❡s ♣❛r❛ I(V)✳ ❊♥tã♦ V é ♥ã♦ s✐♥❣✉❧❛r ✭♦✉ s✉❛✈❡✮ ❡♠ P s❡ ❛ ♠❛tr✐③ m×n

∂f1

∂xj

(P)

16i6m,16j6n

t❡♠ ♣♦st♦ n−dim(V)✳ ❙❡ V é ♥ã♦ s✐♥❣✉❧❛r ❡♠ t♦❞♦ ♣♦♥t♦ ❡♥tã♦ ❞✐③❡♠♦s q✉❡ V é ♥ã♦ s✐♥❣✉❧❛r ✭♦✉ s✉❛✈❡✮✳

❊①❡♠♣❧♦ ✶✳✾✳ ❙❡ V é ❞❛❞❛ ♣♦r ✉♠❛ ú♥✐❝❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ♥ã♦ ❝♦♥st❛♥t❡

f(x1, . . . , xn) = 0.

❊♥tã♦ ❝♦♠♦ dim(V) =n−1✱ t❡♠♦s q✉❡ P ∈V é ♣♦♥t♦ s✐♥❣✉❧❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱

∂f ∂x1

(P) = · · ·= ∂f

∂xn

(P) = 0.

❊①✐st❡ ✉♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ ❝❛r❛❝t❡r✐③❛r♠♦s ❛ s✉❛✈✐❞❛❞❡ ❞❡ ✉♠ ♣♦♥t♦ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡✳ ■ss♦ s❡ ❞❛rá ❡♠ t❡r♠♦s ❞❛s ❢✉♥çõ❡s ♥❛ ✈❛r✐❡❞❛❞❡ ❡ ♠♦str❛rá q✉❡ ❡st❛ ❞❡✜♥✐çã♦ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❡s❝♦❧❤❛ ❞♦s ❣❡r❛❞♦r❡s ❞♦ ✐❞❡❛❧✳ P❛r❛ ❝❛❞❛ ♣♦♥t♦ P V✱ ❞❡✜♥✐♠♦s ♦ s❡❣✉✐♥t❡ ✐❞❡❛❧MP ❞❡K[V] ♣♦r

MP ={f ∈K[V] : f(P) = 0}.

❈❧❛r❛♠❡♥t❡✱MP ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡ ❛❧é♠ ❞✐ss♦✱MP é ✉♠ ✐❞❡❛❧ ♠❛①✐♠❛❧✱ ✈✐st♦ q✉❡ ❡①✐st❡

✉♠ ✐s♦♠♦r✜s♠♦ K[V]/MP →K ❞❛❞♦ ♣♦r f 7→f(P).

❚❡♠♦s t❛♠❜é♠ q✉❡ ♦ q✉♦❝✐❡♥t❡ MP/MP2 é ✉♠ K✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦

✜♥✐t❛✳ ❆♥t❡s ❞❡ ♠♦str❛♠♦s ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ s✉❛✈✐❞❛❞❡ ❡♠ t❡r♠♦s ❞❛s ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ♥❛ ✈❛r✐❡❞❛❞❡✱ ♣r♦✈❛r❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

Pr♦♣♦s✐çã♦ ✶✳✶✵✳ ❙❡❥❛L ❝♦r♣♦ ❡a1, . . . , an ∈L✳ ❊♥tã♦ ♦ ✐❞❡❛❧ ❆P = (x1−a1, . . . , xn−

an)∈L[X] =L[x1, . . . , xn] é ♠❛①✐♠❛❧✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ ✈❡r✐✜❝❛r♠♦s ✐ss♦✱ ❜❛st❛ ♠♦str❛r q✉❡ ❛ ❛♣❧✐❝❛çã♦L[X]/❆P →L❞❛❞❛

♣♦r f +❆P 7→ f(a1, . . . , an) é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s✳ ❆ss✐♠ t❡r❡♠♦s q✉❡ L[X]/❆P é

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✶✶

◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ ❡st❛ ❛♣❧✐❝❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s ❡ q✉❡ é s♦❜r❡❥❡t✐✈❛✳ ❆ ♠❛✐♦r ❞✐✜❝✉❧❞❛❞❡ ✈❡♠ ❞❛ s✉❛ ✐♥❥❡t✐✈✐❞❛❞❡✳

❙❡❥❛ f ∈ L[X] t❛❧ q✉❡ f(a1, . . . , an) = 0✱ ♠♦str❡♠♦s q✉❡ f ∈ ❆P✱ ❞❡ ♦♥❞❡

❝♦♥❝❧✉✐r❡♠♦s q✉❡ ❡st❛ ❛♣❧✐❝❛çã♦ t❡♠ ♥ú❝❧❡♦ tr✐✈✐❛❧✱ ✐♠♣❧✐❝❛♥❞♦ ♥❛ s✉❛ ✐♥❥❡t✐✈✐❞❛❞❡✳ ❚❡♠♦s q✉❡f(a1, . . . , an−1, xn)∈L[xn]t❡♠an❝♦♠♦ r❛í③✳ ❆ss✐♠✱ ❡①✐st❡fn ∈L[xn]

t❛❧ q✉❡

f(a1, . . . , an−1, xn) =fn·(xn−an).

