POLINÔMIOS, EQUAÇÕES ALGÉBRICAS E ESTUDO DAS SUAS RAÍZES REAIS.

❙✉♠ár✐♦

✶ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s ✺

✶✳✶ ❆s♣❡❝t♦s ❤✐stór✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❖ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❯♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛ ❡ ❢♦r♠❛ ❛❧❣é❜r✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹ P❧❛♥♦ ❝♦♠♣❧❡①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✷ P♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s ✶✺

✷✳✶ P♦❧✐♥ô♠✐♦s ❡ ❖♣❡r❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✶✳✶ ❆❞✐çã♦ ❞❡ P♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✶✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ P♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✶✳✸ ❉✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❉✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✸ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ➪❧❣❡❜r❛ ✸✷

✸✳✶ ❆s♣❡❝t♦s ❤✐stór✐❝♦s ❞♦ t❡♦r❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷ ❆ tr❛♥s❧❛çã♦ ♥❛ ✈❛r✐á✈❡❧ p(z+z0) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✸ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❋❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✹ ❋❛t♦r❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✹ ❊q✉❛çõ❡s ❆❧❣é❜r✐❝❛s ✹✶

✹✳✶ ❘❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✶✳✶ ❘❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ t❡r❝❡✐r♦ ❣r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✶✳✷ ❘❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ q✉❛rt♦ ❣r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✷ ❚❡♦r❡♠❛ ❞❛s ❘❛í③❡s ❘❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✸ ❘❡❧❛çã♦ ❡♥tr❡ ❝♦❡✜❝✐❡♥t❡s ❡ r❛í③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✹ ❊q✉❛çõ❡s ❘❡❝í♣r♦❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✺ ❆♥á❧✐s❡ ❞♦ ♥ú♠❡r♦ ❞❡ r❛í③❡s r❡❛✐s ✺✾

✺✳✶ ❚❡♦r❡♠❛ ❞❡ ❉❡s❝❛rt❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✺✳✷ ❚❡♦r❡♠❛ ❞❡ ❇♦❧③❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✺✳✸ ❚❡♦r❡♠❛ ❞❡ ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵

❘❡s✉♠♦

❖ r❡s♣❡❝t✐✈♦ tr❛❜❛❧❤♦ ✈✐s❛ ❝♦♥tr✐❜✉✐r ♣❛r❛ q✉❡ ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s ♣♦ss❛♠ ❛♣r✐♠♦r❛r s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❡♠ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ♣♦❧✐♥ô♠✐♦s ❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳ ■♥✐❝✐❛❧♠❡♥t❡ ❢♦✐ ❛♥❛❧✐s❛❞♦ ♦ ❝♦♥t❡①t♦ ❤✐stór✐❝♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❡♠ s❡❣✉✐❞❛ ❢♦r❛♠ ✈✐st♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s ❝♦♠♦ ♦ ❞❡ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛ ❡ ♣❧❛♥♦ ❝♦♠✲ ♣❧❡①♦✳ ❆❧é♠ ❞✐ss♦✱ ❢♦r❛♠ ❛♣r❡s❡♥t❛❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ♦♣❡r❛çõ❡s ❜ás✐❝❛s ❞♦s ♣♦❧✐♥ô♠✐♦s✱ ♦ ❞✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐✱ ❛tr❛✈és ❞♦ q✉❛❧ ♣♦❞❡♠♦s ♦❜t❡r ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ♣✭①✮ ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦ ❧✐♥✲ ❡❛r✳ P❛rt❡ s✐❣♥✐✜❝❛t✐✈❛ ❞❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞❡ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s✳ ◆❡ss❛ ♣❡rs♣❡❝t✐✈❛✱ ❢♦r❛♠ ❞✐s❝✉t✐❞♦s ❛❧❣✉♥s t❡♦r❡♠❛s ❡ ♠ét♦✲ ❞♦s r❡s♦❧✉t✐✈♦s ❞❡ ❡q✉❛çõ❡s ❝♦♠♦ ♦ ♠ét♦❞♦ ❞❡ ●✉st❛✈♦✱ q✉❡ ♥♦s ❛✉①✐❧✐❛ ♥❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ t❡r❝❡✐r♦ ❡ ❞♦ q✉❛rt♦ ❣r❛✉s✱ ♦ t❡♦r❡♠❛ ❞❛s r❛í③❡s r❛❝✐♦♥❛✐s✱ ❡♥tr❡ ♦✉tr♦s✳ P❛r❛ t❛♥t♦✱ ❢♦✐ ❡ss❡♥❝✐❛❧ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥✲ t❛❧ ❞❛ ➪❧❣❡❜r❛✱ q✉❡ ❛✜r♠❛ q✉❡ t♦❞♦ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ❝♦♥st❛♥t❡ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ✉♠❛ r❛✐③ ❝♦♠♣❧❡①❛✳ ❆❞❡♠❛✐s✱ ♠♦str❛♠♦s ❝♦♠♦ ♣♦❞❡♠♦s ❛♥❛❧✐s❛r ♦ ♥ú♠❡r♦ ❞❡ r❛í③❡s r❡❛✐s ❞❡ ✉♠❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s✳ ◆❡ss❡ s❡♥t✐❞♦✱ ♣r♦✈❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❉❡s❝❛rt❡s✱ q✉❡ ❞✐③ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ r❛í③❡s ♣♦s✐t✐✈❛s ❞❡ ✉♠❛ ❡q✉❛çã♦ ♥ã♦ s✉♣❡r❛ ♦ ♥ú♠❡r♦ ❞❡ ♠✉❞❛♥ç❛s ❞❡ s✐♥❛❧ ♥❛ s❡q✉ê♥❝✐❛ ❞♦s s❡✉s ❝♦❡✜❝✐❡♥t❡s ♥ã♦ ♥✉❧♦s✱ ♣r♦✈❛♠♦s t❛♠❜é♠ ♦ ❚❡♦r❡♠❛ ❞❡ ❇♦❧③❛♥♦✱ q✉❡ ✐♥✈❡st✐❣❛ ♦ ♥ú♠❡r♦ ❞❡ r❛í③❡s r❡❛✐s ❞❡ ✉♠❛ ❡q✉❛çã♦ ♥✉♠ ✐♥t❡r✈❛❧♦ r❡❛❧ ❡✱ ✜♥❛❧♠❡♥t❡✱ ♦ ❚❡♦r❡♠❛ ❞❡ ▲❛❣r❛♥❣❡ q✉❡ ❡st❛❜❡❧❡❝❡ ✉♠ ❧✐♠✐t❡ s✉♣❡r✐♦r ❞❛s r❛í③❡s r❡❛✐s ❞❡ ✉♠❛ ❡q✉❛çã♦✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ◆ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ P♦❧✐♥ô♠✐♦s✱ ❊q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳

