Codificação espaço-temporal

❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

❈♦❞✐✜❝❛çã♦ ❊s♣❛ç♦✲❚❡♠♣♦r❛❧

❚❤✐❛❣♦ ❚❛♠❜❛s❝♦ ▲✉✐③

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜✲ t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r❛

Pr♦❢❛✳ ❉r❛✳ ❈❛r✐♥❛ ❆❧✈❡s

❚❛♠❜❛s❝♦ ▲✉✐③✱ ❚❤✐❛❣♦

❈♦❞✐✜❝❛çã♦ ❊s♣❛ç♦✲❚❡♠♣♦r❛❧✴ ❚❤✐❛❣♦ ❚❛♠❜❛s❝♦ ▲✉✐③✲ ❘✐♦ ❈❧❛r♦✿ ❬s✳♥✳❪✱ ✷✵✶✷✳

✻✻ ❢✳

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳

❖r✐❡♥t❛❞♦r❛✿ ❈❛r✐♥❛ ❆❧✈❡s

✶✳ ❈ó❞✐❣♦ ❞❡ ❖✉r♦✳ ✷✳ á❧❣❡❜r❛s ❞❡ ❞✐✈✐sã♦✳ ✸✳ r❡t✐❝✉❧❛❞♦s ❛❧❣é❜r✐✲ ❝♦s✳ ✹✳ ❝ó❞✐❣♦s ♣❡r❢❡✐t♦s✳ ■✳ ❚ít✉❧♦

❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

❚❤✐❛❣♦ ❚❛♠❜❛s❝♦ ▲✉✐③

❈♦❞✐❢✐❝❛çã♦ ❊s♣❛ç♦✲❚❡♠♣♦r❛❧

❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛✲ ❞♦r❛✿

Pr♦❢❛✳ ❉r❛✳ ❈❛r✐♥❛ ❆❧✈❡s ❖r✐❡♥t❛❞♦r❛

Pr♦❢✳ ❉r✳ ❍❡♥r✐q✉❡ ▲❛③❛r✐ ■●❈❊ ✲ ❯♥❡s♣ ✲ ❘✐♦ ❈❧❛r♦ ✲ ❙P

Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❆♣❛r❡❝✐❞♦ ❉❡ ❆♥❞r❛❞❡ ■❇■▲❈❊ ✲ ❯♥❡s♣ ✲ ❙ã♦ ❏♦sé ❞♦ ❘✐♦ Pr❡t♦ ✲ ❙P

❆❣r❛❞❡❝✐♠❡♥t♦s

❆♣ós ❝♦♥❝❧✉✐r ❡st❡ tr❛❜❛❧❤♦✱ ❛❣r❛❞❡ç♦✿ ❆ ❉❡✉s✳

➚ ♠✐♥❤❛ ♠ã❡✱ ❘♦s♠❛r② ❚❛♠❜❛s❝♦ ▲✉✐③✱ ♠✉❧❤❡r ❢♦rt❡ q✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ❛♦s ❡st✉❞♦s ❡ ♠❡ ❡♥s✐♥♦✉ q✉❡ ❛ ♣❡rs❡✈❡r❛♥ç❛ é ❛ ♣r✐♥❝✐♣❛❧ ✈✐rt✉❞❡ q✉❡ ♣♦❞❡♠♦s t❡r✳

❆♦ ♠❡✉ ♣❛✐✱ ❲❛❧❞✐r ❱✐✈❛❝q✉❛ ▲✉✐③✱ q✉❡ s❡♠♣r❡ ❜❛t❛❧❤♦✉ ♣❡❧♦ ❜❡♠ ❞❛ ❢❛♠í❧✐❛ ❛ q✉❛❧ ❡st❡✈❡ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ❡♠ t♦❞❛s ❛s s✉❛s ❞❡❝✐sõ❡s✳

➚ ♠✐♥❤❛ ✐r♠ã✱ ❚❤❛ís ❚❛♠❜❛s❝♦ ▲✉✐③✱ ♣♦r s❡r ✉♠ ❣r❛♥❞❡ ❡①❡♠♣❧♦ ❞❡ s✉♣❡r❛çã♦ ❡ ❡s❢♦rç♦ ♣❛r❛ ❢❛③❡r ❞❛ ✈✐❞❛ ♥ã♦ ✉♠❛ ♠❡r❛ ♣❛ss❛❣❡♠✱ ♠❛s ✉♠❛ r❡❛❧✐③❛çã♦ ♣❧❡♥❛ ❞❡ s❡✉s s♦♥❤♦s✳

❆ t♦❞♦s ♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s q✉❡ ❝♦♠ t❛♥t♦ ❝❛r✐♥❤♦ ♠❡ ❛♣♦✐❛r❛♠✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❊❧✐é③❡r✱ ❑át✐❛✱ ❍❡♥r✐q✉❡ ❡ ❇r❛③ ♣♦r ✜❝❛r❡♠ ❛♦ ♠❡✉ ❧❛❞♦ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s ❞❡ss❛ ❝❛♠✐♥❤❛❞❛✱ ❡ ♥ã♦ ♠❡ ❞❡✐①❛r❡♠ ❝❛✐r q✉❛♥❞♦ tr♦♣❡❝❡✐✳

❆♦s ❛♠✐❣♦s ❞❛ ❈♦♥❢r❛r✐❛ ♣♦r t♦❞❛s ❛s r✐s❛❞❛s✱ ✈✐❛❣❡♥s ❡ ♥♦✐t❡s ❞❡ s❡①t❛✲❢❡✐r❛ tã♦ ✐♠♣♦rt❛♥t❡s ❡ ♥❡❝❡ssár✐❛s ♣❛r❛ q✉❡ ❝❤❡❣❛ss❡ ❛té ❛q✉✐✳

❆ t♦❞♦s ♦s ♠❡✉s ♦✉tr♦s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦ ♣❡❧♦ ❛♣♦✐♦ ❡ ❢♦rç❛✱ s❡♠♣r❡ ❥✉♥t♦s✳

➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ Pr♦❢❛ _{❉r❛✳ ❈❛r✐♥❛ ❆❧✈❡s ♣❡❧❛ ❛♠✐③❛❞❡✱ ❞❡❞✐❝❛çã♦ ❡①tr❡♠❛ ❡}

❛❥✉❞❛ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ♥❛ ❡①❡❝✉çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳ ➚ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ♣❡❧❛s ❝♦rr❡çõ❡s ❡ s✉❣❡stõ❡s✳

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ♥ós ❛❜♦r❞❛♠♦s ❛❧❣✉♥s ❞♦s ♣r✐♥❝✐♣❛✐s ❛s♣❡❝t♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ❝♦❞✐✲ ✜❝❛çã♦ ❡s♣❛ç♦✲t❡♠♣♦r❛❧ ❡ ❛s ❢❡rr❛♠❡♥t❛s ❛❧❣é❜r✐❝❛s ❡♥✈♦❧✈✐❞❛s ♥❛ ♣r♦❥❡çã♦ ❞❡ ❝ó❞✐❣♦s ❜❛s❡❛❞♦s ❡♠ á❧❣❡❜r❛s ❞❡ ❞✐✈✐sã♦ ❝í❝❧✐❝❛✳ ❆♣r❡s❡♥t❛r❡♠♦s t❛♠❜é♠ ❛ ❝♦♥str✉çã♦ ❞♦ ❈ó❞✐❣♦ ❞❡ ❖✉r♦ ✭❬✾❪✱ ❬✶✵❪✮✱ q✉❡ é ✉♠ ❝ó❞✐❣♦ ❡s♣❛ç♦✲t❡♠♣♦r❛❧ ♣❡r❢❡✐t♦✳

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ ❞✐s❝✉ss s♦♠❡ ♠❛✐♥ ❛s♣❡❝ts r❡❧❛t❡❞ t♦ s♣❛❝❡✲t✐♠❡ ❝♦❞✐♥❣ ❛♥❞ ❛❧❣❡✲ ❜r❛✐❝ t♦♦❧s ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ ❞❡s✐❣♥ ♦❢ ❝♦❞❡s ❜❛s❡❞ ♦♥ ❝②❝❧✐❝ ❞✐✈✐s✐♦♥ ❛❧❣❡❜r❛s✳ ❲❡ ❛❧s♦ ♣r❡s❡♥t t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ●♦❧❞❡♥ ❈♦❞❡ ✭❬✾❪✱ ❬✶✵❪✮✱ ✇❤✐❝❤ ✐s ❛ ♣❡r❢❡❝t s♣❛❝❡✲t✐♠❡ ❝♦❞❡✳

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶✺

✷ ❖ ▼♦❞❡❧♦ ❞♦ ❙✐st❡♠❛ ▼■▼❖ ✶✾

✷✳✶ ❖ ♠♦❞❡❧♦ ❞♦ s✐st❡♠❛ ▼■▼❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ❈r✐tér✐♦s ♣❛r❛ ♠♦❞❡❧❛r ❝ó❞✐❣♦s ❡s♣❛ç♦✲t❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❛❧❣é❜r✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✹ ❈♦r♣♦s ◗✉❛❞rát✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✺ ▼ó❞✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✻ ◆ú♠❡r♦s ♣✲á❞✐❝♦s ❡ ❛♥❡❧ ❞❡ ✈❛❧♦r✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✸ ❘❡t✐❝✉❧❛❞♦s ❆❧❣é❜r✐❝♦s ✸✼

✹ ➪❧❣❡❜r❛s ❈í❝❧✐❝❛s ✹✺

✺ ❖ ❈ó❞✐❣♦ ❞❡ ❖✉r♦ ✺✸

✺✳✶ ❈ó❞✐❣♦s ❡s♣❛ç♦✲t❡♠♣♦ ♣❡r❢❡✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺✳✷ ❖ ❝ó❞✐❣♦ ❞❡ ♦✉r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✺✳✷✳✶ ❖ ❡❧❡♠❡♥t♦γ =i ♥ã♦ é ✉♠❛ ♥♦r♠❛ ❡♠ Q(i,√5) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✺✳✷✳✷ ❖ r❡t✐❝✉❧❛❞♦ Z[i]2 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼} ✺✳✷✳✸ ❖ ❞❡t❡r♠✐♥❛♥t❡ ♠í♥✐♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✺✳✸ ❆❧❣✉♥s ❝♦♠❛♥❞♦s ❡♠ ❑❆❙❍ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✻ ❈♦♥❝❧✉sã♦ ✻✸

✶ ■♥tr♦❞✉çã♦

❉❡s❞❡ ❛s ♣r✐♠❡✐r❛s tr❛♥s♠✐ssõ❡s ❞❡ t❡❧é❣r❛❢♦✱ ❛ tr❛♥s♠✐ssã♦ ❞❡ ❞❛❞♦s ♣♦r ♠❡✐♦s ❡❧étr✐❝♦s t❡♠ r❡✈♦❧✉❝✐♦♥❛❞♦ ♦ ♠♦❞♦ ❝♦♠ ♦ q✉❛❧ ❛s ♣❡ss♦❛s s❡ ❝♦♠✉♥✐❝❛♠ ❡ ❝♦♠♦ ❛ s♦❝✐❡❞❛❞❡ ❧✐❞❛ ❝♦♠ ❛ ✐♥❢♦r♠❛çã♦✳ ◆♦s ❞✐❛s ❛t✉❛✐s✱ é ✐♥❡❣á✈❡❧ ♦ ❢❛t♦ ❞❡ q✉❡ ❛ r♦t✐♥❛ ❞♦ ❞✐❛✲❛✲❞✐❛ s❡ t♦r♥❛ ❝❛❞❛ ✈❡③ ♠❛✐s ❞❡♣❡♥❞❡♥t❡ ❞♦s ♠❡✐♦s ❞❡ ❝♦♠✉♥✐❝❛çã♦ ❡ ❞❡♠❛♥❞❛ ❞❡❧❡s ✉♠❛ t❛①❛ ❞❡ tr❛♥s♠✐ssã♦ ❞❡ ❞❛❞♦s ❝❛❞❛ ✈❡③ ♠❛✐s ❛❧t❛✳ ❉❡s❞❡ ♦s rá❞✐♦s ❝♦♠✉♥✐❝❛❞♦r❡s ❡ ♣❛ss❛♥❞♦ ♣❡❧♦ ❛❞✈❡♥t♦ ❞♦ t❡❧❡❢♦♥❡ ❝❡❧✉❧❛r✱ ❛s tr❛♥s♠✐ssõ❡s ❞❡ ❞❛❞♦s s❡♠ ✜♦ t❡♠ ❣❛♥❤❛❞♦ ✐♠♣♦rtâ♥❝✐❛ s✐❣♥✐✜❝❛t✐✈❛✱ ❡ ❣❛r❛♥t✐r ❛ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❞❡ s✐♥❛❧ é ✉♠ ❞♦s ❞❡s❛✜♦s ❞❡ q✉❡♠ tr❛♥s♠✐t❡ ❡ss❡ t✐♣♦ ❞❡ s✐♥❛❧✳