❆❣♦r❛✱ t❡♠♦s q✉❡ f(a1, . . . , xn−1, xn)−fn ·(xn −an) ∈ (L[xn])[xn−1] t❡♠ an−1

❝♦♠♦ r❛í③✳ ❆ss✐♠✱ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô♠✐♦ fn−1 ∈L[xn−1, xn]♣❛r❛ ♦ q✉❛❧ t❡♠♦s

f(a1, . . . , xn−1, xn)−fn·(xn−an) =fn−1·(xn−1−an−1).

Pr♦❝❡❞❡♥❞♦ ❞❡ ❢♦r♠❛ r❡❝✉rs✐✈❛✱ ♦❜t❡r❡♠♦s q✉❡ f(x1, . . . , xn)♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❛

❢♦r♠❛

f =

n

X

i=1

fi·(xi−ai),

♣❛r❛ ❛❧❣✉♠f1, . . . , fn∈L[X]✳ ❉❡ ♦♥❞❡ s❡❣✉❡ q✉❡ f ∈❆P✳

Pr♦♣♦s✐çã♦ ✶✳✶✶✳ ❙❡❥❛V ✉♠❛ ✈❛r✐❡❞❛❞❡✳ ❯♠ ♣♦♥t♦P ∈V é ♥ã♦ s✐♥❣✉❧❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱

dimK MP/MP2 =dim(V).

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ P = (a1, . . . , an) ∈ V✱ ❡ s❡❥❛ ❆P = (x1−a1, . . . , xn−an) ♦ ❝♦rr❡s✲

♣♦♥❞❡♥t❡ ✐❞❡❛❧ ♠❛①✐♠❛❧ ❞❡ K[X]✳ ❉❡✜♥✐♠♦s ❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r

θ:K[X]Kn θ(f) =

∂f ∂x1

(P), . . . , ∂f ∂x1

(P)

.

❚❡♠♦s q✉❡θ(xi−ai)✱i= 1, . . . , n✱ ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♣❛r❛ K n

✱ ❡ q✉❡ θ[❆2P] ={0}✳ ❆ss✐♠✱

θ ✐♥❞✉③ ✉♠ ✐s♦♠♦r✜s♠♦ ❧✐♥❡❛r θ′ :

P/❆2P →K n

❆❣♦r❛ s❡❥❛ I(V) ♦ ✐❞❡❛❧ ❞❡ V ❡♠ K[X]✱ ❡ s❡❥❛♠ f1, . . . , ft ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛✲

❞♦r❡s ❞❡ I(V)✳ ❊♥tã♦ ♦ ♣♦st♦ ❞❛ ♠❛tr✐③ ❏❛❝♦❜✐❛♥❛ J =

∂f1

∂xj

(P)

16i6t,16j6n

é ✐❣✉❛❧ ❛

❞✐♠❡♥sã♦ ❞❡ θ[I(V)] ❝♦♠♦ s✉❜❡s♣❛ç♦ ❞❡ Kn✳ ❯s❛♥❞♦ ♦ ✐s♦♠♦r✜s♠♦ θ′✱ ❡ss❛ ❞✐♠❡♥sã♦ é

✐❣✉❛❧ ❛ ❞✐♠❡♥sã♦ ❞♦ s✉❜❡s♣❛ç♦(I(V) +❆2P)/❆2P ❞❡ ❆P/❆2P✳

❚❡♠♦s t❛♠❜é♠ q✉❡

MP/MP2 ∼=❆P/(I(V) +❆2P).

❆❧é♠ ❞✐ss♦✱ dimK(❆P/❆2P) =dimK(❆P/(I(V) +❆2P)) +dimK((I(V) +❆

2

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✶✷

❆ss✐♠✱

P♦st♦ (J) =dimK((I(V) +❆2P)/❆2P)

=dimK(❆P/❆2P)−dimK(❆P/(I(V) +❆2P))

=ndimK(MP/MP2).

P♦rt❛♥t♦✱ P é ♥ã♦ s✐♥❣✉❧❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ dimK MP/MP2 =dim(V)✳

❆❣♦r❛ ❞❡✜♥✐r❡♠♦s ♦ ❛♥❡❧ ❧♦❝❛❧ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❡♠ ✉♠ ♣♦♥t♦✳

❉❡✜♥✐çã♦ ✶✳✶✷✳ ❖ ❛♥❡❧ ❧♦❝❛❧ ❞❡ V ❡♠ P✱ ❞❡♥♦t❛❞♦ ♣♦r K[V]P✱ é ❛ ❧♦❝❛❧✐③❛çã♦ ❞❡

K[V] ❡♠ MP✱ ♦✉ s❡❥❛✱

K[V]P ={F ∈K(V) : F =f /g ♣❛r❛ ❛❧❣✉♠ f, g ∈K[V], ❝♦♠ g(P)6= 0}.