❆❜str❛❝t

❚❤❡ r❡s♣❡❝t✐✈❡ ✇♦r❦ ❛✐♠s t♦ ❤❡❧♣ st✉❞❡♥ts ❛♥❞ t❡❛❝❤❡rs t♦ ✐♠♣r♦✈❡ t❤❡✐r ♠❛t❤ s❦✐❧❧s ✐♥ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥s✳ ■♥✐✲ t✐❛❧❧② ✐t ❛♥❛❧②s❡❞ t❤❡ ❤✐st♦r✐❝❛❧ ❝♦♥t❡①t ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs t❤❡♥ ✇❡r❡ s❡❡♥ s♦♠❡ ✐♠♣♦rt❛♥t ❝♦♥❝❡♣ts s✉❝❤ ❛s t❤❡ ❜♦❞② ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✐♠❛❣✐♥❛r② ✉♥✐t ❛♥❞ ❝♦♠♣❧❡① ♣❧❛♥❡✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ♣r♦♣❡rt✐❡s ❛♥❞ ❜❛s✐❝ ♦♣❡r❛t✐♦♥s ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧s ✇❡r❡ ♣r❡s❡♥t❡❞✱ t❤❡ ❇r✐♦t✲❘✉✣♥✐ ❞❡✈✐❝❡✱ t❤r♦✉❣❤ ✇❤✐❝❤ ✇❡ ❝❛♥ ❣❡t t❤❡ q✉♦t✐❡♥t ❛♥❞ r❡♠❛✐♥❞❡r ♦❢ t❤❡ ❞✐✈✐s✐♦♥ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ♣✭①✮ ❜② ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧✳ ❙✐❣♥✐✜❝❛♥t ♣❛rt ♦❢ t❤✐s ✇♦r❦ ✇❛s ❞❡✈♦t❡❞ t♦ t❤❡ st✉❞② ♦❢ ❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s✳ ■♥ t❤✐s ♣❡rs♣❡❝t✐✈❡✱ ✇❡r❡ ❞✐s❝✉ss❡❞ s♦♠❡ t❤❡♦r❡♠s ❛♥❞ ♠❡t❤♦❞s ♦❢ r❡s♦❧✉t✐♦♥ ♦❢ ❡q✉❛t✐♦♥s s✉❝❤ ❛s t❤❡ ♠❡t❤♦❞ ♦❢ ●✉st❛✈♦✱ ✇❤♦ ❤❡❧♣s ✉s ✐♥ t❤❡ r❡s♦❧✉t✐♦♥ ♦❢ ❡q✉❛t✐♦♥s ♦❢ t❤❡ t❤✐r❞ ❛♥❞ ❢♦✉rt❤ ❞❡❣r❡❡s✱ t❤❡ t❤❡♦✲ r❡♠ ♦❢ r❛t✐♦♥❛❧ r♦♦ts✱ ❛♠♦♥❣ ♦t❤❡rs✳ ❋♦r ❜♦t❤✱ ✐t ✇❛s ❡ss❡♥t✐❛❧ t♦ ♣r♦✈❡ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❆❧❣❡❜r❛✱ ✇❤✐❝❤ s❛②s t❤❛t ❛❧❧ ♣♦❧②♥♦♠✐❛❧ ♥♦t ❝♦♥✲ st❛♥t ✇✐t❤ ❝♦♠♣❧❡① ❝♦❡✣❝✐❡♥ts ❤❛s ❛t ❧❡❛st ♦♥❡ ❝♦♠♣❧❡① r♦♦t✳❋✉rt❤❡♠♦r❡✱ ✇❡ s❤♦✇ ❤♦✇ ✇❡ ❝❛♥ ❛♥❛❧②③❡ t❤❡ ♥✉♠❜❡r ♦❢ r❡❛❧ r♦♦ts ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥ ✇✐t❤ r❡❛❧ ❝♦❡✣❝✐❡♥ts✳ ■♥ t❤✐s s❡♥s❡✱ ✇❡ ✇✐❧❧ ♣r♦✈❡ t❤❡ ❚❤❡♦r❡♠ ♦❢ ❉❡s❝❛rt❡s✱ ✇❤✐❝❤ s❛②s t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ r♦♦ts ♦❢ ❛♥ ❡q✉❛t✐♦♥ ❞♦❡s ♥♦t ❡①❝❡❡❞ t❤❡ ♥✉♠❜❡r ♦❢ s✐❣♥❛❧ ❝❤❛♥❣❡s ❢♦❧❧♦✇✐♥❣ ✐ts ♥♦♥✲③❡r♦ ❝♦❡✣❝✐❡♥ts✱ ✇❡ ♣r♦✈❡ t❤❡ t❤❡♦r❡♠ ♦❢ ❇♦❧③❛♥♦✱ ✇❤✐❝❤ ✐♥✈❡st✐❣❛t❡s t❤❡ ♥✉♠❜❡r ♦❢ r❡❛❧ r♦♦ts ♦❢ ❛♥ ❡q✉❛✲ t✐♦♥ ✐♥ ❛ r❡❛❧ ✐♥t❡r✈❛❧ ❛♥❞ ✜♥❛❧❧② t❤❡ t❤❡♦r❡♠ ♦❢ ▲❛❣r❛♥❣❡ t❤❡ ❡st❛❜❧✐s❤❡s ❛♥ ✉♣♣❡r ❧✐♠✐t ♦♥ r♦♦ts ♦❢ ❛♥ ❡q✉❛t✐♦♥✳