❯❧t✐♠❛♠❡♥t❡✱ ❛❧é♠ ❞❡ ❛♣❡♥❛s tr❛♥s♠✐t✐r s✐♥❛✐s ❞❡ ❧✐❣❛çõ❡s t❡❧❡❢ô♥✐❝❛s✱ ♥♦✈♦s t✐♣♦s ❞❡ ❞❛❞♦s ♣❛ss❛r❛♠ ❛ s❡r tr❛♥s♠✐t✐❞♦s ♣♦r ❡ss❡s ❛♣❛r❡❧❤♦s ❝♦♠♦ ♠ús✐❝❛s✱ ❢♦t♦s ❡ ❡t❝✱ ♦ q✉❡ ❞❡♠❛♥❞❛ ✉♠❛ r❡❞❡ q✉❡ ❛❧é♠ ❛❧t❛ q✉❛❧✐❞❛❞❡ ❞❡ s✐♥❛❧ t❡♥❤❛ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ❛❧t❛ t❛♠❜é♠ ♥❛ tr❛♥s♠✐ssã♦ ❞❡ss❡s ❞❛❞♦s✳ ◗✉❛♥❞♦ ❧❡✈❛♠♦s ❡♠ ❝♦♥s✐❞❡r❛çã♦ r❡❞❡s ✸●✱ ✸●✰ ♦✉ ✹●✱ ❛s t❛①❛s ❞❡ tr❛♥s♠✐ssã♦ sã♦ ♥❛ ♦r❞❡♠ ❞❡ ●✐❣❛❜✐ts ♣♦r s❡❣✉♥❞♦ ❡ ❝♦♥s❡❣✉✐r ❛❧✐❛r ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❞❡ s✐♥❛❧ à t❛❧ ✈❡❧♦❝✐❞❛❞❡ é ✉♠❛ t❛r❡❢❛ ❝♦♠♣❧❡①❛✳

❚r❛♥s♠✐t✐r ❞❛❞♦s ♣❡❧♦ ♠❡✐♦ ❛t♠♦s❢ér✐❝♦ ❡♥✈♦❧✈❡ ♠✉✐t♦s ♣r♦❜❧❡♠❛s ✐♥❡r❡♥t❡s ❛ ❡ss❡ ♠❡✐♦✱ ❝♦♠♦ ❞✐❢❡r❡♥ç❛ ❞❡ t❡♠♣❡r❛t✉r❛ ❡♥tr❡ ❝❛♠❛❞❛s ❞❛ ❛t♠♦s❢❡r❛✱ ❢❡♥ô♠❡♥♦s ♠❡t❡♦✲ r♦❧ó❣✐❝♦s✱ ❜❧♦q✉❡✐♦s ❝❛✉s❛❞♦s ♣♦r ❝♦♥str✉çõ❡s✱ ♣❡ss♦❛s✱ ❛♥✐♠❛✐s ❡ ♦✉tr♦s ♦❜❥❡t♦s q✉❡ ❡st❡❥❛♠ ♥♦ ❝❛♠✐♥❤♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞♦ s✐♥❛❧ ❡ ❢❛③❡♠ ❝♦♠ q✉❡ ❡❧❡ s❡ ❡♥❢r❛q✉❡ç❛✱ ❛❧é♠ ❞❛ ♣❡r❞❛ ♥❛t✉r❛❧ ❞❡ ❡♥❡r❣✐❛ q✉❡ ♦❝♦rr❡ ❞✉r❛♥t❡ ❛ ♣r♦♣❛❣❛çã♦ ❞❛ ♦♥❞❛✳ P❛r❛ ♦❜t❡r ✉♠ ❜♦♠ r❡s✉❧t❛❞♦✱ ✉♠❛ ♦♣çã♦ s❡r✐❛ ✉t✐❧✐③❛r ✉♠❛ ❢❛✐①❛ ❞❡ ❢r❡qüê♥❝✐❛ ❣r❛♥❞❡✱ ♠❛s ❝♦♠♦ ❛s ❢❛✐①❛s ❞❡ ❢r❡qüê♥❝✐❛s sã♦ ❡s❝❛ss❛s ❡ ❝❛r❛s✱ ❡ss❛ ♦♣çã♦ ♥ã♦ é ✈✐á✈❡❧✳ ❯♠ t✐♣♦ ❞❡ s✐st❡♠❛ q✉❡ t❡♠ s✐❞♦ ❜❛st❛♥t❡ ✉t✐❧✐③❛❞♦ ❡ ✈✐st♦ ❝♦♠♦ ♠✉✐t♦ ♣r♦♠✐ss♦r é ♦ ▼■▼❖ ✲ ▼✉❧t✐♣❧❡ ■♥♣✉t ▼✉❧t✐♣❧❡ ❖✉t♣✉t ✲ q✉❡ ❝♦♥s✐st❡ ♥♦ ✉s♦ ❞❡ ♠ú❧t✐♣❧❛s ❛♥t❡♥❛s ♣❛r❛ ❡♥✈✐♦ ❡ r❡❝❡❜✐♠❡♥t♦ ❞❡ s✐♥❛❧✳ ❖s s✐st❡♠❛s ❞❡ ❝♦♠✉♥✐❝❛çã♦ ▼■▼❖ ❡stã♦ s❡♥❞♦ ❛♠♣❧❛♠❡♥t❡ ❡①♣❧♦r❛❞♦s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦r ❢♦r♥❡❝❡r ❣❛♥❤♦s ♥❛ tr❛♥s♠✐ssã♦ ❞❡ s✐♥❛❧✳

❖s ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❡♠ ❛ tr❛♥s♠✐ssã♦ ❞❡ ✉♠ s✐♥❛❧ s❡♠ ✜♦ ❛♣❛r❡❝❡♠ ✐♥❞❡♣❡♥❞❡♥✲ t❡♠❡♥t❡ ❞♦ s✐st❡♠❛ ✉t✐❧✐③❛❞♦ ♣♦✐s sã♦ ♣ró♣r✐♦s ❞♦ ♠❡✐♦✳ ❖s ❡❢❡✐t♦s ♥♦❝✐✈♦s ❞♦ ❛♠❜✐❡♥t❡ sã♦ r❡❢❡r❡♥t❡s ❛ ❞❡s✈❛♥❡❝✐♠❡♥t♦s ❞❡ ❧❛r❣❛ ❡s❝❛❧❛ ❡ ♣❡q✉❡♥❛ ❡s❝❛❧❛✳ ❉❡s✈❛♥❡❝✐♠❡♥t♦s ❞❡ ❧❛r❣❛ ❡s❝❛❧❛ ❡stã♦ ❛ss♦❝✐❛❞♦s à ♣❡r❞❛ ❞❛ q✉❛❧✐❞❛❞❡ ❞♦ s✐♥❛❧ ❞❡✈✐❞♦ à ♣❡r❞❛ ❞❡ ♣♦tê♥❝✐❛ ❞♦ s✐♥❛❧ ❝♦♥❢♦r♠❡ ❡ss❡ s❡ ♣r♦♣❛❣❛ ❡ à ♦❜stá❝✉❧♦s q✉❡ ❡ss❡ s✐♥❛❧ ♣♦❞❡ ❡♥❢r❡♥t❛r ❛♦ ❧♦♥❣♦ ❞♦ ♣❡r❝✉rs♦ ❝♦♠♦ ❡❞✐❢í❝✐♦s✱ ár✈♦r❡s✱ ♦✉tr❛s ❛♥t❡♥❛s✱ ❡t❝✱ ♦✉ s❡❥❛✱ ❛t✉❛♠ ♥❛ ♦r❞❡♠ ❞❡

✶✻ ■♥tr♦❞✉çã♦

✈ár✐♦s ❝♦♠♣r✐♠❡♥t♦s ❞❡ ♦♥❞❛ ❞♦ s✐♥❛❧✳ ❖s ❞❡s✈❛♥❡❝✐♠❡♥t♦s ❞❡ ♣❡q✉❡♥❛ ❡s❝❛❧❛ sã♦ ♦s q✉❡ ❛t✉❛♠ ♥❛ ♦r❞❡♠ ❞❡ ✉♠ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ❞♦ s✐♥❛❧ ❡ s❡ ❞ã♦ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ♦ ♠❡s♠♦ s✐♥❛❧ ♣♦❞❡ ❝❤❡❣❛r ❛♦ r❡❝❡♣t♦r ♣♦r ♣❡r❝✉rs♦s ❞✐❢❡r❡♥t❡s✱ ❝♦♠ ❢❛s❡s ❞✐❢❡r❡♥t❡s ❡ ❛♠♣❧✐t✉❞❡s ❞✐❢❡r❡♥t❡s✳ ❆ s♦❜r❡♣♦s✐çã♦ ❞❡ss❡s s✐♥❛✐s é ❝❤❛♠❛❞❛ ❞❡ ❝♦♠♣♦♥❡♥t❡ ❞❡ ♠✉❧t✐♣❡r❝✉s♦✳ ❖s ♠✉❧t✐♣❡r❝✉rs♦s ❣❡r❛♠ ✈❛r✐❛çõ❡s rá♣✐❞❛s ♥❛ ❛♠♣❧✐t✉❞❡ ❞♦ s✐♥❛❧ r❡✲ ❝❡❜✐❞♦✱ ✉♠❛ ✈❡③ q✉❡ ♦s s✐♥❛✐s q✉❡ ❝❤❡❣❛♠ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❛♦ r❡❝❡♣t♦r ❝♦♠❜✐♥❛♠✲s❡ ❝♦♥str✉t✐✈❛ ❡✴♦✉ ❞❡str✉t✐✈❛♠❡♥t❡✳ ❯♠❛ ♠❛♥❡✐r❛ ❞❡ ❛✉♠❡♥t❛r ❛ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❞♦ s✐♥❛❧ é ❡♥✈✐❛r ♦ ♠❡s♠♦ s✐♥❛❧ ❛tr❛✈és ❞❡ ♠ú❧t✐♣❧❛s ❛♥t❡♥❛s✱ ♦✉ s❡❥❛✱ ✉s❛r ❛ r❡❞✉♥❞â♥❝✐❛ ❝♦♠♦ ❢♦r♠❛ ❞❡ ❛✉♠❡♥t❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ q✉❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s s✐♥❛✐s ❡♥✈✐❛❞♦s ❝❤❡❣✉❡ ❛♦ r❡❝❡♣t♦r s❡♠ ❞❡s✈❛♥❡❝✐♠❡♥t♦ ♦✉ ✐♥t❡r❢❡rê♥❝✐❛✳

❙❡ ❝♦♥s✐❞❡r❛r♠♦s q✉❡ ❡①✐st❡♠ nt ❛♥t❡♥❛s tr❛♥s♠✐ss♦r❛s ❡ nr ❛♥t❡♥❛s r❡❝❡♣t♦r❛s✱

❤❛✈❡r✐❛ nt· nr ❧✐❣❛çõ❡s ❡♥tr❡ ♦ tr❛♥s♠✐ss♦r ❡ ♦ r❡❝❡♣t♦r✳ ❊ss❡ ❣❛♥❤♦ ♦❜t✐❞♦ é ❝❤❛✲

♠❛❞♦ ❞❡ ❣❛♥❤♦ ❞❡ ❞✐✈❡rs✐❞❛❞❡ ❡✱ ♥❡st❡ ❝❛s♦✱ ❞✐③✲s❡ q✉❡ ❤á ✉♠❛ ♣r♦t❡çã♦ ❞❡ ♦r❞❡♠ nt ·nr ❝♦♥tr❛ ❞❡s✈❛♥❡❝✐♠❡♥t♦s ❞♦ s✐♥❛❧✱ ♦✉ s❡❥❛✱ ❛s ❝❤❛♥❝❡s ❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠ s✐♥❛❧