◆♦t❡ q✉❡ s❡ F =f /g K[V]P✱ ❡♥tã♦ F(P) =f(P)/g(P) ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳ ❆s ❢✉♥çõ❡s

❡♠ K[V]P sã♦ ❝❤❛♠❛❞❛s r❡❣✉❧❛r❡s ✭♦✉ ❞❡✜♥✐❞❛s✮ ❡♠ P✳

✶✳✶✳✷ ❱❛r✐❡❞❛❞❡s Pr♦❥❡t✐✈❛s

❖s ❡s♣❛ç♦s ♣r♦❥❡t✐✈♦s ❢♦r❛♠ ♣❡♥s❛❞♦s ❝♦♠♦ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❛❞✐❝✐♦♥❛r ✏♣♦♥t♦s ♥♦ ✐♥✜♥✐t♦✑ ❛♦ ❡s♣❛ç♦ ❛✜♠✳ ❉❡✜♥✐♠♦s ♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ ❝♦♠♦ ❛ ❝♦❧❡çã♦ ❞❡ r❡t❛s ♣❛ss❛♥❞♦ ♣❡❧❛ ♦r✐❣❡♠ ♥♦ ❡s♣❛ç♦ ❛✜♠ ❞❡ ❞✐♠❡♥sã♦ ♠❛✐♦r q✉❡ ✉♠✳

❉❡✜♥✐çã♦ ✶✳✶✸✳ ❖ n✲❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ ✭s♦❜r❡ K✮✱ ❞❡♥♦t❛❞♦ ♣♦r Pn ♦✉ Pn(K)✱ é ♦

❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s (n+ 1)✲✉♣❧❛s (x0, x1, . . . , xn)∈An+1 t❛❧ q✉❡✱ ❛♦ ♠❡♥♦s ✉♠ xi é ♥ã♦

♥✉❧♦✱ ♠ó❞✉❧♦ ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛

(x0, x1, . . . , xn)∼(y0, y1, . . . , yn)

s❡ ❡①✐st❡ ✉♠ λ∈K∗ t❛❧ q✉❡ xi =λyi ♣❛r❛ t♦❞♦ i✳ ❯♠❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛

{(λx0, λx1, . . . , λxn) : λ∈K

}

é ❞❡♥♦t❛❞❛ ♣♦r [x0, x1, . . . , xn]✱ ❡ ♦s x0, x1, . . . , xn ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡ sã♦ ❝❤❛♠❛❞❛s ❝♦♦r✲

❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ♣❛r❛ ♦ ❝♦rr❡♣♦♥❞❡♥t❡ ♣♦♥t♦ ❡♠ Pn✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s

K✲r❛❝✐♦♥❛✐s ❡♠ Pn é ♦ ❝♦♥❥✉♥t♦

Pn(K) ={[x0, x1, . . . , xn]∈Pn: ❡①✐st❡♠y0, . . . , yn∈K,[x0, x1, . . . , xn] = [y0, y1, . . . , yn]}.

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✶✸

❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s

[x0, x1, . . . , xn]τ = [xτ0, xτ1, . . . , xτn].

❊st❛ ❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s✱ ❞❛❞♦ q✉❡

[λx0, λx1, . . . , λxn]τ = [λτxτ0, λτxτ1, . . . , λτxτn] = [x0τ, xτ1, . . . , xτn].

❆ ✜♠ ❞❡ ❞❡✜♥✐r♠♦s ❛s ✈❛r✐❡❞❛❞❡s ♣r♦❥❡t✐✈❛s✱ s❡❣✉✐♥❞♦ ❛ ✐❞é✐❛ ❞❡ t♦♠❛r ♣♦♥t♦s ❞♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ q✉❡ s❡❥❛♠ ③❡r♦s ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ♣r❡❝✐s❛♠♦s q✉❡ ❡st❛ ❞❡✜♥✐çã♦ ✐♥❞❡✲ ♣❡♥❞❛ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ❛ss♦❝✐❛❞❛ ❛♦ ♣♦♥t♦✳ P❛r❛ ✐ss♦ ♣r❡❝✐s❛♠♦s ❞❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳

❉❡✜♥✐çã♦ ✶✳✶✹✳ ❯♠ ♣♦❧✐♥ô♠✐♦ f ∈K[X] =K[x0, x1, . . . , xn] é ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ d

s❡

f(λx0, λx1, . . . , λxn) =λd·f(x0, x1, . . . , xn), ∀λ∈K.

❯♠ ✐❞❡❛❧ I K[X] é ❤♦♠♦❣ê♥❡♦ s❡ é ❣❡r❛❞♦ ♣♦r ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s✳

❙❡❥❛ f ✉♠ ♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦ ❡ s❡❥❛ P Pn✳ ❆❣♦r❛ ❢❛③ s❡♥t✐❞♦ ♣❡r❣✉♥t❛r s❡

f(P) = 0✱ ✉♠❛ ✈❡③ q✉❡ ❛ r❡s♣♦st❛ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❡s❝♦❧❤❛ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s

♣❛r❛ P✳

P❛r❛ ❝❛❞❛ ✐❞❡❛❧ ❤♦♠♦❣ê♥❡♦ I✱ ❛ss♦❝✐❛♠♦s ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ Pn ❞❛❞♦ ♣♦r

V(I) = {P Pn : f(P) = 0, ♣❛r❛ t♦❞♦ ♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦ f I}.