❑❡②✇♦r❞s✿ ❈♦♠♣❧❡① ♥✉♠❜❡rs✱ P♦❧②♥♦♠✐❛❧s✱ P♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥s✳

■♥tr♦❞✉çã♦

➱ ❢❛t♦ q✉❡ ❛ ❛❜♦r❞❛❣❡♠ ❞♦ ❝♦♥t❡ú❞♦ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ♣♦❧✐♥ô♠✐♦s ❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✱ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❛t✉❛✐s✱ ♠❡r❡❝❡♠ ✉♠❛ ❛t❡♥çã♦ ❡s♣❡❝✐❛❧ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ♠✉✐t❛s ✈❡③❡s sã♦ ✐♥s❡r✐❞♦s ♥♦s ♠❡s♠♦s ✉♠❛ s❡r✐❡ ❞❡ ❞❡✜♥✐çõ❡s✱ ❝♦♥❝❡✐t♦s ❡ ❡①❡r❝í❝✐♦s ♦s q✉❛✐s ♥ã♦ ❡stã♦ ❜❡♠ ❝♦♥❝❛t❡♥❛❞♦s✱ ♣♦r ✈❡③❡s✱ ❢♦r❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❧ó❣✐❝❛ ❡ ❞❡ ✉♠ ❝♦♥t❡①t♦ ❤✐stór✐❝♦✱ ♥♦s r❡♠❡t❡♥❞♦ ❛ ✉♠ ♠❛t❡r✐❛❧ ❞✐❞át✐❝♦ ♠❡r❛♠❡♥t❡ ❡♥❝✐❝❧♦♣é❞✐❝♦✱ ♦ q✉❡ ✈❡♠ ❞✐✜❝✉❧t❛♥❞♦ ♦ ❛♣r❡♥❞✐③❛❞♦ ❞♦ ❡❞✉❝❛♥❞♦✳

◆✉♠❛ t❡♥t❛t✐✈❛ ❞❡ ❢♦r♥❡❝❡r ✉♠ ♠❛t❡r✐❛❧ ❞✐❞át✐❝♦ r❛③♦❛❧✈❡❧♠❡♥t❡ ❡str✉t✉✲ r❛❞♦ ❡ ❝♦♥❝✐s♦ ❡ ✉t✐❧✐③❛♥❞♦ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❛❝❡ssí✈❡❧ ❛ ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s✱ ♦❜❥❡t✐✈❡✐ ❡s❝r❡✈❡r ❛ r❡s♣❡✐t♦ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ♣♦❧✐♥ô♠✐♦s ❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✱ ❝♦♠ ❞❡st❛q✉❡ ♣❛r❛ ❡st❡ ú❧t✐♠♦✱ ♣♦ré♠✱ ❞❡ ❢♦r♠❛ ♠❛✐s ❧♦❣✐❝❛✲ ♠❡♥t❡ ❡❧❛❜♦r❛❞❛✱ ✐♥❝❧✉✐♥❞♦ ❡♠ ❛❧❣✉♥s ♠♦♠❡♥t♦s ♦s ❛s♣❡❝t♦s ❤✐stór✐❝♦s ❡♥✲ ✈♦❧✈✐❞♦s✱ ❛❝r❡s❝❡♥t❛♥❞♦ ❛❧❣✉♥s t❡♦r❡♠❛s ❡ ♣r♦♣♦s✐çõ❡s ♠❛✐s ❛♣r♦❢✉♥❞❛❞♦s q✉❡ ♦s ✉s✉❛✐s ❡ s✉❛s r❡s♣❡❝t✐✈❛s ❞❡♠♦♥str❛çõ❡s✱ ♣❡r♠✐t✐♥❞♦ ❛ss✐♠ ✉♠ ❝❡rt♦ ❛♣r✐♠♦r❛♠❡♥t♦ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❞❛q✉❡❧❡s q✉❡ s❡ ✐♥t❡r❡ss❛♠ ♣❡❧♦ ❛ss✉♥t♦✳ ❖ ❝❛♣ít✉❧♦ ✶ é ❞❡st✐♥❛❞♦ ❛♦ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♦♥❞❡ ✐♥✐❝✐❛❧✲ ♠❡♥t❡ ❢♦✐ ❞✐s❝✉t✐❞♦ ♦ ❝♦♥t❡①t♦ ❤✐stór✐❝♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ❊♠ s❡❣✉✐❞❛✱ ♣r♦✈❛♠♦s q✉❡ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s s❛t✐s❢❛③❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s q✉❡ ♦ ❝❛r❛❝t❡r✐③❛♠ ❝♦♠♦ ❝♦r♣♦✱ ❛❧é♠ ❞✐ss♦✱ ❞✐s❝♦rr❡♠♦s s♦❜r❡ ❛ ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛✱ ❢♦r♠❛ ❛❧❣é❜r✐❝❛ ❞❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ❡ ♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✳ ❆ ✐♥t❡♥çã♦ ♣r✐♥❝✐♣❛❧ ❞❡ss❡ ❝❛♣ít✉❧♦ ❢♦✐ ❞❡ ❢♦r♥❡❝❡r s✉❜sí❞✐♦s ♣❛r❛ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ➪❧❣❡❜r❛ ❡ ♥❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s✳