tr❛♥s♠✐t✐❞♦ ❝❤❡❣❛r ❛ ✉♠ r❡❝❡♣t♦r ❝♦♠ q✉❛❧✐❞❛❞❡ é ❛✉♠❡♥t❛❞❛✳ ❆ ♦r❞❡♠ ❞♦ ❣❛♥❤♦ ❞❡ ❞✐✈❡rs✐❞❛❞❡ é ❞❡✜♥✐❞❛ ❝♦♠♦ s❡♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❧✐❣❛çõ❡s ✐♥❞❡♣❡♥❞❡♥t❡s ❡♥tr❡ r❡❝❡♣t♦r❡s ❡ tr❛♥s♠✐ss♦r❡s✱ s❡♥❞♦ q✉❡ sã♦ ❝♦♥s✐❞❡r❛❞❛s ❧✐❣❛çõ❡s ✐♥❞❡♣❡♥❞❡♥t❡s ❛q✉❡❧❛s q✉❡ ♥ã♦ sã♦ r❡❞✉♥❞❛♥t❡s✱ ♦✉ s❡❥❛✱ ♥ã♦ tr❛♥s♠✐t❡♠ ♦ ♠❡s♠♦ s✐♥❛❧✳ ❊ss❡ ♠ét♦❞♦✱ ❛♣❡s❛r ❞❡ ❣❛r❛♥t✐r t❛❧ ❛✉♠❡♥t♦ ❞❛ q✉❛❧✐❞❛❞❡ ❞♦ s✐♥❛❧✱ ♥ã♦ ♦t✐♠✐③❛ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ tr❛♥s♠✐ssã♦ ❞❛ ✐♥❢♦r♠❛çã♦ ♣♦✐s ✈ár✐❛s ❛♥t❡♥❛s ❡♥✈✐❛♠ ♦ ♠❡s♠♦ s✐♥❛❧ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ✐st♦ é✱ ❛ ✐♥❢♦r♠❛çã♦ ❞❡♠♦r❛ ♠❛✐s ♣❛r❛ s❡r tr❛♥s♠✐t✐❞❛ ♣♦r ❝♦♠♣❧❡t♦ ❞♦ q✉❡ ❞❡♠♦r❛r✐❛ s❡ ❝❛❞❛ ❧✐❣❛çã♦ tr❛♥s♠✐t✐ss❡ ✉♠ s✐♥❛❧ ❞✐❢❡r❡♥t❡✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ❛ ✐♥❢♦r♠❛çã♦ ❢♦r ❞✐✈✐❞✐❞❛ ❡♠ ♣❡q✉❡♥♦s ❜❧♦❝♦s ❡ ❝❛❞❛ ❛♥t❡♥❛ ❞♦ s✐st❡♠❛ tr❛♥s♠✐t✐r ✉♠ ❞❡ss❡s ❜❧♦❝♦s✱ t♦❞❛s ❛s ❧✐❣❛çõ❡s ❡♥tr❡ tr❛♥s♠✐ss♦r❡s ❡ r❡❝❡♣t♦r❡s t♦r♥❛♠✲s❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ tr❛♥s♠✐ssã♦ ❞❡ ❞❛❞♦s ❛✉♠❡♥t❛ ♠✉✐t♦✱ ♠❛s ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ q✉❡ ❛❧❣✉♥s s✐♥❛✐s s❡✲ ❥❛♠ ♣❡r❞✐❞♦s ♦✉ ✐♥❝♦rr❡t❛♠❡♥t❡ ✐♥t❡r♣r❡t❛❞♦s ♣♦r ✉♠ r❡❝❡♣t♦r ❞❡✈✐❞♦ à ✈ár✐♦s ❢❛t♦r❡s ♣r❡s❡♥t❡s ♥♦ ♠❡✐♦ ❞❡ ♣r♦♣❛❣❛çã♦ é ❛❧t❛✳ ❊ss❡ ❣❛♥❤♦ ♦❜t✐❞♦ ♥❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ tr❛♥s✲ ♠✐ssã♦ ❞❡✈✐❞♦ à ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❞❛s ❧✐❣❛çõ❡s é ❝❤❛♠❛❞♦ ❞❡ ❣❛♥❤♦ ❞❡ ♠✉❧t✐♣❧❡①❛❣❡♠✳ ❖ ❣❛♥❤♦ ❞❡ ♠✉❧t✐♣❧❡①❛❣❡♠ ❞❡♣❡♥❞❡ ❞✐r❡t❛♠❡♥t❡ ❞♦ ♥ú♠❡r♦ ❞❡ ❛♥t❡♥❛s tr❛♥s♠✐ss♦r❛s ❞♦ s✐st❡♠❛ ❥á q✉❡ q✉❛♥t♦ ♠❛✐s ❛♥t❡♥❛s✱ ❡♠ ♠❛✐s ❜❧♦❝♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❛ ✐♥❢♦r♠❛çã♦ ♣♦❞❡ s❡r ❞✐✈✐❞✐❞❛ ❡ ❝♦♥s❡qü❡♥t❡♠❡♥t❡ ♠❛✐s rá♣✐❞♦ ❛ ✐♥❢♦r♠❛çã♦ s❡rá tr❛♥s♠✐t✐❞❛✳ P❛r❛ ♠❛①✐♠✐③❛r ♦s ❣❛♥❤♦s t❛♥t♦ ❞❡ ❞✐✈❡rs✐❞❛❞❡ q✉❛♥t♦ ❞❡ ♠✉❧t✐♣❧❡①❛çã♦ ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ♦ ✉s♦ ❞♦s ❝ó❞✐❣♦s ❡s♣❛ç♦✲t❡♠♣♦r❛✐s✳ ❉❡ss❛ ❢♦r♠❛✱ ❛s ❛♥t❡♥❛s tr❛♥s♠✐t❡♠ ❛ ♠❡s♠❛ ✐♥✲ ❢♦r♠❛çã♦ ❝♦♠ ✉♠❛ ♣❡q✉❡♥❛ ❞✐❢❡r❡♥ç❛ ❞❡ t❡♠♣♦ ❡♥tr❡ ❛s tr❛♥s♠✐ssõ❡s✱ ❞❡ ❢♦r♠❛ q✉❡ ❤❛❥❛ r❡❞✉♥❞â♥❝✐❛✱ ❣❛r❛♥t✐♥❞♦ ❛ q✉❛❧✐❞❛❞❡ ❞❡ s✐♥❛❧✱ ♠❛s q✉❡ ❛s ❧✐❣❛çõ❡s s❡❥❛♠ ✐♥❞❡✲ ♣❡♥❞❡♥t❡s✱ ❛✉♠❡♥t❛♥❞♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ tr❛♥s♠✐ssã♦✳ ❉❡♥tr❡ ♦s t✐♣♦s ❞❡ ❝♦❞✐✜❝❛çã♦ ❡s♣❛ç♦✲t❡♠♣♦r❛❧ ❞❡st❛❝❛♠✲s❡ ♦s ❝ó❞✐❣♦s ❞❡ ❜❧♦❝♦ ❡s♣❛ç♦✲t❡♠♣♦r❛✐s ✭❙❚❇❈✮ ♣♦r s✉❛ ❢❛❝✐❧✐❞❛❞❡ ❞❡ ❝♦❞✐✜❝❛çã♦✴❞❡❝♦❞✐✜❝❛çã♦✳

✶✼

❛❧❣é❜r✐❝♦ q✉❡ ❢♦r♥❡❝❡ ♥❛t✉r❛❧♠❡♥t❡ ✉♠ ❝♦♥❥✉♥t♦ ❧✐♥❡❛r ❞❡ ♠❛tr✐③❡s ✐♥✈❡rsí✈❡✐s ❡ ❛ss✐♠✱ ♦ ❝r✐tér✐♦ ❞♦ ♣♦st♦ é s❛t✐s❢❡✐t♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♦r❣❛♥✐③❛♠♦s ♦ r❡st❛♥t❡ ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦ ❝♦♥❢♦r♠❡ ❞❡❧✐♥❡❛♠♦s ♥❛ s❡q✉ê♥❝✐❛✳

◆♦ ❈❛♣ít✉❧♦ ✷✱ ❛♣r❡s❡♥t❛♠♦s ♦ ♠♦❞❡❧♦ ❞♦ s✐st❡♠❛ ▼■▼❖✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❛❧❣é❜r✐❝♦s q✉❡ s❡r✈✐r❛♠ ❞❡ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ♦ ❞❡s❡♥✲ ✈♦❧✈✐♠❡♥t♦ ❞♦s ❞❡♠❛✐s ❝❛♣ít✉❧♦s✳

◆♦ ❈❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛♠♦s ❛ t❡♦r✐❛ ❞❡ r❡t✐❝✉❧❛❞♦s ❛❧❣é❜r✐❝♦s✳

◆♦ ❈❛♣ít✉❧♦ ✹✱ ❛♣r❡s❡♥t❛♠♦s ❛s á❧❣❡❜r❛s ❝í❝❧✐❝❛s ❡ ❡①♣❧♦r❛♠♦s ❛ s✉❛ r❡❧❛çã♦ ❝♦♠ ❛ ❝♦♥str✉çã♦ ❞❡ ❝ó❞✐❣♦s ❡s♣❛ç♦✲t❡♠♣♦✳

◆♦ ❈❛♣ít✉❧♦ ✺✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❝♦♥str✉çã♦ ❞♦ ❈ó❞✐❣♦ ❞❡ ❖✉r♦ q✉❡ é ✉♠ ❙❚❇❈ ♣❡r❢❡✐t♦✳

✷ ❖ ▼♦❞❡❧♦ ❞♦ ❙✐st❡♠❛ ▼■▼❖

➱ ♥❡❝❡ssár✐♦ ❝r✐❛r ✉♠ ♠♦❞❡❧♦ ❛❞❡q✉❛❞♦ ❞❡ ❝❛♥❛❧ ▼■▼❖ q✉❡ ❛❜r❛♥❥❛ ❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❝❛♥❛❧ s❡♠ ✜♦✳ ◆❡st❡ s❡♥t✐❞♦✱ ❞❡s❝r❡✈❡r❡♠♦s ❝r✐tér✐♦s ♣❛r❛ q✉❡ ❛ ♣r♦✲ ❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡rr♦ s❡❥❛ ♠✐♥✐♠✐③❛❞❛✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s s♦❜r❡ ❛ t❡♦r✐❛ ❛❜♦r❞❛❞❛ ❛q✉✐✱ ♣♦❞❡♠ s❡r ❝♦♥s✉❧t❛❞❛s ❛s r❡❢❡rê♥❝✐❛s ❬✶❪✱ ❬✸❪ ❡ ❬✾❪✳

✷✳✶ ❖ ♠♦❞❡❧♦ ❞♦ s✐st❡♠❛ ▼■▼❖

❯♠❛ ú♥✐❝❛ ❛♥t❡♥❛ tr❛♥s♠✐ss♦r❛ ❞✉r❛♥t❡ ✉♠ ú♥✐❝♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ❡♥✈✐❛ ✉♠ s✐♥❛❧ q✉❡ ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞♦ ❝♦♠♦ ✉♠ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ x _∈ C✳ ❉✉r❛♥t❡ ♦

♣❡r❝✉rs♦ ❞♦ s✐♥❛❧ ❛té ❛ ❛♥t❡♥❛ r❡❝❡♣t♦r❛✱ ❡❧❡ s♦❢r❡rá ❛❧❣✉♠❛ ❞✐st♦rçã♦ q✉❡ ♣♦❞❡ s❡r ♠♦❞❡❧❛❞❛ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ x ♣♦r ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ h ❝❤❛♠❛❞♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❞❡s✈❛♥❡❝✐♠❡♥t♦✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❛♥t❡♥❛ r❡❝❡♣t♦r❛ ❝❛♣t❛rá ❛❧❣✉♠ r✉í❞♦ ♣r❡s❡♥t❡ ♥♦ ♠❡✐♦ ❞❡ ♣r♦♣❛❣❛çã♦ q✉❡ ♣♦❞❡ s❡r ♠♦❞❡❧❛❞♦ ♣❡❧❛ ❛❞✐çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z✳ ❆ss✐♠✱ ♦ s✐♥❛❧ r❡❝❡❜✐❞♦ s❡rá y =hx+z✳

❙✉♣♦♥❞♦ ❛❣♦r❛ q✉❡ ❤❛❥❛ nt ❛♥t❡♥❛s tr❛♥s♠✐t✐♥❞♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ✉♠ s✐♥❛❧ ♣❛r❛

✉♠❛ ú♥✐❝❛ ❛♥t❡♥❛ r❡❝❡♣t♦r❛✳ ❙❡❥❛ xj ∈ C ♦ s✐♥❛❧ ❡♥✈✐❛❞♦ ♣❡❧❛ j✲és✐♠❛ ❛♥t❡♥❛ (1 ≤

j _≤nt)✳ ❈♦♠♦ ❡❧❛s tr❛♥s♠✐t❡♠ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ❝❛❞❛ ✉♠❛ ❞❡ss❛s ❛♥t❡♥❛s t❡♠ ♣❛rt✐✲

❝✐♣❛çã♦ ♥♦ s✐♥❛❧ ❞❡t❡❝t❛❞♦ ♥♦ r❡❝❡♣t♦r✳ ❖ s✐♥❛❧ r❡❝❡❜✐❞♦ y _∈ C s❡rá ♣♦rt❛♥t♦ ❛❧❣✉♠❛

❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ t❛✐s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ♠❛✐s ✉♠ ❝❡rt♦ r✉í❞♦✿ y=h1x1+h2x2 +...+hntxnt+z

♦♥❞❡ hj ∈C é ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❞❡s✈❛♥❡❝✐♠❡♥t♦ ❡♥tr❡ ❛j✲és✐♠❛ ❛♥t❡♥❛ tr❛♥s♠✐ss♦r❛ ❡ ♦

r❡❝❡♣t♦r✳

❆✉♠❡♥t❛♥❞♦ ❛❣♦r❛ ♦ ♥ú♠❡r♦ ❞❡ ❛♥t❡♥❛s r❡❝❡♣t♦r❛s ♣❛r❛nr✱ t❡r❡♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s