❉❡✜♥✐çã♦ ✶✳✶✺✳ ❯♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ ✭♣r♦❥❡t✐✈♦✮ é q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❞❛ ❢♦r♠❛ V(I) ♣❛r❛ ❛❧❣✉♠ ✐❞❡❛❧ ❤♦♠♦❣ê♥❡♦ I✳ ❙❡ V é ✉♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ ♣r♦❥❡t✐✈♦✱ ♦ ✐❞❡❛❧ ✭❤♦♠♦❣ê♥❡♦✮ ❞❡ V✱ ❞❡♥♦t❛❞♦ ♣♦r I(V)✱ é ♦ ✐❞❡❛❧ ❞❡ K[X] ❣❡r❛❞♦ ♣♦r

{f K[X] : f é ❤♦♠♦❣ê♥❡♦ ❡ f(P) = 0,P V}.

❉✐③❡♠♦s q✉❡ V é ❞❡✜♥✐❞♦ s♦❜r❡ K✱ ❞❡♥♦t❛❞♦ ♣♦r V /K✱ s❡ s❡✉ ✐❞❡❛❧ I(V) ♣♦❞❡ s❡r

❣❡r❛❞♦ ♣♦r ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ❡♠K[X]✳ ❙❡ V é ❞❡✜♥✐❞♦ s♦❜r❡ K✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❞❡ V é ♦ ❝♦♥❥✉♥t♦

V(K) =V Pn(K). ❈♦♠♦ ✉s✉❛❧✱ V(K) ♣♦❞❡ t❛♠❜é♠ s❡r ❞❡s❝r✐t♦ ❝♦♠♦

V(K) = {P V : Pτ =P, τ Gal(K/K)}.

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✶✹

❞❡ ✈❛r✐❡❞❛❞❡ ♣r♦❥❡t✐✈❛✳

❉❡✜♥✐çã♦ ✶✳✶✻✳ ❯♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ ♣r♦❥❡t✐✈♦ V é ❝❤❛♠❛❞♦ ✉♠❛ ✈❛r✐❡❞❛❞❡ ✭♣r♦❥❡✲ t✐✈❛✮ s❡ s❡✉ ✐❞❡❛❧ ❤♦♠♦❣ê♥❡♦ I(V) é ✉♠ ✐❞❡❛❧ ♣r✐♠♦ ❡♠ K[X]✳

❆❣♦r❛✱ ❞❡ ❢♦r♠❛ ♥❛t✉r❛❧ ♣♦❞❡♠♦s ❡♥①❡r❣❛r ♦ ❡s♣❛ç♦ ❛✜♠ An ❝♦♠♦ s✉❜❝♦♥❥✉♥t♦

❞♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦Pn✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳

❋✐①❛❞❛ ✉♠❛ ❝♦♦r❞❡♥❛❞❛ 06i6n ❝♦♥s✐❞❡r❛♠♦s ❛ ✐♥❝❧✉sã♦

ϕi :An −→Pn

(x1, . . . , xn)7−→[x1, . . . , xi−1,1, xi, . . . , xn].

❆❣♦r❛ ❞❡♥♦t❛♠♦s ♣♦r Hi ❛♦ ❤✐♣❡r♣❧❛♥♦ ❡♠ Pn ❞❡✜♥✐❞♦ ♣❡❧❛ ❡q✉❛çã♦xi = 0✱ ✐st♦

é✱

Hi ={P = [x0, . . . , xn]∈Pn:xi = 0}.

❚❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦ ♥❛t✉r❛❧ ❡♥tr❡An ❡ ♦ ❝♦♠♣❧❡♠❡♥t♦ ❞❡H

i✱ ❞❡♥♦t❛❞♦

♣♦rUi✱ ♦✉ s❡❥❛✱ s❡♥❞♦ Ui ={[x1, . . . , xn]∈Pn:xi 6= 0}t❡♠♦s ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛

ϕ−i 1 :Ui −→An

[x0, . . . , xn]7−→

x0

xi

, . . . ,xi−1 xi

,xi+1 xi

, . . . ,xn xi

.

❆ss✐♠✱ ♣❛r❛ ✉♠ i ✜①❛❞♦✱ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r An ❝♦♠ U

i ❡♠ Pn✳ ❙❡ V é ✉♠❛

✈❛r✐❡❞❛❞❡ ♣r♦❥❡t✐✈❛ ❡♠Pn t❡♠♦s q✉❡ V An✱ q✉❡ ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ϕ−1

i (V ∩Ui) ♣❛r❛

❛❧❣✉♠i ✜①❛❞♦✱ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛✜♠ ❝✉❥♦ ✐❞❡❛❧ I(V ∩An)K[Y] é ❞❛❞♦ ♣♦r

I(V An) ={f(y0, y1, . . . , yi−1,1, yi+1, . . . , yn) :f(x0, . . . , xn)∈I(V)}.