❖ ❝❛♣ít✉❧♦ ✷ tr❛t❛ ❞❛s ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ ♦♣❡r❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s✱ ❛❞❡♠❛✐s✱ ♣r♦✈❛ ♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✲ ✈✐sã♦ ♣❛r❛ ♣♦❧✐♥ô♠✐♦s ♠♦str❛♥❞♦ ❛❧❣✉♠❛s ❝♦♥s❡q✉ê♥❝✐❛s ❞❡ss❡ ❛❧❣♦r✐t♠♦ ❡ ♣♦r ✜♠ é ✈✐st♦ ❝♦♠♦ ♣♦❞❡♠♦s ♦❜t❡r ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ♣♦r ♣♦❧✐♥ô♠✐♦ ❧✐♥❡❛r✱ ♦ ❝♦♥❤❡❝✐❞♦ ❞✐s♣♦s✐t✐✈♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐✳ ◆♦ ❝❛♣ít✉❧♦ ✸ ❝♦♠❡ç❛♠♦s ❞❡s❝r❡✈❡♥❞♦ ❛❧❣✉♠❛s t❡♥t❛t✐✈❛s ❤✐stór✐❝❛s ❞❡ ❞❡♠♦♥str❛r ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ➪❧❣❡❜r❛✳ Pr❡s❝✐♥❞✐♥❞♦ ❞❛s ♥♦çõ❡s ❜ás✐❝❛s ❞❡ t♦♣♦❧♦❣✐❛✱ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❡ ❛♥á❧✐s❡ ❝♦♠♣❧❡①❛✱ ♣r♦✈❛♠♦s ♦ ❚❋❆ ❞❡ ❢♦r♠❛ q✉❡ ♥ã♦ ❢♦ss❡ ♥❡❝❡ssár✐♦ ❣r❛♥❞❡ tr❛q✉❡❥♦ ❛❧❣é❜r✐❝♦ ♣❛r❛ ❡♥t❡♥❞ê✲ ❧♦✳ ❖ ❝❛♣ít✉❧♦ ✹ é ❞❡✈♦t❛❞♦ ❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s ❡ ❢♦✐ ❛ ❡st❡ ❝❛♣ít✉❧♦ q✉❡ t❡♥t❡✐ ❞❛r ♠❛✐♦r ❞❡st❛q✉❡✱ ❥✉st❛♠❡♥t❡ ♣❡❧♦ ❢❛t♦ ❞❡ ♠✉✐t♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❛❞♦t❛❞♦s ❤♦❥❡ tr❛t❛r❡♠ ♦ ❛ss✉♥t♦ ❞❡ ❢♦r♠❛ ❜❛st❛♥t❡ s✉♣❡r✜❝✐❛❧✳ ❉❡ss❡ ♠♦❞♦✱ ❢♦r❛♠ ❛♣r❡s❡♥t❛❞♦s ♦s ♠ét♦❞♦s ❞❡ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ t❡r❝❡✐r♦ ❡ q✉❛rt♦ ❣r❛✉s✱ ♦ t❡♦r❡♠❛ ❞❛s r❛í③❡s r❛❝✐♦♥❛✐s✱ q✉❡ ♥♦s ❢♦r♥❡❝❡ ❛s ♣♦s✲ sí✈❡✐s r❛í③❡s r❛❝✐♦♥❛✐s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❝♦❡✜❝✐❡♥t❡s ✐♥t❡✐r♦s ❡ ♠❛✐s ❛❞✐❛♥t❡ ❝♦♠♦ r❡s♦❧✈❡r ❛❧❣✉♠❛s ❡q✉❛çõ❡s ❝♦♠ ❝❛r❛❝t❡ríst✐❝❛s ❡s♣❡❝✐❛✐s✱ ❛s ❝❤❛♠❛❞❛s