❞❡ ❞❡s✈❛♥❡❝✐♠❡♥t♦hij (1≤i≤nr,1≤j ≤nt)✱ ❝❛❞❛ ✉♠ ❝♦rr❡s♣♦♥❞❡♥❞♦ à tr❛♥s♠✐ssã♦

❡♥tr❡ ❛j✲és✐♠❛ ❛♥t❡♥❛ tr❛♥s♠✐ss♦r❛ ❡ ❛i✲és✐♠❛ ❛♥t❡♥❛ r❡❝❡♣t♦r❛❀ t❛♠❜é♠ ❤❛✈❡rá r✉í❞♦ ❝❛♣t❛❞♦ ♣♦r ❝❛❞❛ r❡❝❡♣t♦r✱ q✉❡ s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦r z1, z2, ..., znr ∈ C. P❛r❛ ❝❛❞❛

❛♥t❡♥❛ r❡❝❡♣t♦r❛ i_{∈ {}1,_{· · ·} , nr}, t❡r❡♠♦s ❛ ❡q✉❛çã♦

yi =hi1x1+hi2x2+...+hintxnt +zi✳

✷✵ ❖ ▼♦❞❡❧♦ ❞♦ ❙✐st❡♠❛ ▼■▼❖

P♦❞❡♠♦s✱ ❡♥tã♦✱ ❝♦♥s✐❞❡r❛r ♦ ❝❛♥❛❧ ❝♦♠♦ ✉♠ t♦❞♦ ❡①♣r❡ss❛♥❞♦ ❛s nr ❡q✉❛çõ❡s

❛tr❛✈és ❞❡ ✉♠❛ ❡q✉❛çã♦ ♠❛tr✐❝✐❛❧✿

      y1 y2 ✳✳✳ ynr

      =      

h11 h12 · · · h1nt

h21 h22 · · · h2nt

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ hnr1 hnr2 · · · hnrnt

            x1 x2 ✳✳✳ xnt

      +       z1 z2 ✳✳✳ znr

     

q✉❡ ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛ ❛❜r❡✈✐❛❞❛ ~y =H~x+~z ✳

❖ s✐st❡♠❛ ▼■▼❖ ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ ❝♦♠❜❛t❡r ♦ ❞❡s✈❛♥❡❝✐♠❡♥t♦ ✉s❛♥❞♦ té❝♥✐❝❛s ❞❡ ❞✐✈❡rs✐❞❛❞❡✱ ✐st♦ é✱ ❞✐❢❡r❡♥t❡s ré♣❧✐❝❛s ❞♦ s✐♥❛❧ sã♦ tr❛♥s♠✐t✐❞❛s r❡❞✉♥❞❛♥t❡♠❡♥t❡ ❛tr❛✈és ❞❡ ❝❛♥❛✐s q✉❡ s♦❢r❡♠ ❞❡s✈❛♥❡❝✐♠❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡①✐st✐♥❞♦ ❞❡st❛ ❢♦r♠❛ ✉♠❛ ❛❧t❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ q✉❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❝❛♥❛✐s ♥ã♦ s❡❥❛ ❛❢❡t❛❞♦ ♣❡❧♦ ❞❡s✈❛♥❡✲ ❝✐♠❡♥t♦✳

❱✐s❛♥❞♦ ❢♦r♥❡❝❡r ❞✐✈❡rs✐❞❛❞❡✱ ✈ár✐♦s t✐♣♦s ❞❡ ❡sq✉❡♠❛s tê♠ s✐❞♦ s✉❣❡r✐❞♦s✳ ❯♠ ❞❡❧❡s sã♦ ♦s ❝ó❞✐❣♦s ❞❡ ❜❧♦❝♦ ❡s♣❛ç♦✲t❡♠♣♦ ✭❙❚❇❈✮✱ ♦s q✉❛✐s ❢♦r♥❡❝❡♠ r❡❞✉♥❞â♥❝✐❛ ❡♠ ❡s♣❛ç♦✱ ❛tr❛✈és ❞♦ ✉s♦ ❞❡ ♠ú❧t✐♣❧❛s ❛♥t❡♥❛s✱ ❡ r❡❞✉♥❞â♥❝✐❛ ♥♦ t❡♠♣♦✱ ❛tr❛✈és ❞❛ ❝♦❞✐✜❝❛çã♦ ❞❡ ❝❛♥❛❧✳

❖ ❝r✐tér✐♦ ♣❛r❛ ♠♦❞❡❧❛r ❝ó❞✐❣♦s ❡s♣❛ç♦✲t❡♠♣♦ ❞❡♣❡♥❞❡ ❞♦ t✐♣♦ ❞❡ r❡❝❡♣t♦r q✉❡ é ❝♦♥s✐❞❡r❛❞♦✳ ❉✉❛s ♣r✐♥❝✐♣❛✐s ❝❧❛ss❡s ❞❡ r❡❝❡♣t♦r❡s t❡♠ s✐❞♦ ❝♦♥s✐❞❡r❛❞❛s ♥❛ ❧✐t❡r❛t✉r❛✿ ❝♦❡r❡♥t❡ ❡ ♥ã♦✲❝♦❡r❡♥t❡✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ❝♦♥s✐❞❡r❛❞♦ ♥❡st❡✱ ♦ r❡❝❡♣t♦r r❡❝✉♣❡r❛ ❛ ✐♥❢♦r♠❛çã♦ ❡①❛t❛ s♦❜r❡ ♦ ❡st❛❞♦ ❞♦ ❝❛♥❛❧ ✭✐st♦ é t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♣❡r❢❡✐t♦ ❡st❛❞♦ ❞❡ ✐♥❢♦r♠❛çã♦ ❞♦ ❝❛♥❛❧ ✭❈❙■✮✳ ◆❛ ♣rát✐❝❛ ✐st♦ ♣♦❞❡ s❡r ♦❜t✐❞♦ ✐♥tr♦❞✉③✐♥❞♦ ❛❧❣✉♠ sí♠❜♦❧♦ ❣✉✐❛ q✉❡ ♣❡r♠✐t❡ ✉♠❛ ❡st✐♠❛t✐✈❛ ♣r❡❝✐s❛ ❞♦ ❝❛♥❛❧✱ ❛ss✐♠✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ ❛ ♠❛tr✐③ ❞♦ ❝❛♥❛❧ H é ❝♦♥❤❡❝✐❞❛ ♥♦ r❡❝❡♣t♦r✳ P❛r❛ ♦ ❝❛s♦ ♥ã♦✲❝♦❡r❡♥t❡ ♣♦❞❡♠ s❡r ❝♦♥s✉❧t❛❞❛s ❛s r❡❢❡rê♥❝✐❛s ❬✶✶❪ ❡ ❬✶✷❪✳

❆té ❛❣♦r❛ ❛✉♠❡♥t❛♠♦s ❛ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❡✴♦✉ t❛①❛ ❞❡ tr❛♥s♠✐ssã♦ ❛❞✐❝✐♦♥❛♥❞♦ ❛♥t❡✲ ♥❛s✱ ♣r♦♣♦r❝✐♦♥❛♥❞♦ ♠❛✐s ❝❛♠✐♥❤♦s ♣❛r❛ ❧❡✈❛r ❛ ✐♥❢♦r♠❛çã♦ ❞♦ tr❛♥s♠✐ss♦r ❛♦ r❡❝❡♣t♦r✱ ♣♦❞❡♠♦s ❝❤❛♠❛r ✐ss♦ ❞❡ ❞✐✈❡rs✐❞❛❞❡ ❡s♣❛❝✐❛❧✳ ❖✉tr♦ ❣❛♥❤♦ ✐♠♣♦rt❛♥t❡ ❣❡r❛❞♦ ♣❡❧♦ ✉s♦ ❞❡ ❝ó❞✐❣♦s ❡s♣❛ç♦✲t❡♠♣♦ ♣♦❞❡ s❡r ❝❤❛♠❛❞♦ ❞❡ ❞✐✈❡rs✐❞❛❞❡ t❡♠♣♦r❛❧✿ ❝♦♠♦ ♥✉♠ ❝ó❞✐❣♦ ❞❡ ❜❧♦❝♦ ❝♦♠✉♠✱ ❝♦♥s✐❞❡r❡♠♦s q✉❡ ❝❛❞❛ ❛♥t❡♥❛ tr❛♥s♠✐ss♦r❛ ✭s✐♥❝r♦♥✐③❛❞❛ ❝♦♠ ❛s ♦✉✲ tr❛s✮ ❡♥✈✐❡ ✉♠❛ str✐♥❣ ❞❡T ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ❱❛♠♦s tr❛t❛r ♦ t❡♠♣♦ ❝♦♠♦ ❞✐s❝r❡t♦ ❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ s❡rá tr❛♥s♠✐t✐❞♦ ❞✉r❛♥t❡ ❝❛❞❛ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✳ P♦rt❛♥t♦ ♣❛r❛ ❝❛❞❛ t❡♠♣♦ k ♥♦ ❜❧♦❝♦ _{1, ..., T_}✱ ❛ j✲és✐♠❛ ❛♥t❡♥❛ tr❛♥s♠✐ss♦r❛ (1 _≤ j _≤ nt)

❡♥✈✐❛ ✉♠ sí♠❜♦❧♦ xjk ∈ C❀ ❛♥❛❧♦❣❛♠❡♥t❡✱ ❝❛❞❛ t❡♠♣♦ k✱ ❛ i✲és✐♠❛ ❛♥t❡♥❛ r❡❝❡♣t♦r❛

(1_≤i_≤nr)❝❛♣t❛rá ✉♠ sí♠❜♦❧♦ yik.

P♦❞❡♠♦s ❛ss✉♠✐r q✉❡ ♦ ❝❛♥❛❧ é q✉❛s❡✲❡stát✐❝♦❀ ✐st♦ é✱ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❞❡s✈❛♥❡✲ ❝✐♠❡♥t♦ hj ♣❡r♠❛♥❡❝❡♠ ❝♦♥st❛♥t❡s ❞✉r❛♥t❡ ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ T q✉❡ ✉♠ ❜❧♦❝♦ ❞❡

❖ ♠♦❞❡❧♦ ❞♦ s✐st❡♠❛ ▼■▼❖ ✷✶

❙❡y~k ❞❡♥♦t❛ ♦ ✈❡t♦r (y1k, y2k, ..., ynrk)

T _{❞❡ sí♠❜♦❧♦s r❡❝❡❜✐❞♦s ♥♦ t❡♠♣♦ k ❡ ❛♥❛❧♦✲}

❣❛♠❡♥t❡ x~k = (x1k, x2k, ..., xntk)

T _{❞❡♥♦t❛ ♦ ✈❡t♦r ❞❡ sí♠❜♦❧♦s tr❛♥s♠✐t✐❞♦ ♥♦ t❡♠♣♦ k✱}

♣♦❞❡♠♦s ❡s❝r❡✈❡r

~

yk =H ~xk+z~k✱

♦♥❞❡ z~k é ♦ ✈❡t♦r ❞❡ r✉í❞♦ ♥♦ t❡♠♣♦ k✳ P♦❞❡♠♦s ❡♥tã♦ r❡♣r❡s❡♥t❛r ❡ss❛s T ❡q✉❛çõ❡s

♥✉♠❛ ú♥✐❝❛ ❡q✉❛çã♦ ♠❛tr✐❝✐❛❧

(y~1y~2 · · · y~T) =H(x~1x~2 · · · x~T) + (z~1z~2 · · · z~T);

q✉❡ ♣♦❞❡ s❡r ❛❜r❡✈✐❛❞❛ ♣❛r❛

Y =H·X+Z✳

▲❡♠❜r❛♥❞♦ q✉❡ Y é ✉♠❛ ♠❛tr✐③ nr× T ♥❛ q✉❛❧ yik r❡♣r❡s❡♥t❛ ♦ sí♠❜♦❧♦ r❡❝❡❜✐❞♦

♣❡❧❛ ❛♥t❡♥❛ i ♥♦ t❡♠♣♦ k❀ ❛ ♠❛tr✐③ ❞♦ ❝❛♥❛❧ H t❡♠ ❞✐♠❡♥sã♦ nr×nt ❝♦♠♦ ❛♥t❡s❀ X

é ✉♠❛ ♠❛tr✐③ nt×T ❡♠ q✉❡ xjk r❡♣r❡s❡♥t❛ ♦ sí♠❜♦❧♦ tr❛♥s♠✐t✐❞♦ ♣❡❧❛ ❛♥t❡♥❛ j ♥♦

t❡♠♣♦ k❀ ❡ ❛ ♠❛tr✐③ ❞❡ r✉í❞♦ Z t❡♠ ❞✐♠❡♥sã♦ nr ×T✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❢♦❝❛r❡♠♦s ♦

❝❛s♦ ♦♥❞❡ nt=nr =T =n ❡ ❡♥tã♦ ♣♦❞❡♠♦s ❝♦❞✐✜❝❛r n2 sí♠❜♦❧♦s ❞❡ ✐♥❢♦r♠❛çã♦✳

❋✐❣✉r❛ ✷✳✶✿ ❊①❡♠♣❧♦ ❞❡ ❈❛♥❛❧ ▼■▼❖2×2

X :♠❛tr✐③ ❞❡ s✐♥❛❧ tr❛♥s♠✐t✐❞♦ nt×T

Y :♠❛tr✐③ ❞❡ s✐♥❛❧ r❡❝❡❜✐❞♦ nr×T

H :♠❛tr✐③ ❞❡ ❝❛♥❛❧ nr×nt

Z : ♠❛tr✐③ r✉í❞♦ ●❛✉ss✐❛♥❛ ❝♦♠♣❧❡①❛ nr×T

Y =HX +Z

❉❡✜♥✐çã♦ ✷✳✶✳ ❯♠ ST BC é ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ C ❞❡ nt×T ♠❛tr✐③❡s ❝♦♠♣❧❡①❛s X ❡

é ❧✐♥❡❛r s❡

✷✷ ❖ ▼♦❞❡❧♦ ❞♦ ❙✐st❡♠❛ ▼■▼❖

✷✳✷ ❈r✐tér✐♦s ♣❛r❛ ♠♦❞❡❧❛r ❝ó❞✐❣♦s ❡s♣❛ç♦✲t❡♠♣♦

❙♦❜ ❛ s✉♣♦s✐çã♦ ❞❡ ♣❡r❢❡✐t♦ ❈❙■✱ ❛ ❞❡❝♦❞✐✜❝❛çã♦ ♣♦r ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭▼▲✮ ❝♦rr❡s♣♦♥❞❡ ❛ ❡s❝♦❧❤❡r ✉♠❛ ♣❛❧❛✈r❛✲❝ó❞✐❣♦ X q✉❡ ♠✐♥✐♠✐③❛✿

min

X∈C||Y −HX|| 2_.