◗✉❛♥❞♦ ❡st✐✈❡r♠♦s ❢❛③❡♥❞♦ r❡❢❡rê♥❝✐❛ à ✐♥❞❡♥t✐✜❝❛çã♦ An ❝♦♠ U

i✱ ✉s❛r❡♠♦s ❛

♥♦t❛çã♦An i✳

❈♦♠♦ ♦s ❝♦♥❥✉♥t♦s U0, . . . , Un ❝♦❜r❡♠ ♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦Pn✱ t♦❞❛ ✈❛r✐❡❞❛❞❡ ♣r♦✲

❥❡t✐✈❛V é ❝♦❜❡rt❛ ♣❡❧❛ ❝♦❧❡çã♦ ❞❡ ✈❛r✐❡❞❛❞❡s ❛✜♥s V ∩An

0, . . . , V ∩Ann✱ ✈✐❛ ❛s ❛♣❧✐❝❛çõ❡s

ϕi✳

❙❡ ✜①❛r♠♦s i✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣r♦❥❡t✐✈❛ V ❝♦♠♦ ❛ ✉♥✐ã♦ ❞❡ s✉❛ ♣❛rt❡ ❛✜♠V An

i ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s s❡✉s ♣♦♥t♦s ♥♦ ✐♥✜♥✐t♦V ∩Hi✱ ❡ ❞❡♥♦t❛r❡♠♦s

Hi ♣♦r H∞✳

❊ss❡ ♣r♦❝❡ss♦ ❞❡ ❝♦♥s✐❞❡r❛r ♦ ♣♦❧✐♥ô♠✐♦f(y0, y1, . . . , yi−1,1, yi+1, . . . , yn)♥♦ ❧✉❣❛r

❞❡ f(x0, . . . , xn) é ❝❤❛♠❛❞♦ ❞❡s♦♠♦❣❡♥❡✐③❛çã♦ ❝♦♠ r❡s♣❡✐t♦ ❛ xi✳ ❊ ❛ ✐♥✈❡rsã♦ ❞❡st❡

♣r♦❝❡ss♦ é ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✱ ❞❛❞♦ f(y1, . . . , yn)∈K[y1, . . . , yn]✱ ❞❡✜♥✐♠♦s

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✶✺

♦♥❞❡ d =deg(f) é ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ t❛❧ q✉❡ f∗ é ✉♠ ♣♦❧✐♥ô♠✐♦✳ ❈❤❛♠❛r❡♠♦s f

❞❡ ❛ ❤♦♠♦❣❡♥❡✐③❛çã♦ ❞❡f ❝♦♠ r❡s♣❡✐t♦ à xi✳

❉❡✜♥✐çã♦ ✶✳✶✼✳ ❙❡❥❛V ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣r♦❥❡t✐✈❛ ❡ ❡s❝♦❧❤❛it❛❧ q✉❡V∩An

i 6=∅✳ ❉❡✜♥✐♠♦s

❛ ❞✐♠❡♥sã♦ ❞❡ V ❝♦♠♦ ❛ ❞✐♠❡♥sã♦ ❞❛ ✈❛r✐❡❞❛❞❡ ❛✜♠V ∩An i✳

❚❡♠♦s q✉❡ ❡st❛ ❞❡✜♥✐çã♦ ✐♥❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞♦j ♣❛r❛ ♦ q✉❛❧V An

j 6=∅✳ P❛r❛

✐ss♦ ♣♦❞❡♠♦s t❛♠❜é♠ ❞❡✜♥✐r ♦ ❝♦r♣♦ ❞❡ ❢✉♥çõ❡s ❞❡ V ❝♦♠♦ s❡♥❞♦ ♦ ❝♦r♣♦ ❞❛s ❢✉♥çõ❡s r❛❝✐♦♥❛✐s F(X) = f(X)

g(X) t❛✐s q✉❡

✭✐✮ f ❡ g sã♦ ❤♦♠♦❣ê♥❡♦s ❞❡ ♠❡s♠♦ ❣r❛✉✳

✭✐✐✮ g 6∈I(V)✳

✭✐✐✐✮ ❡ ✐❞❡♥t✐✜❝❛♠♦s ❞✉❛s ❢✉♥çõ❡s f1

g1 ❡

f2

g2 s❡

f1g2−f2g1 ∈I(V)✳

❊①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ❝♦r♣♦s ❞❛❞♦s ♣♦r ❡ss❛s ❞✉❛s ❞❡✜♥✐çõ❡s✳ ❆ss✐♠ t❡♠♦s q✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐♠❡♥sã♦ ❞❡ V ✐♥❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞♦j t❛❧ q✉❡ V An