❡q✉❛çõ❡s r❡❝í♣r♦❝❛s✳ ❖ ú❧t✐♠♦ ❝❛♣ít✉❧♦ ✐♥✈❡st✐❣❛ ♦ ♥ú♠❡r♦ ❞❡ r❛í③❡s r❡❛✐s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s✳ ◆❡ss❛ ♣❡rs♣❡❝t✐✈❛✱ é ❞❡♠♦♥str❛❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❉❡s❝❛rt❡s✱ q✉❡ ❢♦r♥❡❝❡ ✐♥❢♦r♠❛çõ❡s ❛ r❡s♣❡✐t♦ ❞♦ ♥ú♠❡r♦ ❞❡ r❛í③❡s ♣♦s✐t✐✈❛s ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ❛ ♣❛rt✐r ❞♦ ♥ú♠❡r♦ ❞❡ tr♦❝❛s ❞❡ s✐♥❛✐s ♥❛ s❡q✉ê♥❝✐❛ ❞♦s s❡✉s ❝♦❡✜❝✐❡♥t❡s ♥ã♦ ♥✉❧♦s✳ ❆❞❡♠❛✐s✱ ❡①♣❧♦r❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❇♦❧③❛♥♦ q✉❡ ♥♦s ✐♥❞✐❝❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ r❛í③❡s r❡❛✐s ♥✉♠ ❝❡rt♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ r❡❛❧ ❡ ✜♥❛❧♠❡♥t❡✱ ♦ ❚❡♦r❡♠❛ ❞❡ ▲❛❣r❛♥❣❡ q✉❡ ❡st❛❜❡❧❡❝❡ ✉♠ ❧✐♠✐t❡ s✉♣❡r✐♦r ❞❛s r❛í③❡s r❡❛✐s ❞❡ ✉♠❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧✳

✶ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s

✶✳✶ ❆s♣❡❝t♦s ❤✐stór✐❝♦s

❖ ❛♣❛r❡❝✐♠❡♥t♦ ❞♦s ❝❤❛♠❛❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ✲ ♥ú♠❡r♦s ❞❛ ❢♦r♠❛ a+

b√₋1❝♦♠ ❛ ❡ ❜ ♥ú♠❡r♦s r❡❛✐s ✲ ❞❛t❛ ❞♦ sé❝✉❧♦ ❳❱■✱ ❡ é ❣❡r❛❧♠❡♥t❡ ❛tr✐❜✉í❞♦

❛ ●✐r♦❧❛♠♦ ❈❛r❞❛♥♦ ✭✶✺✵✶✲✶✺✼✻✮✳ ◆❛ s✉❛ ♦❜r❛ ❆rt✐s ▼❛❣♥❛❡ ✭ ❣❡r❛❧♠❡♥t❡ r❡❢❡r✐❞❛ ❝♦♠♦ ❆rs ▼❛❣♥❛✮✱ ❞❛t❛❞❛ ❞❡ ✶✺✹✺✱ ❈❛r❞❛♥♦ ❝♦♥s✐❞❡r❛ ❡q✉❛çõ❡s q✉❛❞rát✐❝❛s q✉❡✱ t❛✐s ❝♦♠♦ ❛ ❡q✉❛çã♦

x2+ 2x+ 2 = 0

♣♦r ❡①❡♠♣❧♦✱ ♥ã♦ ❛❞♠✐t❡♠ s♦❧✉çõ❡s r❡❛✐s✳ ❆ ❢ór♠✉❧❛ r❡s♦❧✈❡♥t❡ ❞á ❡①♣r❡ssõ❡s ✧❢♦r♠❛✐s✧ ♣❛r❛ ❛s ❞✉❛s s♦❧✉çõ❡s ❞❡ss❛ ❡q✉❛çã♦✱ ♠❛s ❡♥✈♦❧✈❡ r❛í③❡s ❞❡ ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s✿

−2_±√₋4

2 =−1±2

√ −1.

❊♠❜♦r❛ t❛✐s ♥ú♠❡r♦s s✉r❥❛♠✱ ❞❡ ❢❛t♦✱ ♥❛ r❡❢❡r✐❞❛ ♦❜r❛ ❞❡ ❈❛r❞❛♥♦✱ ❡❧❡ ♣ró♣r✐♦ s❡ ❛♣r❡ss❛ ❛ ❞❡s✈❛❧♦r✐③á✲❧♦s✱ r❡❢❡r✐♥❞♦✲s❡✲❧❤❡s ❝♦♠♦ ✧tã♦ s✉❜t✐s q✉❛♥t♦ ✐♥út❡✐s✧✳ ❉❡ ❢❛t♦✱ ♣❛r❛ ❈❛r❞❛♥♦✱ t❛❧ ❝♦♠♦ ♣❛r❛ ♦s r❡st❛♥t❡s ♠❛t❡♠át✐❝♦s ❞♦ s❡✉ t❡♠♣♦✱ ❝♦♠ ✉♠❛ ❝♦♥❝❡♣çã♦ ❞❛ ♠❛t❡♠át✐❝❛ ❛✐♥❞❛ ❤❡r❞❛❞❛ ❞♦s ●r❡❣♦s✱ ♦ q✉❡ ✐♥t❡r❡ss❛✈❛ ❡r❛♠✱ ❡ss❡♥❝✐❛❧♠❡♥t❡✱ ♦s ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s❀ ♥❡ss❡ s❡♥✲ t✐❞♦✱ ✉♠❛ ❡q✉❛çã♦ t❛❧ ❝♦♠♦ ❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r✱ ♥ã♦ t✐♥❤❛ ✐♥t❡r❡ss❡ ♣♦r s✐ ♣ró♣r✐❛✱ s✉r❣✐♥❞♦ ❛ss♦❝✐❛❞❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❛♦ ♣r♦❜❧❡♠❛ ❣❡♦♠étr✐❝♦ ❞❛ ❞❡t❡r✲ ♠✐♥❛çã♦ ❞❛ ✐♥t❡rs❡çã♦ ❞❛ ♣❛rá❜♦❧❛ y=x2 ❝♦♠ ❛ r❡t❛y=₋2x₋2,♣r♦❜❧❡♠❛ ❡ss❡ s❡♠ s♦❧✉çã♦✳ ❊♠ ✶✺✶✷✱ ❘❛❢❛❡❧ ❇♦♠❜❡❧❧✐✱ ❞✐s❝í♣✐❧♦ ❞❡ ❈❛r❞❛♥♦✱ ❧✐❞❛✱ ♥♦ s❡✉ ❧✐✈r♦ ❆❧❣❡❜r❛✱ ❝♦♠ ❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❝ú❜✐❝❛s ❞♦ t✐♣♦