❯♠❛ ❡st✐♠❛t✐✈❛ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡rr♦ ♣♦❞❡ s❡r ♦❜t✐❞❛ ✉s❛♥❞♦ ❛ ✉♥✐ã♦ ❧✐♠✐t❛❞❛

P(e)_≤ 1

|C| X

x∈C

X

ˆ

X6=X

P(X_−→X)ˆ ✭✷✳✶✮

♦♥❞❡ P(X −→X)ˆ é ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡rr♦ ♣♦♥t♦ ❛ ♣♦♥t♦✱ ✐st♦ é✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ q✉❡✱ q✉❛♥❞♦ ✉♠❛ ♣❛❧❛✈r❛✲❝ó❞✐❣♦ X é tr❛♥s♠✐t✐❞❛✱ ♦ r❡❝❡♣t♦r ▼▲ ❞❡❝✐❞❡ ❡rr♦♥❡❛♠❡♥t❡ ❡♠ ❢❛✈♦r ❞❡ ♦✉tr❛ ♣❛❧❛✈r❛✲❝ó❞✐❣♦ _X,ˆ _{❛ss✉♠✐♥❞♦ q✉❡ s♦♠❡♥t❡ X ❡X}ˆ _{❡stã♦ ♥♦ ❝ó❞✐❣♦✳}

◆♦ ❝❛s♦ ❞❡ ❞❡s✈❛♥❡❝✐♠❡♥t♦ ❘❛②❧❡✐❣❤ ✐♥❞❡♣❡♥❞❡♥t❡ (hij ∼Nc(0,1)),♣♦❞❡♠♦s ❡s❝r❡✲

✈❡r

P(X _−→X)ˆ _≤det

"

Int+

(X₋X)(Xˆ ₋X)ˆ †

4N0

#−nr

, ✭✷✳✷✮

♦♥❞❡ † ❞❡♥♦t❛ ❛ ♠❛tr✐③ tr❛♥s♣♦st❛ ❝♦♥❥✉❣❛❞❛✳

❉❡♥♦t❛♥❞♦ ♣♦r r ♦ ♣♦st♦ ❞❛ ♠❛tr✐③ ❞✐❢❡r❡♥ç❛ ❞❛ ♣❛❧❛✈r❛✲❝ó❞✐❣♦✱ s❡ r = nt ♣❛r❛

t♦❞♦s ♦s ♣❛r❡s (X,X),ˆ ❞✐③❡♠♦s q✉❡ ♦ ❝ó❞✐❣♦ t❡♠ ♣♦st♦ t♦t❛❧✳ ❙❡ ❞❡♥♦t❛r♠♦s ♣♦r λj, j = 1,· · · , r ♦s ❛✉t♦✈❛❧♦r❡s ♥ã♦ ♥✉❧♦s ❞❛ ♠❛tr✐③ ❞✐stâ♥❝✐❛ ❞❛ ♣❛❧❛✈r❛✲❝ó❞✐❣♦

A = (X₋X)(Xˆ ₋X)ˆ † _{✭✷✳✸✮}

♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ✭✷✳✸✮ ❝♦♠♦

P(X _−→X)ˆ _≤

r

Y

j=1

1 + λj 4N0

−nr

. ✭✷✳✹✮

P❛r❛ ❛❧t❛ r❡❧❛çã♦ s✐♥❛❧✲r✉í❞♦ ✭N0 ♣❡q✉❡♥♦✮✱ t❡♠♦s

P(X _−→X)ˆ _≤δ−nr

1 4N0

−rnr

, ✭✷✳✺✮

♦♥❞❡ δ=

r

Y

j=1

λj.❊♥tã♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

P(X _−→X)ˆ _≤

δ1/r

4N0

−rnr

. ✭✷✳✻✮

◆♦ ❝❛s♦ ❞❡ ❝ó❞✐❣♦s ❞❡ ♣♦st♦ ♠á①✐♠♦ (r = nt)✱ t❡♠♦s δ = det(A) = nt

Y

j=1

λj 6= 0

❈r✐tér✐♦s ♣❛r❛ ♠♦❞❡❧❛r ❝ó❞✐❣♦s ❡s♣❛ç♦✲t❡♠♣♦ ✷✸

◆♦ ❝❛s♦ ❞❡ ❝ó❞✐❣♦s ❝♦♠ ❞✐✈❡rs✐❞❛❞❡ t♦t❛❧✱ ♦ t❡r♠♦ ❞♦♠✐♥❛♥t❡ ♥❛ ✉♥✐ã♦ ❧✐♠✐t❛❞❛ ✭✷✳✶✮ é ❞❛❞♦ ♣❡❧♦ ❡♥tã♦ ❝❤❛♠❛❞♦ ❞❡t❡r♠✐♥❛♥t❡ ♠í♥✐♠♦ ❞♦ ❝ó❞✐❣♦✱

δmin = min X6= ˆX

det(A). ✭✷✳✼✮

❖ t❡r♠♦ (δmin)1/nt é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❣❛♥❤♦ ❞❡ ❝♦❞✐✜❝❛çã♦ ❬✶✸❪✳

◆♦ ❝❛s♦ ❞❡ ❝ó❞✐❣♦s ❧✐♥❡❛r❡s ❛ s♦♠❛ ♦✉ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ q✉❛❧q✉❡r ♣❛r ❞❡ ♣❛❧❛✈r❛s✲❝ó❞✐❣♦ é ✉♠❛ ♣❛❧❛✈r❛✲❝ó❞✐❣♦✱ ♣♦rt❛♥t♦ ❛ ✉♥✐ã♦ ❧✐♠✐t❛❞❛ r❡❞✉③ ❛

P(e)≤ 1 |C|

X

X6=0

P(0−→X) ✭✷✳✽✮

❡ t❡♠♦s

δmin = min X6=0_nt×T

det(XX†_{). ✭✷✳✾✮}

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣❛r❛ ♠✐♥✐♠✐③❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡rr♦ é ♥❡❝❡ssár✐♦ ❝♦♥s✐❞❡r❛r ❞♦✐s ❝r✐tér✐♦s✳

✶✳ ❖ ❝r✐tér✐♦ ❞♦ ♣♦st♦✿ ❞❡ ❢❛t♦✱ ♣❛r❛ ❛t✐♥❣✐r ❛ ❞✐✈❡rs✐❞❛❞❡ ♠á①✐♠❛ntnr,❛ ♠❛tr✐③

(X−X)ˆ ❞❡✈❡ t❡r ♣♦st♦ t♦t❛❧ ♣❛r❛ q✉❛❧q✉❡r ♣❛r ❞❡ ♣❛❧❛✈r❛s✲❝ó❞✐❣♦X❡X.ˆ ❈ó❞✐❣♦s q✉❡ ❛t✐♥❣❡♠ ❛ ❞✐✈❡rs✐❞❛❞❡ ♠á①✐♠❛ sã♦ ❝❤❛♠❛❞♦s t♦t❛❧♠❡♥t❡ ❞✐✈❡rs♦s✳

✷✳ ❖ ❝r✐tér✐♦ ❞♦ ❞❡t❡r♠✐♥❛♥t❡✿ s❡ ❛ ❞✐✈❡rs✐❞❛❞❡ ❞❡ntnré ❛t✐♥❣✐❞❛✱ ❡♥tã♦ ♦ ❞❡t❡r✲

♠✐♥❛♥t❡ ♠í♥✐♠♦ ❞❡ A t♦♠❛❞♦ s♦❜r❡ t♦❞♦s ♦s ♣❛r❡s ❞❡ ♣❛❧❛✈r❛s✲❝ó❞✐❣♦ ❞✐st✐♥t❛s ❞❡✈❡ s❡r ♠❛①✐♠✐③❛❞♦✳

❆❧é♠ ❞✐ss♦✱ ♣❛r❛ ♠❡❧❤♦r❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞❡ ❝ó❞✐❣♦s ❡s♣❛ç♦✲t❡♠♣♦✱ ❡①✐❣✐♠♦s q✉❡ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s s❡❥❛♠ s❛t✐s❢❡✐t❛s✿

• ❉❡t❡r♠✐♥❛♥t❡ ♥ã♦✲♥✉❧♦✳ ❉✐③❡♠♦s q✉❡ ♦ ❝ó❞✐❣♦ t❡♠ ✉♠ ❞❡t❡r♠✐♥❛♥t❡ ♥ã♦✲♥✉❧♦

s❡✱ ❛♥t❡s ❞❛ ♥♦r♠❛❧✐③❛çã♦ ❙◆❘✱ ❡①✐st❡ ✉♠ ❧✐♠✐t❛♥t❡ ✐♥❢❡r✐♦r s♦❜r❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ♠í♥✐♠♦ q✉❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ t❛♠❛♥❤♦ ❞❛ ❝♦♥st❡❧❛çã♦✳

• ❋♦r♠❛✳ P❛❧❛✈r❛s✲❝ó❞✐❣♦ ❡ s✐♥❛✐s ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ♣♦r ♠❡✐♦ ❞❡ ❡sq✉❡♠❛s

✷✹ ❖ ▼♦❞❡❧♦ ❞♦ ❙✐st❡♠❛ ▼■▼❖

❝♦♠♦M v, ♦♥❞❡v é ♦ ✈❡t♦r ❝♦♥t❡♥❞♦ ♦s sí♠❜♦❧♦s ❞❡ ✐♥❢♦r♠❛çã♦ QAM ♦✉HEX, ❡♥q✉❛♥t♦ M é ✉♠❛ ♠❛tr✐③ q✉❡ ❞❡❝♦❞✐✜❝❛ ♦s sí♠❜♦❧♦s ❞❡♥tr♦ ❞❡ ❝❛❞❛ ❝❛♠❛❞❛✳ ❉❡ ♠♦❞♦ ❛ ♦❜t❡r ❝ó❞✐❣♦s ❝♦♠ ❡♥❡r❣✐❛ ❡✜❝✐❡♥t❡✱ ❡①✐❣✐♠♦s q✉❡ ❛ ♠❛tr✐③ M s❡❥❛ ✉♥✐tár✐❛✳ ❘❡❢❡r✐♠♦s ❛ ❡st❡ t✐♣♦ ❞❡ ❢♦r♠❛ ❞❡ ❝♦♥st❡❧❛çã♦ ❝♦♠♦ ❢♦r♠❛ ❝ú❜✐❝❛✱ ♣♦✐s ✉♠❛ ♠❛tr✐③ ✉♥✐tár✐❛ ❛♣❧✐❝❛❞❛ s♦❜r❡ ✉♠ ✈❡t♦r ❝♦♥t❡♥❞♦ ✈❛❧♦r❡s ❞✐s❝r❡t♦s ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ♣♦♥t♦s ❣❡r❛❞♦r❡s ❞♦ r❡t✐❝✉❧❛❞♦✳ P♦r ❡①❡♠♣❧♦✱ s❡ ✉s❛r♠♦s sí♠❜♦❧♦sQAM, ♦❜t❡♠♦s ♦ r❡t✐❝✉❧❛❞♦ ✭❝ú❜✐❝♦✮ Zn.