j 6=∅✳

✶✳✷ ❈✉r✈❛s ❆❧❣é❜r✐❝❛s

❈♦♠♦ ❥á ❢♦✐ ♠❡♥❝✐♦♥❛❞♦ ♥♦ ✐♥í❝✐♦ ❞❡st❡ ❝❛♣ít✉❧♦✱ ♣r❡t❡♥❞❡♠♦s ❞❡✜♥✐r ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s q✉❡ s❡rã♦ ♥❡❝❡ssár✐♦s ♥♦s ❞❡♠❛✐s ❝❛♣ít✉❧♦s✳ ◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥✲ t❛♠♦s ❛s ❞❡✜♥✐çõ❡s ❞❡ ❝✉r✈❛s ❛❧❣é❜r✐❝❛s✱ ♠❛♣❛s ❡♥tr❡ ❝✉r✈❛s✱ ❞✐✈✐s♦r❡s ❡ ❞✐❢❡r❡♥❝✐❛✐s✱ ❡ ❡♥✉♥❝✐❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲❘♦❝❤✳

❆♦ ❛❞♦t❛r♠♦s ❛ ❧✐♥❣✉❛❣❡♠ ❡ ♥♦t❛çã♦ ✉s❛❞❛s ❡♠ ❬❙✐❧✈❡r♠❛♥❪✱ ♦♣t❛♠♦s ♣♦r ♥ã♦ ❞❡✲ ✜♥✐r ❝♦♥❝❡✐t♦s ❝♦♠♦ ❛ ❚♦♣♦❧♦❣✐❛ ❞❡ ❩❛r✐s❦✐✱ ❡s♣❛ç♦s ♠✉❧t✐❤♦♠♦❣ê♥❡♦s ❡ ✈❛r✐❡❞❛❞❡s q✉❛s❡✲ ♣r♦❥❡t✐✈❛s✳ ◆❡ss❛ ❛❜♦r❞❛❣❡♠ ♠❛✐s ❣❡r❛❧✱ ♠✉♥✐♠♦s ♦s ❡s♣❛ç♦s ❛✜♥s✱ ♣r♦❥❡t✐✈♦s ❡ ♠❛✐s ❣❡r❛❧♠❡♥t❡ ♦s ♠✉❧t✐♣r♦❥❡t✐✈♦s✱ ❝♦♠ ✉♠❛ t♦♣♦❧♦❣✐❛ ♥❛ q✉❛❧ ♦s ❢❡❝❤❛❞♦s sã♦ ♦s ❝♦♥❥✉♥t♦s ❛❧❣é❜r✐❝♦s✱ ❡ ❛s ✈❛r✐❡❞❛❞❡s s❡rã♦ ♦s ❢❡❝❤❛❞♦s ✐rr❡❞✉tí✈❡✐s✱ s❡♥❞♦ q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ Y ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦X é ❞✐t♦ ✐rr❡❞✉tí✈❡❧✱ q✉❛♥❞♦ ♥ã♦ ♣✉❞❡r s❡r ❡s❝r✐t♦ ❝♦♠♦ ❛ ✉♥✐ã♦ ❞❡ ❞♦✐s s✉❜❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s r❡❧❛t✐✈♦s ❡ ♣ró♣r✐♦s✱ ✐st♦ é✱ s❡ Y = Y1 ∪Y2 ❝♦♠ Y1 ❡ Y2

❢❡❝❤❛❞♦s ❡♠ Y✱ ❡♥tã♦ Y = Y1 ♦✉ Y = Y2✳ ❆ss✐♠✱ ❞❡✜♥✐❞❛ ❛ t♦♣♦❧♦❣✐❛ ♣♦r ♠❡✐♦ ❞♦s

❢❡❝❤❛❞♦s✱ ♣♦❞❡♠♦s ❢❛❧❛r ❞❡ ❛❜❡rt♦s ❡ ❛ss✐♠ t❡♠♦s ✉♠❛ ♥♦çã♦ ♣r❡❝✐s❛ ❞❡ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ♣♦♥t♦ ♣♦r ♠❡✐♦ ❞♦s ❛❜❡rt♦s q✉❡ ❝♦♥té♠ ❡st❡ ♣♦♥t♦✳ ❚♦❞❛s ❡ss❛s ♥♦çõ❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❧✐✈r♦s ❝♦♠♦ ❬❍❛rts❤♦r♥❡❪✱ ❬❙❤❛❢❛r❡✈✐❝❤❪ ❡ ❬❋✉❧t♦♥❪✳

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✶✻

❉❡✜♥✐çã♦ ✶✳✶✽✳ ❆s ❝✉r✈❛s ❛❧❣é❜r✐❝❛s sã♦ ✈❛r✐❡❞❛❞❡s ♣r♦❥❡t✐✈❛s ❞❡ ❞✐♠❡♥sã♦ ✶✳

❊①❡♠♣❧♦ ✶✳✶✾✳ ❙❡❥❛ V P2 ❛ ✈❛r✐❡❞❛❞❡ ♣r♦❥❡t✐✈❛ ❢♦r♠❛❞❛ ♣❡❧♦s ♣♦♥t♦s q✉❡ s❛t✐s❢❛③❡♠

yz =x2.