x3 = 3px+ 2q. ♣❡❧❛ ❛♣❧✐❝❛çã♦ ❞❛ ❝❤❛♠❛❞❛ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦✿

x= 3

q

q+pq2_−p3 ₊ 3

q

❆♦ ❛♣❧✐❝❛r t❛❧ ❢ór♠✉❧❛ à r❡s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ x3 = 15x+ 4

✭ q✉❡ ❝♦rr❡s♣♦♥❞❡✱ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✱ ❛ ❞❡t❡r♠✐♥❛r ❛ ✐♥t❡rs❡çã♦ ❞❛ ❝ú❜✐❝❛ y =x3 ❝♦♠ ❛ r❡t❛ ❞❡ ❡q✉❛çã♦ y= 15x+ 4 ♦❜té♠ ❝♦♠♦ s♦❧✉çã♦

x= 3

q

2 + 11√₋1 + 3

q

2₋11√₋1,

✐st♦ é✱ é ✉♠❛ ❡①♣r❡ssã♦ ❡♥✈♦❧✈❡♥❞♦ r❛í③❡s ❞❡ ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s✳ ◆❡st❡ ❝❛s♦✱ ♥♦ ❡♥t❛♥t♦✱ ❇♦♠❜❡❧❧✐ s❛❜❡ ✭♣♦r ✐♥s♣❡çã♦✮ ❡①✐st✐r s♦❧✉çã♦ r❡❛❧ ① ❂ ✹ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✱ ♦ q✉❡ ♣❛r❡❝❡✱ ♣♦rt❛♥t♦✱ s❡r ♣❛r❛❞♦①❛❧✳ ➱✱ ❡♥tã♦✱ q✉❡ ❧❤❡ s✉r❣❡ ❛ ✐❞❡✐❛ q✉❡✱ ❡❧❡ ♣ró♣r✐♦✱ ❝♦♥s✐❞❡r❛ ❞❡ ✧❧♦✉❝❛✧✿ ❊ s❡

3

q

2 + 11√₋1 f or da f orma 2 +n√₋1

❡

3

q

2₋11√₋1 f or da f orma 2₋n√₋1,

❞❡ ♠♦❞♦ q✉❡✱ ❛♦ s♦♠❛r✱ s❡ ♦❜t❡♥❤❛✱ ❞❡ ❢❛t♦ ① ❂ ✹❄❆❞♠✐t✐♥❞♦ q✉❡ t❛❧ ❢❛t♦ é ✈❡r❞❛❞❡✱ ❡❧❡✈❛♥❞♦ ❢♦r♠❛❧♠❡♥t❡ ❛♦ ❝✉❜♦ ❝❛❞❛ ✉♠❛ ❞❡ss❛s ❡①♣r❡ssõ❡s✭✉s❛♥❞♦