• ❯♥✐❢♦r♠✐❞❛❞❡ ♠é❞✐❛ ❞❛ ❡♥❡r❣✐❛ tr❛♥s♠✐t✐❞❛ ♣♦r ❛♥t❡♥❛✳ ❆i✲és✐♠❛ ❛♥t❡♥❛ ❞♦ s✐st❡♠❛ tr❛♥s♠✐t✐rá ✉♠ s✐♥❛❧ xik ♣❛r❛ ♦ t❡♠♣♦k. ◆ós ❡①✐❣✐♠♦s q✉❡ ❛ ❡♥❡r❣✐❛

❞❡ ❝❛❞❛ ❡♥tr❛❞❛ ❞❛ ♣❛❧❛✈r❛✲❝ó❞✐❣♦ s❡❥❛ ❝♦♥st❛♥t❡✱ ❛ss✐♠✱ t❡♠♦s ✉♠❛ r❡♣❛rt✐çã♦ ❡q✉✐❧✐❜r❛❞❛ ❞❛ ❡♥❡r❣✐❛ ♣❛r❛ ♦ tr❛♥s♠✐ss♦r✳

✷✳✸ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❛❧❣é❜r✐❝♦s

❉❡✜♥✐çã♦ ✷✳✷✳ ❙❡❥❛♠ L ❡ K ❞♦✐s ❝♦r♣♦s✳ ❉✐③❡♠♦s q✉❡ L é ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❝♦r♣♦s s❡ K _⊆L✱ ♦✉ s❡❥❛✱ K é s✉❜❝♦r♣♦ ❞❡ L✳

◆♦t❛çã♦✿ L/K.

❉❡✜♥✐çã♦ ✷✳✸✳ ❙❡❥❛V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡K✳ ❙❡V ❛❞♠✐t❡ ✉♠❛ ❜❛s❡ ✜♥✐t❛ ❡♥tã♦ ❝❤❛♠❛♠♦s ❞❡ ❞✐♠❡♥sã♦ ❞❡ V ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ t❛❧ ❜❛s❡✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❞✐③❡♠♦s q✉❡ ❛ ❞✐♠❡♥sã♦ ❞❡ ❱ é ✐♥✜♥✐t❛✳

❉❡✜♥✐çã♦ ✷✳✹✳ ❙❡❥❛ L/K ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❝♦r♣♦s✳ ❆ ❞✐♠❡♥sã♦ ❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ L s♦❜r❡ K é ❝❤❛♠❛❞❛ ❣r❛✉ ❞❛ ❡①t❡♥sã♦ ❞❡ L s♦❜r❡ K ❡ é ❞❡♥♦t❛❞❛ ♣♦r [L:K]. ❙❡ [L:K] é ✜♥✐t♦✱ ❞✐③❡♠♦s q✉❡ ❛ ❡①t❡♥sã♦ L/K é ✉♠❛ ❡①t❡♥sã♦ ✜♥✐t❛✳

◆♦t❛çã♦✿ [L:K]<_∞.

❉❡✜♥✐çã♦ ✷✳✺✳ ❯♠❛ ❡①t❡♥sã♦ ✜♥✐t❛ s♦❜r❡ ♦ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ Q, é ❝❤❛✲ ♠❛❞❛ ❞❡ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s✳

❖❜s❡r✈❛çã♦ ✷✳✶✳ ❖s ❝♦r♣♦s ❞❡ ♥ú♠❡r♦s Q(i) = _{a+bi_|a, b_∈Q_} ❡ Q(j) = _{a+bj_|a, b_∈Q_}, ♦♥❞❡ j é ❛ r❛✐③ t❡r❝❡✐r❛ ♣r✐♠✐t✐✈❛ ❞❛ ✉♥✐❞❛❞❡ ✱ ✐st♦ é✱ j3 _{= 1 ❡ j}n _{6= 1,1 ≤ n ≤ 2✮ sã♦ ❞❡ ♣❛rt✐❝✉❧❛r ✐♥t❡r❡ss❡✳ ❉❡ ❢❛t♦✱ r❡str✐♥❣✐♥❞♦ a ❡ b}

❡♠ Z ♣♦❞❡♠♦s ♦❜t❡r ♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s Z[i] = _{a+bi _|a, b_∈Z_} ❡ ♦

❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ❞❡ ❊✐s❡✐♥st❡✐♥ Z[j] =_{a+bj_|a, b_∈Z_}✳

❚❡♦r❡♠❛ ✷✳✶✳ ✭❚❡♦r❡♠❛ ❞❛ ♠✉❧t✐♣❧✐❝❛t✐✈✐❞❛❞❡ ❞♦s ❣r❛✉s✮

❙❡❥❛♠ K✱ M ❡ L ❝♦r♣♦s t❛✐s q✉❡ K _⊆ M _⊆ L ❡ [L:K] < _∞✳ ❊♥tã♦✱ [L:K] = [L:M]_·[M :K]

❙✉♣♦♥❤❛ [L:M] = m ❡ [M : K] = n✳ ❙❡❥❛♠ B1 = {α1, . . . , αm} ✉♠❛ ❜❛s❡ ❞❡ L

s♦❜r❡ M ❡B2 ={β1, . . . , βn} ✉♠❛ ❜❛s❡ ❞❡M s♦❜r❡ K✳

❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❛❧❣é❜r✐❝♦s ✷✺

✶✳ [B] =L✳

❉❡ ❢❛t♦✿ α∈L⇒ ∃a1, a2,· · · , am ∈M t❛✐s q✉❡ α= m

X

i=1 aiαi ✳

ai ∈M ⇒ ∃bi1, bi2,· · ·, bin ∈Kt❛✐s q✉❡ai = n

X

j=1

bijβj✳ ▲♦❣♦✱α= m

X

i=1

n

X

j=1

bijαiβj✳

✷✳ B é ▲✳■✳s♦❜r❡ K✳ ❉❡ ❢❛t♦✱

m

X

i=1

n

X

j=1

bijαiβj = 0⇒ m

X

i=1

n

X

j=1 bijβj

!

αi❂✵✳

❈♦♠♦ B1 é ▲✳■✳ ✱ t❡♠♦s

n

X

j=1

bijβj = 0✳ ❈♦♠♦ B2 é ▲✳■✳ ✱ t❡♠♦s bij = 0,∀i, j.

P♦rt❛♥t♦ B é ✉♠❛ ❜❛s❡ ❞❡ L/K ❝♦♠ mn ❡❧❡♠❡♥t♦s✳ P❡❧❛ ❉❡✜♥✐çã♦ ✭✷✳✸✮✱ [L:K] =mn= [L:M].[M :K].

❊①❡♠♣❧♦ ✷✳✶✳ ◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r Q √2,√3:Q✳

❆✜r♠❛çã♦ ■ ✿ {1,√2} é ✉♠❛ ❜❛s❡ ❞❡ Q(√2)s♦❜r❡ Q✳

❉❡ ❢❛t♦✱

✶✳ {1,√2} ❣❡r❛ Q(√2) s♦❜r❡Q✳

❈♦♠♦ Q(√2) = _{x _| x = α+β√2, α, β _∈ Q_}✱ ❡♥tã♦ _∀x _∈ Q(√2), x = α.1 + β√2, α, β _∈Q✱ ♦✉ s❡❥❛✱x é ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ _{1,√2_}✳

✷✳ {1,√2_} é ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡✳

❙✉♣♦♥❤❛ q✉❡ α.1 +β√2 = 0, α, β,_∈ Q✳ ❙❡ β ₆= 0✱ ❡♥tã♦ √2 = −α

β ✱ ♦ q✉❡ é ❛❜s✉r❞♦ ♣♦✐s√2é ✐rr❛❝✐♦♥❛❧✳ P♦rt❛♥t♦✱β = 0 ❡ ❞❛í t❡♠♦s q✉❡α.1 + 0.√2 = 0⇒

α= 0✳ ❉❡ss❛ ❢♦r♠❛✱{1,√2}é ✉♠❛ ❜❛s❡ ❞❡ Q(√2) s♦❜r❡ Q✳

▲♦❣♦ [Q(√2) :Q] = 2✳

❆✜r♠❛çã♦ ■■ ✿ {1,√3_} é ✉♠❛ ❜❛s❡ ❞❡ Q(√3)s♦❜r❡ Q(√2) ✶✳ ❉❡ ❢❛t♦✱ {1,√3} ❣❡r❛ Q(√2,√3) s♦❜r❡ Q(√2)✳

❈♦♠♦Q(√3,√2) =_{x_|x=α+β√3, α, β_∈Q(√2)_}✱ ❡♥tã♦_∀x_∈Q(√3,√2), x= α.1 +β√3, α, β _∈Q(√2)✱ ♦✉ s❡❥❛✱x é ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ _{1,√3_}✳

✷✳ {1,√3_} é ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡✳

❙✉♣♦♥❤❛ q✉❡ α.1 +β√3 = 0, α, β,∈ Q(√2)✳ ❆ss✐♠ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ✐❣✉❛❧✲ ❞❛❞❡ ❝♦♠♦

✷✻ ❖ ▼♦❞❡❧♦ ❞♦ ❙✐st❡♠❛ ▼■▼❖

❙❡ β = (r +s√2) ₆= 0✱ ❡♥tã♦ √3 = −(p+q

√

2) r+s√2 =

−(p+q√2) r+s√2 .

r₋s√2 r−s√2 =

−(pr−ps√2 +qr√2−2qs) r2 _−2s =

−pr+ 2qs+ (ps−qr)√2 r2_−2s =

−pr+ 2qs r2_−2s +

(ps−qr)√2 r2_−2s = a+b√2, a, b_∈Q✳ ❆ss✐♠✱ t❡♠♦s q✉❡

(√3)2 = (a+b√2)2 3 = a2+ 2ab√2 + 2b2 3−a2−2b2 = 2ab√2

3−a2 _−2b2 2ab =

√

2,

✭✷✳✶✵✮ ♦ q✉❡ é ❛❜s✉r❞♦ ♣♦✐s √2 é ✐rr❛❝✐♦♥❛❧✳

P♦rt❛♥t♦✱r+s√2 = β = 0❡ ❞❛í t❡♠♦s q✉❡p+q√2+0.√3 = 0_⇒p+q√2 =α = 0✳ ❉❡ss❛ ❢♦r♠❛✱ {1,√3_} é ✉♠❛ ❜❛s❡ ❞❡ Q(√3)s♦❜r❡ Q(√2)✳

▲♦❣♦ [Q(√2,√3) : Q(√2)] = 2✳

P❡❧♦ ❚❡♦r❡♠❛ ❞❛ ♠✉❧t✐♣❧✐❝❛t✐✈✐❞❛❞❡ ❞♦s ❣r❛✉s✱

[Q(√2,√3) :Q] = [Q(√2,√3) :Q(√2)].[Q(√2) :Q] = 2.2 = 4 ❡ {1,√2,√3,√6}é ✉♠❛ ❜❛s❡ ❞❡ Q(√2,√3)s♦❜r❡ Q✳

❉❡✜♥✐çã♦ ✷✳✻✳ ❙❡❥❛L/K✉♠❛ ❡①t❡♥sã♦ ❞❡ ❝♦r♣♦s ❡α∈L✳ ❉✐③❡♠♦s q✉❡αé ❛❧❣é❜r✐❝♦ s♦❜r❡ K s❡ ❡①✐st❡ f ∈ K[x]∗ _{= K[x]\0 ✭K[x]✿ ❝♦♥❥✉♥t♦ ❞♦s ♣♦❧✐♥ô♠✐♦s s♦❜r❡ K ♥❛}

✐♥❞❡t❡r♠✐♥❛❞❛ ①✮ t❛❧ q✉❡ f(α) = 0✳ ❈❛s♦ ♥ã♦ ❡①✐st❛ f ❝♦♠ t❛✐s ❝♦♥❞✐çõ❡s✱ ❞✐③❡♠♦s q✉❡ α é tr❛♥s❝❡♥❞❡♥t❡✳

❖❜s❡r✈❛çã♦ ✷✳✷✳ ❙❡K =Q✱ ❞✐③❡♠♦s s✐♠♣❧❡s♠❡♥t❡ q✉❡ α é ❛❧❣é❜r✐❝♦ ♦✉ tr❛♥s❝❡♥✲ ❞❡♥t❡✳

❊①❡♠♣❧♦ ✷✳✷✳ ➱ ❢á❝✐❧ ❞❛r ❡①❡♠♣❧♦s ❞❡ ♥ú♠❡r♦s ❛❧❣é❜r✐❝♦s s♦❜r❡ Q ✿

• t♦❞♦s ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s • √2

• √2 +√5

• i✳

◆♦ ❡♥t❛♥t♦✱ é ❞✐❢í❝✐❧ ❞❛r ❡①❡♠♣❧♦s ❞❡ ♥ú♠❡r♦s tr❛♥s❝❡♥❞❡♥t❡s✳ ❊♠ ✶✽✼✸✱ ❍❡r♠✐t❡ ❞❡♠♦♥str♦✉ q✉❡ ♦ ♥ú♠❡r♦

e= lim

n→∞

1 + 1 n+ 1

❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❛❧❣é❜r✐❝♦s ✷✼

é tr❛♥s❝❡♥❞❡♥t❡ ❡ ❡♠ ✶✽✽✷ ▲✐♥❞❡♠❛♥♥ ❞❡♠♦♥str♦✉ q✉❡ π é tr❛♥s❝❡♥❞❡♥t❡ ❬✶✹❪✳

❉❡✜♥✐çã♦ ✷✳✼✳ ❯♠❛ ❡①t❡♥sã♦ L s♦❜r❡ K é ❛❧❣é❜r✐❝❛ s❡ t♦❞♦ α ∈L é ❛❧❣é❜r✐❝♦ s♦❜r❡ K✳