❚♦♠❛♥❞♦ z = 1✱ ♦❜t❡♠♦s

V A23 ={(x, y) :y=x2}.

❆ss✐♠✱ K(V) =K(x+ (yx2), y+ (yx2))✱ ♦✉ s❡❥❛✱ K(V) = Kx,y˜) ♦♥❞❡ y˜

❡ x˜ sã♦ ❢✉♥çõ❡s s❛t✐s❢❛③❡♥❞♦ y˜= ˜x2✳ ❚❡♠♦s✿

K tr❛♥s❝❡♥❞❡♥t❡֒ K(˜x) ❛❧❣é❜r✐❝❛֒ K(˜x,y˜).

❉❡ ♦♥❞❡ t❡♠♦s q✉❡ ♦ ❣r❛✉ ❞❡ tr❛♥s❝❡♥❞ê♥❝✐❛ ❞❛ ❡①t❡♥sã♦ K(V)/K é ✐❣✉❛❧ ❛ ✶✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ V é ✉♠❛ ❝✉r✈❛✳ ❉❡ ♠♦❞♦ ❣❡r❛❧✱ t❡♠♦s q✉❡ ♣❛r❛ q✉❛❧q✉❡r f K[x, y, z]

♣r✐♠♦ ❡ ❤♦♠♦❣ê♥❡♦✱ ❛ ✈❛r✐❡❞❛❞❡ V(f) = {P ∈P2 :f(P) = 0} é ✉♠❛ ❝✉r✈❛✳

❆❣♦r❛ ♠♦str❛r❡♠♦s ✉♠❛ ✐♠♣♦rt❛♥t❡ ♣r♦♣r✐❡❞❛❞❡ ❞♦s ❛♥é✐s ❧♦❝❛✐s ❞❡ ✉♠❛ ❝✉r✈❛ ❡♠ ✉♠ ♣♦♥t♦ ♥ã♦ s✐♥❣✉❧❛r✳ ❊st❛ ♣r♦♣r✐❡❞❛❞❡ é ❛ ❞❡ q✉❡ ❡st❡s sã♦ ❞♦♠í♥✐♦s ❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✳

❉❡✜♥✐çã♦ ✶✳✷✵✳ ❙❡❥❛K ✉♠ ❝♦r♣♦✳ ❈❤❛♠❛r❡♠♦s ❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛ ❡♠ K ❛ ✉♠❛ ❛♣❧✐❝❛çã♦ s♦❜r❡❥❡t✐✈❛

v :K∗ Z

s❛t✐s❢❛③❡♥❞♦✿

✐✮ v(xy) =v(x) +v(y), ♦✉ s❡❥❛✱v é ✉♠ ❤♦♠♦♠♦r✜s♠♦✳

✐✐✮ v(x+y)>min{v(x), v(y)}.

➱ ❝♦♥✈❡♥✐❡♥t❡ ❡st❡♥❞❡r v à K ❝♦❧♦❝❛♥❞♦v(0) = +✱ ♦♥❞❡+é t❛❧ q✉❡a+= + ❡ +>a, aZ∪ {+∞}.

❚❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s x K t❛✐s q✉❡ v(x) > 0 é ✉♠ ❞♦♠í♥✐♦✱ ❝❤❛♠❛❞♦ ♦

❞♦♠í♥✐♦ ❞❡ ✈❛❧♦r✐③❛çã♦ ❞❡ v✳

❊①❡♠♣❧♦ ✶✳✷✶✳ ❙❡❥❛ K =Q✳ ❉❛❞♦p ♣r✐♠♦ ✜①❛❞♦✱ t❡♠♦s q✉❡ qQ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❡ ❢♦r♠❛ ú♥✐❝❛ ❞❛ ❢♦r♠❛ q =pay✱ ♦♥❞❡ a Z ❡ t❛♥t♦ ♦ ♥✉♠❡r❛❞♦r q✉❛♥t♦ ♦ ❞❡♥♦♠✐♥❛❞♦r

❞❡ y ♥ã♦ ❞✐✈✐❞❡♠ p✳ ❉❡✜♥✐♠♦s vp(q) = a✳ ❊st❛ é ❝❤❛♠❛❞❛ ❞❡ ✈❛❧♦r✐③❛çã♦ p✲á❞✐❝❛✳

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✶✼

Pr♦♣♦s✐çã♦ ✶✳✷✷✳ ❙❡ D é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✱ ❡♥tã♦D é ✉♠ ❛♥❡❧ ❧♦❝❛❧✱ ❡ s❡✉ ✐❞❡❛❧ ♠❛①✐♠❛❧ m é ♦ ❝♦♥❥✉♥t♦ ❞♦s x∈D✱ t❛✐s q✉❡ v(x)>0✳

❉❡♠♦♥str❛çã♦✳ ❱❡r ✭Pr♦♣♦s✐çã♦ ✺✳✶✽✱ ❬❆t✐②❛❤❪✮ ♣❛r❛ ♠♦str❛r q✉❡ D é ❧♦❝❛❧✳ ❊ ❜❛st❛ ♠♦str❛r q✉❡ ♦ ✐❞❡❛❧ ♠❛①✐♠❛❧ ❞❡D ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ✐❞❡❛❧ m={x∈D:v(x)>0}.