(√₋1)2 _{=−1✮ ❡ ✐❣✉❛❧❛♥❞♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛}

2 + 11√₋1 e 2₋11√₋1, ♦❜té♠✲s❡ q✉❡

3

q

2 + 11√₋1 = 2 +√₋1 e 3

q

2₋11√₋1 = 2₋√₋1

♦✉ s❡❥❛✱ ✈❡♠ ♥ ❂ ✶✦

❈♦♠♦ ❢❛❝✐❧♠❡♥t❡ s❡ ❡♥t❡♥❞❡✱ ❇♦♠❜❡❧❧✐ ♥ã♦ ❛❝❡✐t❛ ❛✐♥❞❛ ♦s ♥ú♠❡r♦s ❝♦♠✲ ♣❧❡①♦s ❝♦♠♦ ♥ú♠❡r♦s ❞❡ ♣❧❡♥♦ ❞✐r❡✐t♦✱ ❝♦♥t✐♥✉❛♥❞♦ ❛ ❝♦♥s✐❞❡rá✲❧♦s ♠✐st❡✲ r✐♦s♦s❀ ♥♦ ❡♥t❛♥t♦✱ é ❡❧❡ ♦ ♣r✐♠❡✐r♦ ❛ ❡s❝r❡✈❡r ❡①♣❧✐❝✐t❛♠❡♥t❡ ❛s r❡❣r❛s ♣❛r❛ ❛ ❛❞✐çã♦✱ s✉❜tr❛çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡ ❛ ♠♦str❛r ❝♦♠♦✱ ✉s❛♥❞♦ ❡ss❡ t✐♣♦ ❞❡ ♥ú♠❡r♦s ❝♦♠ ❛ ❛r✐t♠ét✐❝❛✱ é ♣♦ssí✈❡❧ ♦❜t❡r s♦❧✉çõ❡s r❡❛✐s ♣❛r❛ ❛ ❝ú❜✐❝❛✱ q✉❛♥❞♦ s❡ ✉s❛ ❛ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦✲❚❛rt❛❣❧✐❛✳ ➱ t❛♠❜é♠ ❡❧❡ q✉❡ ✐♥tr♦❞✉③ ✉♠❛ ♥♦t❛çã♦ ♣ró♣r✐❛ ♣❛r❛√−1,❝❤❛♠❛♥❞♦✲❧❤❡ ✧ ♣✐ú ❞❡ ♠❡♥♦✧✳ ➚ ♠❡❞✐❞❛ q✉❡ s❡ ❞❡s❡♥✈♦❧✈❡♠ ♦✉tr❛s ♠❛♥✐♣✉❧❛çõ❡s ❝♦♠ ♥ú♠❡r♦s ❝♦♠✲ ♣❧❡①♦s ❡ sã♦ ✐♥tr♦❞✉③✐❞❛s ❢✉♥çõ❡s ❝♦♠♣❧❡①❛s ❞❡ ✈❛r✐á✈❡❧ ❝♦♠♣❧❡①❛✱ ✈❛✐✲s❡ t♦r♥❛♥❞♦ ❝❧❛r❛ s✉❛ ✉t✐❧✐❞❛❞❡ ❡ ❞❡s❝♦❜r✐♥❞♦ ❝♦♠♦ ❛ s✉❛ ✉t✐❧✐③❛çã♦ ♣♦❞❡ ❝♦♥✲ tr✐❜✉✐r s✐❣♥✐✜❝❛♠❡♥t❡ ♣❛r❛ ❛ s✐♠♣❧✐✜❝❛çã♦ ❞❡ ♠✉✐t♦s ♣r♦❜❧❡♠❛s✳

❆♣❡s❛r ❞✐ss♦✱ ❞✉r❛♥t❡ q✉❛s❡ três sé❝✉❧♦s✱ ❡st❡s ♥ú♠❡r♦s ✈ã♦ s❡♥❞♦ ✉s❛✲ ❞♦s✱ ♠❛s ♥ã♦ sã♦ ♥✉♥❝❛ ❝♦♥s✐❞❡r❛❞♦s ♥ú♠❡r♦s ✈❡r❞❛❞❡✐r❛♠❡♥t❡ ✧❧❡❣ít✐♠♦s✧✳

❊♠ ✶✻✸✼✱ ♥❛ s✉❛ ♦❜r❛ ▲❛ ●é♦♠étr✐❡✭❝♦♥t✐❞❛ ♥♦ ❉✐s❝♦✉rs ❞❡ ❧❛ ♠ét❤♦❞❡✮✱ ❉❡s❝❛rt❡s ❢❛③ ❛ ❞✐st✐♥çã♦❡♥tr❡ ♥ú♠❡r♦s ✧r❡❛✐s✧ ❡ ♥ú♠❡r♦s ✧✐♠❛❣✐♥ár✐♦s✧✭♦✉ q✉❡ ❡①✐st❡♠ ❛♣❡♥❛s ♥❛ ✐♠❛❣✐♥❛çã♦✮✱ ✐♥t❡r♣r❡t❛♥❞♦ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ✐♠❛❣✐♥ár✐❛s ♣❛r❛ ✉♠ ❝❡rt♦ ♣r♦❜❧❡♠❛ ❝♦♠♦ ✉♠ s✐♥❛❧ ❞❡ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❡♠ ❝❛✉s❛ ♥ã♦ t❡♠ s♦❧✉çã♦ ❣❡♦♠étr✐❝❛✳ ❊st❛ ♦♣✐♥✐ã♦ é✱ ✉♠ sé❝✉❧♦ ♠❛✐s t❛r❞❡✱ ❛✐♥❞❛ ❝♦♠♣❛rt✐❧❤❛❞❛ ♣♦r ❊✉❧❡r✱ ❛♣❡s❛r ❞❡ ❡st❡ t❡r ❝♦♥tr✐❜✉í❞♦ ❞❡ ❢♦r♠❛ s✐❣✲ ♥✐✜❝❛t✐✈❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❞❛s ❢✉♥çõ❡s ❝♦♠♣❧❡①❛s✳ ❉❡ ❢❛t♦✱ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s só ❝♦♠❡ç❛♠ ❛ s❡r ❛❝❡✐t♦s ❝♦♠♦ ♥ú♠❡r♦s ❞❡ ✧♣❧❡♥♦ ❞✐r❡✐t♦✧ ❡♠ ♣❧❡♥♦ sé❝✉❧♦ ❳■❳✱ q✉❛♥❞♦ s✉r❣❡ ❛ ✐❞❡✐❛ ❞❡ ♦s ✐❞❡♥t✐✜❝❛r ❝♦♠ ♣♦♥t♦s ❞♦ ♣❧❛♥♦✳