❊①❡♠♣❧♦ ✷✳✸✳ ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡ ❛ ❡①t❡♥sã♦ Q(√2) s♦❜r❡ Q é ❛❧❣é❜r✐❝❛✳

α∈Q(√2)⇒α=a+b√2, a, b∈Q✳

x=a+b√2⇒(x−a)2 _{= 2b ⇒x}2_−2ax+a2_{−2b = 0.}

▲♦❣♦✱ α é r❛✐③ ❞❡ x2_−2ax+a2_{−2b ∈Q[x] ❡✱ ❞❡ss❛ ❢♦r♠❛✱ α é ❛❧❣é❜r✐❝♦ s♦❜r❡Q} ❡✱ ♣♦rt❛♥t♦✱ ❛ ❡①t❡♥sã♦ Q(√2)/Q é ❛❧❣é❜r✐❝❛✳

❉❡✜♥✐çã♦ ✷✳✽✳ ❉✐③❡♠♦s q✉❡ α é ✐♥t❡✐r♦ ❛❧❣é❜r✐❝♦ s❡ ❡①✐st❡ f(x)∈Z[x]∗ _=Z[x]\0✱

♠ô♥✐❝♦✱ t❛❧ q✉❡ f(α) = 0✳ ❖ ❝♦♥❥✉♥t♦ OK = {α ∈ K | α é ✐♥t❡✐r♦ ❛❧❣é❜r✐❝♦ } é ✉♠

❛♥❡❧ ❝❤❛♠❛❞♦ ❛♥❡❧ ❞♦s ✐♥t❡✐r♦s ❞❡ ❑✳

❊①❡♠♣❧♦ ✷✳✹✳ ❖ ❡❧❡♠❡♥t♦ α = √2 +√3 é ✐♥t❡✐r♦ s♦❜r❡ Z, ♣♦✐s é r❛✐③ ❞♦ s❡❣✉✐♥t❡ ♣♦❧✐♥ô♠✐♦ x4_−10x2_{+ 1 ∈Z[x].}

❉❡✜♥✐çã♦ ✷✳✾✳ ❙❡❥❛ L/K ❡①t❡♥sã♦ ❞❡ ❝♦r♣♦s ❞❡ α ❛❧❣é❜r✐❝♦ s♦❜r❡ K✳ ❖ ♣♦❧✐♥ô♠✐♦ p(x) ∈ K[x]∗_{✱ ♠ô♥✐❝♦ ❡ ❞❡ ♠❡♥♦r ❣r❛✉ t❛❧ q✉❡ p(α) = 0 é ❝❤❛♠❛❞♦ ♣♦❧✐♥ô♠✐♦}

♠✐♥✐♠❛❧ ❞❡ α s♦❜r❡ K✳ ◆♦t❛çã♦ ✿ p=irrα |K✳

❉❡✜♥✐çã♦ ✷✳✶✵✳ ❙❡❥❛ f(x)_∈K[x]t❛❧ q✉❡∂f _≥1✳ ❉✐③❡♠♦s q✉❡f(x)é ✉♠ ♣♦❧✐♥ô♠✐♦ ✐rr❡❞✉tí✈❡❧ s♦❜r❡ K s❡ t♦❞❛ ✈❡③ q✉❡ f(x) =g(x).h(x), f(x), g(x)_∈K[x]✱ ❡♥tã♦f(x) = a ♦✉ h(x) = b✱ a, b _∈ K✱ a, b ❝♦♥st❛♥t❡s✳ ❙❡ f(x) ♥ã♦ é ✐rr❡❞✉tí✈❡❧✱ ❞✐③❡♠♦s q✉❡ é r❡❞✉tí✈❡❧ s♦❜r❡ K✳

❚❡♦r❡♠❛ ✷✳✷✳ ✭❈r✐tér✐♦ ❞❡ ❊✐s❡♥st❡✐♥✮

❙❡❥❛ f(x) =anxn+· · ·+a1x+a0 ∈Z[x]✳ ❙❡ ❡①✐st✐r p∈Z ✱ p ♣r✐♠♦✱ t❛❧ q✉❡✿

• p_|a0, ..., an−1

• p∤an

• p2 _∤a 0

❊♥tã♦ f(x) é ✐rr❡❞✉tí✈❡❧ ❡♠Z[x]✳

▲❡♠❛ ✷✳✶✳ ✭▲❡♠❛ ❞❡ ●❛✉ss ❬✶✺❪✱ ♣✳ ✹✹✮ ❙❡❥❛ f(x) _∈ Z[x] t❛❧ q✉❡ f(x) é ✐rr❡❞✉tí✈❡❧ s♦❜r❡ Z ✳ ❊♥tã♦ f(x) é ✐rr❡❞✉tí✈❡❧ s♦❜r❡ Q✳

❖❜s❡r✈❛çã♦ ✷✳✸✳ ❙❡❥❛ p(x)_∈K[x]∗ _{❡α∈K t❛❧ q✉❡p(x)é ♦ ♣♦❧✐♥ô♠✐♦ ♠✐♥✐♠❛❧ ❞❡α}

s♦❜r❡ K ✳ ❊♥tã♦ p(x) é ✐rr❡❞✉tí✈❡❧ s♦❜r❡K✳

❉❡ ❢❛t♦✿ ❙✉♣♦♥❤❛ p(x) = f(x).g(x)✱ f(x), g(x) _∈ K[x]∗ _{❡ ∂f(x) ≥ 1 ❡ ∂g(x) ≥ 1}

✷✽ ❖ ▼♦❞❡❧♦ ❞♦ ❙✐st❡♠❛ ▼■▼❖

❊①❡♠♣❧♦ ✷✳✺✳ ❙❡❥❛ α=√2 +√3 ❡K =Q✳

α =√2+√3⇒(α−√2)2 _{= (}√₃₎2 _⇒α2_−2α√_{2+2 = 3⇒(α}2₋₁₎2 _{= (2α}√₂₎2 _⇒ (α2₋₁₎2 _{= 8α}2 _⇒α4_−10α2_{+ 1 = 0.❆ss✐♠✱ p(x) = x}4_−10x2_{+ 1 = 0}

❱❛♠♦s ✈❡r✐✜❝❛r s❡ p(x) é ✐rr❡❞✉tí✈❡❧ s♦❜r❡ Z✳

❙✉♣♦♥❤❛ q✉❡ ❤❛❥❛ ❢❛t♦r ❞❡ ❣r❛✉ ✶✱(x−a),t❛❧ q✉❡p= (x−a)(a3x3+a2x2+a1x+a0)✱ a3, a2, a1, a0 ∈Z✳

❆♣❧✐❝❛♥❞♦ ❛ ❞✐str✐❜✉t✐✈❛✱ t❡♠♦s q✉❡ ♦ t❡r♠♦ ❝♦♥st❛♥t❡a.a0 = 1✳ ❈♦♠♦a ∈Z❡♥tã♦ a =±1✳ ▲♦❣♦ ±1 s❡r✐❛ r❛✐③ ❞❡ p(x)✱ ♠❛s p(±1)6= 0✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ ♥ã♦ ❤á ❢❛t♦r

❞❡ ❣r❛✉ ✶✱ ❡ ♣♦r ❝♦♥s❡q✉ê♥❝✐❛ ♥❡♠ ❞❡ ❣r❛✉ ✸✱ ♣❛r❛ p(x)✳

P♦r ♦✉tr♦ ❧❛❞♦✱ p(x) = (x2 _{−(−5 + 2}√_6))(x2_{+ (5 + 2}√_{6)) ❡ t❡♠♦s q✉❡ ❝❛❞❛ ❢❛t♦r} ♥ã♦ t❡♠ r❛✐③ ❡♠ Z❡✱ ❛ss✐♠✱ p(x)♥ã♦ ♣♦ss✉✐ ❢❛t♦r❡s ❞❡ ❣r❛✉ ✷ ❝♦♠ r❛í③❡s ❡♠ Z✳

❉❡ss❛ ❢♦r♠❛✱ p(x) é ✐rr❡❞✉tí✈❡❧ s♦❜r❡ Z ❡✱ ♣❡❧♦ ▲❡♠❛ ❞❡ ●❛✉ss✱ é ✐rr❡❞✉tí✈❡❧ s♦❜r❡ Q✳

P♦rt❛♥t♦✱ p=irrα |Q✳

❊①❡♠♣❧♦ ✷✳✻✳ ❙❡❥❛ α=i∈C ❡K =R. α =i⇒α2 _=−1⇒α2_{+ 1 = 0.}

❆ss✐♠✱ s❡❥❛ p(x) =x2_{+ 1✳ ❱❛♠♦s ✈❡r✐✜❝❛r s❡ p(x)é ✐rr❡❞✉tí✈❡❧ s♦❜r❡ R✳}

❈♦♠♦ p(x) = (x+i)(x−i)✱ t❡♠♦s q✉❡ ❝❛❞❛ ❢❛t♦r ♥ã♦ t❡♠ r❛✐③ ❡♠ R❡ ❛ss✐♠✱ p(x) é ✐rr❡❞✉tí✈❡❧ s♦❜r❡R✳ P♦rt❛♥t♦✱ p(x) =irrα |R✳

❚❡♦r❡♠❛ ✷✳✸✳ ❙❡❥❛ L/K ❡①t❡♥sã♦ ❞❡ ❝♦r♣♦s ❡ α _∈ L ❛❧❣é❜r✐❝♦ s♦❜r❡ K✱ ❝♦♠ p(x) = irrα | K t❛❧ q✉❡ ∂p = n✳ ❊♥tã♦ {1, a, ..., an−1} é ❜❛s❡ ❞❡ K(α) s♦❜r❡ K ❡

[K(α) :K] =∂p=n✳

❙❡❥❛ K[α] =K(α) = _{f(α)_|f(x)_∈K[x]_}✳

f(x), p(x) _∈ K[x] _⇒❡①✐st❡♠ q(x), r(x) _∈ K[x] t❛✐s q✉❡ f(x) = p(x).q(x) +r(x) ❝♦♠ r(x) = 0 ♦✉ ∂r(x) < ∂p(x)✳ ▲♦❣♦✱ ❡①✐st❡♠ a0, a1,· · · , an−1 ∈ K t❛✐s q✉❡ r(x) = a0+a1x+· · ·+an−1xn−1✳

❆ss✐♠✱ s❡ f(α) =p(α)q(α) +r(α)✳

❈♦♠♦ p(α) = 0✱ f(α) =a0+a1α+· · ·+an−1αn−1 ⇒K[α] = [1, a,· · · , an−1]✳ ❱❛♠♦s ✈❡r✐✜❝❛r s❡ {1, a,_{· · ·}, an−1_{} é ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡✳}

❉❡ ❢❛t♦✱ a0 +a1α+· · ·+an−1αn−1 = 0⇒f(α) = 0✱ ♣❛r❛ f(x) =a0 +a1x+· · ·+ an−1xn−1.