Pr♦♣♦s✐çã♦ ✶✳✷✸✳ ❙❡❥❛D✉♠ ❞♦♠í♥✐♦ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦ q✉❡ ♥ã♦ é ✉♠ ❝♦r♣♦✱ s❡❥❛ ms❡✉

✐❞❡❛❧ ♠❛①✐♠❛❧ ❡ k =D/m s❡✉ ❝♦r♣♦ r❡s✐❞✉❛❧✳ ❆s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✳

✐✮ D é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✳

✐✐✮ m é ✉♠ ✐❞❡❛❧ ♣r✐♥❝✐♣❛❧✳

✐✐✐✮ dimk(m/m2) = 1✳

❉❡♠♦♥str❛çã♦✳ ❱❡r ✭Pr♦♣♦s✐çã♦ ✾✳✷✱ ❬❆t✐②❛❤❪✮✳

Pr♦♣♦s✐çã♦ ✶✳✷✹✳ ❙❡❥❛♠ C ✉♠❛ ❝✉r✈❛ ❡ P ∈ C ✉♠ ♣♦♥t♦ ♥ã♦ s✐♥❣✉❧❛r✳ ❚❡♠♦s q✉❡ K[C]P é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✳

❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡ K[C]P é ❧♦❝❛❧✱ ◆♦❡t❤❡r✐❛♥♦ ❡ ♥ã♦ é ✉♠ ❝♦r♣♦✳ ❙❡✉ ✐❞❡❛❧

♠❛①✐♠❛❧ é mP = MP ✭✈✐st♦ ♥♦ ❛♥❡❧ ❧♦❝❛❧✐③❛❞♦ K[C]P✮✳ ❆ss✐♠✱ ❝♦♠♦ K ∼= K[C]P/mP

dimKMP/MP2 = dim(C) = 1✱ t❡♠♦s ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ q✉❡ K[C]P é ✉♠ ❞♦♠í♥✐♦

❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✳

❊ss❛ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛ ✭✉♥✐❢♦r♠✐③❛❞❛✮ é ❞❛❞❛ ♣♦r

ordP :K[C]P −→ {0,1,2, . . .} ∪+∞

f 7−→sup{d:f md

P}.

❊ ✉s❛♥❞♦ordP(f /g) =ordP(f)−ordP(g), ♣♦❞❡♠♦s ❡st❡♥❞❡rordP àK(C)✳

❉✐③❡♠♦s q✉❡ ✉♠ ✉♥✐❢♦r♠✐③❛♥t❡ ♣❛r❛ C ❡♠ P✱ é ✉♠ ❣❡r❛❞♦r ❞❡ mP✱ ✐st♦ é✱ ✉♠

tK(C)♣❛r❛ ♦ q✉❛❧ ordP(t) = 1✳

❉❡✜♥✐çã♦ ✶✳✷✺✳ ❙❡❥❛♠ C ❝✉r✈❛ ❡ P ∈C ♥ã♦ s✐♥❣✉❧❛r✳ ❈❤❛♠❛♠♦s ❞❡ ♦r❞❡♠ ❞❡ f ❡♠ P à ordP(f)✳ ❙❡ ordP(f) > 0✱ ❞✐③❡♠♦s q✉❡ f t❡♠ ✉♠ ③❡r♦ ❡♠ P✱ ❡ s❡ ordP(f) < 0✱

❡♥tã♦ f t❡♠ ✉♠ ♣ó❧♦ ❡♠ P✳

Pr♦♣♦s✐çã♦ ✶✳✷✻✳ ❙❡❥❛ C ✉♠❛ ❝✉r✈❛ s✉❛✈❡ ❡ f ∈ K(C) ❝♦♠ f 6= 0✳ ❊♥tã♦ ❡①✐st❡♠

❛♣❡♥❛s ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ♣♦♥t♦s P ❞❡ C t❛✐s q✉❡ f t❡♠ ③❡r♦ ♦✉ ♣ó❧♦ ❡♠ P✳ ▼❛✐s ❞♦ q✉❡ ✐ss♦✱ s❡f ♥ã♦ t❡♠ ♣ó❧♦s✱ ❡♥tã♦ f ∈K✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ ❛ ✜♥✐t✉❞❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣ó❧♦s✱ ✈❡r ✭❈❛♣ít✉❧♦ ✶✱ ▲❡♠❛ ✶✳✺✱ ❬❍❛rts❤♦r♥❡❪✮✳ ❆ss✐♠✱ ♣❛r❛ ♦❜t❡r ❛ ✜♥✐t✉❞❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ③❡r♦s✱ ❜❛st❛ ✉s❛r ❛ ✜♥✐t✉❞❡ ❞♦s ♣ó❧♦s ♣❛r❛

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