P❡❧❛ ♠✐♥✐♠❛❧✐❞❛❞❡ ❞♦ ❣r❛✉ ❞❡ p(x)✱ f(x) = 0, ♦✉ s❡❥❛✱ a0 = a1 = · · · = an−1 = 0✳ ▲♦❣♦✱ {1, a, ..., an−1_{} é ❜❛s❡ ❞❡ K(α) s♦❜r❡ K ❡ [K(α) :K] =n✳}

❊①❡♠♣❧♦ ✷✳✼✳ ❙❡❥❛ α = √3

2, p=x3 _{−2✳ ❆♣❧✐❝❛♥❞♦ ♦ ❝r✐tér✐♦ ❞❡ ❊✐s❡✐♥st❡✐♥ ❡♠ p(x)} ❝♦♠ ♦ ♣r✐♠♦p= 2 t❡♠♦s q✉❡ p(x) =irrα |Q✳

❆ss✐♠✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✭✷✳✸✮✱ [Q(α) :Q] = 3❡_{1,√3

2,√3

4_} é ❜❛s❡ ❞❡ Q(√3

2)s♦❜r❡Q✳

❚❡♦r❡♠❛ ✷✳✹✳ ✭❚❡♦r❡♠❛ ❞♦ ❡❧❡♠❡♥t♦ ♣r✐♠✐t✐✈♦✮

❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❛❧❣é❜r✐❝♦s ✷✾

❊①❡♠♣❧♦ ✷✳✽✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s Q(i,√5)✳ ▼♦str❛r❡♠♦s q✉❡Q(i,√5) =

Q(i+√5)✳

➱ ❝❧❛r♦ q✉❡Q(i+√5)⊆Q(i,√5)✳

❆❣♦r❛✱ (1 +√5)3 _{= 14i+ 2}√_5∈Q(i+√_5)✳

❆✐♥❞❛✱ (1 +√5)3_−2(i+√_{5) = 12i∈Q(i+}√_{5)❡✱ ♣♦rt❛♥t♦✱ i∈ Q(i+}√_{5) ❡ ♣♦r} ❝♦♥s❡q✉ê♥❝✐❛√5t❛♠❜é♠✳ P❡❧♦ ❚❡♦r❡♠❛ ✭✷✳✸✮✱ ✉♠❛ ❜❛s❡ ❞❡Q(i,√5)/Q♣♦❞❡ s❡r ❞❛❞❛

♣♦r {1, i+√5,(i+√5)2_,(i+√₅₎3_}

❱❡❥❛♠♦s ❛❣♦r❛ ❝♦♠♦ ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s K ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦✱ ❞✐③❡♠♦s ♥❛ ✈❡r❞❛❞❡ ♠❡r❣✉❧❤❛❞♦ ❡♠ C✳

▲❡♠❜r❡♠♦s q✉❡ ❞❛❞♦s A ❡ B ❛♥é✐s✱ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s é ✉♠❛ ❢✉♥çã♦ ϕ :A→B q✉❡ s❛t✐s❢❛③✱ ♣❛r❛ t♦❞♦ a, b∈A✿

• ϕ(a+b) =ϕ(a) +ϕ(b)

• ϕ(a.b) =ϕ(a).ϕ(b)

• ϕ(1) = 1

❉❡✜♥✐çã♦ ✷✳✶✶✳ ❙❡❥❛♠ L/K ❡ K/Q ❡①t❡♥sõ❡s ❞❡ ❝♦r♣♦s✳

❉✐③❡♠♦s q✉❡ ϕ :K →L é ✉♠ Q✲❤♦♠♦♠♦r✜s♠♦ s❡ ϕ s❛t✐s❢❛③ ϕ(a) =a,∀a ∈Q✱

♦✉ s❡❥❛✱ ✜①❛ Q✳

❉❡✜♥✐çã♦ ✷✳✶✷✳ ❯♠ Q✲❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s ϕ:K →C é ❝❤❛♠❛❞♦ ✉♠ ♠❡r❣✉❧❤♦

❞❡ K ❡♠ C✳

❚❡♦r❡♠❛ ✷✳✺✳ ✭❬✶✽❪ ♣✳ ✸✸✮ ❙❡❥❛ K = Q(θ) ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s ❡ [Q(θ) : Q] = n✳ ❊①✐st❡♠ ❡①❛t❛♠❡♥t❡ n ♠❡r❣✉❧❤♦s ❞✐st✐♥t♦s ❞❡ K ❡♠ C✱

σi : K → C

θ _7→ σi(θ) = θi, i= 1, ..., n,

♦♥❞❡ θi sã♦ ❛s r❛í③❡s ❞✐st✐♥t❛s ❞♦ ♣♦❧✐♥ô♠✐♦ ♠✐♥✐♠❛❧ ❞❡ θ s♦❜r❡ C✳

◆♦t❡♠♦s q✉❡ ✉♠ ❞♦s σi✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r σ1✱ é ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡✱ ✐st♦ é✱ σ1(x) = x,∀x∈K✳

◗✉❛♥❞♦ ❛♣❧✐❝❛♠♦s ♦ ♠❡r❣✉❧❤♦ σi ♥✉♠ ❡❧❡♠❡♥t♦ q✉❛❧q✉❡r x ∈ K✱

x=

n

X

k=1

akθk, ak ∈Q✱ t❡♠♦s✱ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦sQ✲❤♦♠♦♠♦r✜s♠♦s✱

σi(x) =σi n

X

k=1 akθk

!

=

n

X

k=1

σi(ak)σi(θ)k = n

X

k=1

ak(θi)k ∈C.

❊①❡♠♣❧♦ ✷✳✾✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦r♣♦ Q(√3

2)❡ ♦ ♣♦❧✐♥ô♠✐♦ ♠✐♥✐♠❛❧ p(x) =x3_−2. ❈♦♠♦ ❛s r❛í③❡s ♥✲és✐♠❛s ❞❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ♥ã♦ ♥✉❧♦ ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❝♦♠♦ ♦ ♣r♦❞✉t♦ ❞❡ ✉♠❛ ❞❡ s✉❛s r❛í③❡s ♣❛rt✐❝✉❧❛r❡s ♣❡❧❛s r❛í③❡s ♥✲és✐♠❛s ❞❛ ✉♥✐✲ ❞❛❞❡ 1, ω, ..., ωn−1_{✱ ♦♥❞❡ ω = cos} 2π

n

✸✵ ❖ ▼♦❞❡❧♦ ❞♦ ❙✐st❡♠❛ ▼■▼❖

{√3

2, ω√3

2, ω2√3

2}✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♦❜❡r✈❛çã♦ ✭✷✳✶✮✱ ♣❛r❛ n = 3✱ t❡♠♦s q✉❡ w = j✳ ▲♦❣♦✱

(x3_{−2) = (x−}√3

2)(x−j√3

2)(x−j2√3

2)✳

❆ss✐♠ t❡♠♦s três ♠❡r❣✉❧❤♦s✿

σ1(x) : Q( 3

√

2) _7−→ C

3

√

2 _−→ √3

2

σ2(x) : Q( 3

√

2) _7−→ C

3

√

2 _−→ j√3

2

σ3(x) : Q( 3

√

2) 7−→ C

3

√

2 −→ j2√3

2

❉❡✜♥✐çã♦ ✷✳✶✸✳ ❙❡❥❛♠ K ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s ❡ x_∈K✳

❖s ❡❧❡♠❡♥t♦s σ1(x), σ2(x),· · ·, σn(x) sã♦ ❝❤❛♠❛❞♦s ❝♦♥❥✉❣❛❞♦s ❞❡ x ❡

N(x) =

n

Y

i=1

σi(x) é ❝❤❛♠❛❞♦ ❞❡ ◆♦r♠❛ ❞❡ x ❡

T r(x) =

n

X

i=1

σi(x) é ❝❤❛♠❛❞♦ ❞❡ ❚r❛ç♦ ❞❡ x✳

❚❡♦r❡♠❛ ✷✳✻✳ ✭❬✶✽❪ ♣✳ ✸✽✮ ❙❡❥❛ K ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s✳ P❛r❛ t♦❞♦ x _∈ K✱ t❡♠♦s N(x)_∈Q ❡ T r(x)_∈Q✳ ❙❡ x_{∈ O}K✱ t❡♠♦s N(x)∈Z ❡ T r(x)∈Z✳

❉❡✜♥✐çã♦ ✷✳✶✹✳ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s ❛❧❣é❜r✐❝♦ ❞❡ ❣r❛✉ n s♦❜r❡ Q ❡ _OK

s❡✉ ❛♥❡❧ ❞♦s ✐♥t❡✐r♦s✳ ❉✐③❡♠♦s q✉❡ w1, w2, ..., wn é ✉♠❛ ❜❛s❡ ✐♥t❡❣r❛❧ ❞❡ K s❡ wi ∈

K,_∀i, i= 1, ...n ❡ _OK =Zw1+Zw2+...+Zwn✳

❊①❡♠♣❧♦ ✷✳✶✵✳ ❙❡❥❛ K =Q(√2)✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s✳

❖ ♣♦❧✐♥ô♠✐♦ ♠✐♥✐♠❛❧ ❞❡√2s♦❜r❡Qéx2_{−2❡ s✉❛s r❛í③❡s sã♦θ} 1 =

√

2❡θ2 =−

√

2. ❆ss✐♠✱

σ1(

√

2) = √2❡ σ2(

√

2) =−√2.

P❛r❛ x_∈Q(√2)✱ t❡♠♦s x=a+b√2, a, b _∈Q✳

σ1(a+b

√

2) = a+b√2❡ σ2(a+b

√

2) = a−b√2

❆ ♥♦r♠❛ ❞❡ x é✿

❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❛❧❣é❜r✐❝♦s ✸✶

N(x) = (a+b√2) + (a−b√2) = 2a∈Q.

P❛r❛ ❡①t❡♥sõ❡s ❞♦ t✐♣♦ L/K t❡♠♦s ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿

❚❡♦r❡♠❛ ✷✳✼✳ ❙❡❥❛ L/K ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❝♦r♣♦s✳ P❛r❛ t♦❞♦x_∈L✱ t❡♠♦sNL/K(x)∈

K ❡ T rL/K(x)∈K✳ ❙❡ x∈ OL t❡♠♦s q✉❡ NL/K(x)∈ OK ❡ NL/K(x)∈ OK✳

❉❡✜♥✐çã♦ ✷✳✶✺✳ ❙❡❥❛ {w1, w2, ..., wn} ✉♠❛ ❜❛s❡ ✐♥t❡❣r❛❧ ❞❡ K✳

❖ ❞✐s❝r✐♠✐♥❛♥t❡ ❞❡ K é ❞❡✜♥✐❞♦ ♣♦r

dK =det[σj(wi)]2, i, j = 1,2, ..., n✳

❚❡♦r❡♠❛ ✷✳✽✳ ❖ ❞✐s❝r✐♠✐♥t❛♥t❡ ❞❡ ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s ♣❡rt❡❝❡ ❛ Z✳

❉❡✜♥✐çã♦ ✷✳✶✻✳ ❯♠ ✐❞❡❛❧ I ❞❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❆ é ✉♠ s✉❜❣r✉♣♦ ❛❞✐t✐✈♦ ❞❡ ❆

q✉❡ é ❢❡❝❤❛❞♦ ❡♠ r❡❧❛çã♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❆✱ ✐st♦ é✱ aI ⊆ I ♣❛r❛ t♦❞♦ a∈A. ❊♥tr❡ t♦❞♦s ♦s ✐❞❡❛✐s ❞❡ ✉♠ ❛♥❡❧✱ ❛❧❣✉♥s ❞❡❧❡s t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❡s♣❡❝✐❛❧ ❞❡ s❡r❡♠ ❣❡r❛❞♦s ♣♦r s♦♠❡♥t❡ ✉♠ ❡❧❡♠❡♥t♦✳ ❊st❡s s❡rã♦ ❞❡ ♣❛rt✐❝✉❧❛r ✐♥t❡r❡ss❡ ♥❡st❡ tr❛❜❛❧❤♦✳ ❉❡✜♥✐çã♦ ✷✳✶✼✳ ❯♠ ✐❞❡❛❧ I é ♣r✐♥❝✐♣❛❧ s❡ ❡❧❡ é ❞❛ ❢♦r♠❛

I =hxiA={xy, y∈A}, x∈ I.

❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r I =hxi.

❊①❡♠♣❧♦ ✷✳✶✶✳ nZ é ✉♠ ✐❞❡❛❧ ♣r✐♥❝✐♣❛❧ ❞❡Z ♣❛r❛ t♦❞♦ n.

❉❡✜♥✐çã♦ ✷✳✶✽✳ ❯♠ ❞♦♠í♥✐♦ A é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐❞❡❛✐s ♣r✐♥❝✐♣❛✐s s❡ t♦❞♦ ✐❞❡❛❧ ❞❡ A é ♣r✐♥❝✐♣❛❧✳

❉❡✜♥✐çã♦ ✷✳✶✾✳ ❯♠ ❛♥❡❧ A é ❝❤❛♠❛❞♦ ✐♥t❡❣r❛❧♠❡♥t❡ ❢❡❝❤❛❞♦ s❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞♦ s❡✉ ❝♦r♣♦ ❞❡ ❢r❛çõ❡s q✉❡ é ✐♥t❡✐r♦ s♦❜r❡ A ❡stá ❡♠ A✳

Pr♦♣♦s✐çã♦ ✷✳✶✳ ❙❡ A é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐❞❡❛✐s ♣r✐♥❝✐♣❛✐s ❡♥tã♦ A é ✐♥t❡❣r❛❧♠❡♥t❡ ❢❡❝❤❛❞♦✳

❊①❡♠♣❧♦ ✷✳✶✷✳ ❖ ❛♥❡❧ Z ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s é ✐♥t❡❣r❛❧♠❡♥t❡ ❢❡❝❤❛❞♦✱ ♣♦✐s é ✉♠

❞♦♠í♥✐♦ ❞❡ ✐❞❡❛✐s ♣r✐♥❝✐♣❛✐s✳

P♦❞❡♠♦s ❞❡✜♥✐r ❛ ♥♦r♠❛ ❞❡ ✉♠ ✐❞❡❛❧✳ ◆♦ ❝❛s♦ ❞❡ ✉♠ ✐❞❡❛❧ ♣r✐♥❝✐♣❛❧✱ ✐ss♦ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛ ♥♦r♠❛ ❞♦ ❣❡r❛❞♦r ❞♦ ✐❞❡❛❧✱ ❞❡✜♥✐❞♦ ❡♠ ✭✷✳✶✸✮✳

❉❡✜♥✐çã♦ ✷✳✷✵✳ ❙❡❥❛ I =_hx_iOL ✉♠ ✐❞❡❛❧ ♣r✐♥❝✐♣❛❧ ❞❡OL.❙✉❛ ♥♦r♠❛ é ❞❡✜♥✐❞❛ ♣♦r