❊♠ ✶✼✾✼✱ ♦ ♥♦r✉❡❣✉ês ❈❛s♣❛r ❲❡ss❡❧ ✭✶✼✹✺✲✶✽✶✽✮ ❛♣r❡s❡♥t❛ à ❘❡❛❧ ❆❝❛❞❡ ♠✐❛ ❞❡ ❈✐ê♥❝✐❛s ❉✐♥❛♠❛rq✉❡s❛ ✉♠ ❛rt✐❣♦ ✐♥t✐t✉❧❛❞♦ ✧ ❖♠ ❉✐r❡❦t✐♦♥❡♥s ❛♥❛✲ ❧②t✐s❦❡ ❇❡t❡❣♥✐♥❣✧ ✭✧❙♦❜r❡ ❛ ❘❡♣r❡s❡♥t❛çã♦ ❆♥❛❧ít✐❝❛ ❞❡ ❉✐r❡çã♦✧✮✱ ♥♦ q✉❛❧ ❞❡s❝r❡✈❡ ♣♦r♠❡♥♦r✐③❛❞❛♠❡♥t❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ ♥ú♠❡r♦s ❝♦♠✲ ♣❧❡①♦s✳ ❚❛❧ ❛rt✐❣♦ é ♣✉❜❧✐❝❛❞♦ ✭ ❡♠ ❞✐♥❛♠❛rq✉ês✮ ♥❛s ▼❡♠ór✐❛s ❞❛ r❡❢❡r✐❞❛ ❆❝❛❞❡♠✐❛✱ ❡♠ ✶✼✾✾✱ ♠❛s ♣❡r♠❛♥❡❝❡ t♦t❛❧♠❡♥t❡ ❞❡s❝♦♥❤❡❝✐❞♦ ❛té à s✉❛ tr❛❞✉✲ çã♦ ♣❛r❛ ❢r❛♥❝ês✱ ❝❡r❝❛ ❞❡ ❝❡♠ ❛♥♦s ♠❛✐s t❛r❞❡✳ ❊♥tr❡t❛♥t♦✱ ❛ ✐❞❡✐❛ é ❛tr✐❜✉í❞❛ ❛♦ s✉íç♦ ❏❡❛♥ ❆r❣❛♥❞✱ q✉❡ ❛ ❛♣r❡s❡♥t❛✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ❡♠ ✶✽✵✻✳ ❉❡s❞❡ ❛í✱ ❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ é ❝♦♥✲ ❤❡❝✐❞❛ ✈✉❧❣❛r♠❡♥t❡ ♣♦r ❞✐❛❣r❛♠❛ ❞❡ ❆r❣❛♥❞✳

➱✱ ♥♦ ❡♥t❛♥t♦✱ ❛♣❡♥❛s ❝♦♠ ❛ ♣✉❜❧✐❝❛çã♦ ❡♠ ✶✽✸✶ ❞❡ ✉♠ tr❛❜❛❧❤♦ ❞❡ ●❛✉ss✱ ♥♦ q✉❛❧ é ❢❡✐t♦ ✉♠ ❡st✉❞♦ ♣♦r♠❡♥♦r✐③❛❞♦ ❞❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ q✉❡ ❡st❡s ❝♦♠❡ç❛♠ ❛ ❣❛♥❤❛r ❝❡rt❛ r❡s♣❡✐t❛❜✐❧✐❞❛❞❡✱ ❡ s❡ ❛❝❡✐t❛ q✉❡✱ ❞❡ ❢❛t♦✱ ♥ã♦ ❤á ♥❛❞❛ ❞❡ ✧✐♠❛❣✐♥ár✐♦✧ ❛❝❡r❞❛ ❞❡❧❡s✳ ➱ ❛ ●❛✉ss q✉❡ s❡ ❞❡✈❡ t❛♠❜é♠ ❛ ✐♥tr♦❞✉çã♦ ❞❛ ❞❡s✐❣♥❛çã♦ ✧♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✧✳

❋✐♥❛❧♠❡♥t❡✱ ❡♠ ✶✽✸✼✱ q✉❛s❡ três sé❝✉❧♦s ❞❡♣♦✐s ❞♦ s❡✉ ❛♣❛r❡❝✐♠❡♥t♦ ❝♦♠ ❈❛r❞❛♥♦✱ ❍❛♠✐❧t♦♥ ♣✉❜❧✐❝❛ ❛ ❞❡✜♥✐çã♦ ❢♦r♠❛❧ ❡ ❝♦♠♣❧❡t❛ ❞♦ s✐st❡♠❛ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❝♦♠♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s ❞❡ ♥ú♠❡r♦s r❡❛✐s ❝♦♠ ❞✉❛s ♦♣❡r❛çõ❡s ✭ ✉♠❛ ❛❞✐çã♦ ❡ ✉♠❛ ♠✉❧t✐♣❧✐❝❛çã♦✮ ❜❡♠ ❞❡✜♥✐❞❛s✳

❯♠❛ ✈❡③ ❛❝❡✐t♦s t♦t❛❧♠❡♥t❡ ❡st❡s ♥ú♠❡r♦s✱ ❛ ❛♥á❧✐s❡ ❝♦♠♣❧❡①❛✱ ♦✉ s❡❥❛✱ ♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s ❝♦♠♣❧❡①❛s ❞❡ ✈❛r✐á✈❡❧ ❝♦♠♣❧❡①❛✱ ❞❡s❡♥✈♦❧✈❡✉✲s❡ ❞❡ ✉♠❛ ❢♦r♠❛ ❡①tr❡♠❛♠❡♥t❡ rá♣✐❞❛ ♥♦ sé❝✉❧♦ ❳■❳✱ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❞❡✈✐❞♦ ❛♦s tr❛✲ ❜❛❧❤♦s ❞❡ ❈❛✉❝❤②✳ ❆✐♥❞❛ ❞✉r❛♥t❡ ❡st❡ sé❝✉❧♦✱ ❡st❛ t❡♦r✐❛ ✈❛✐ s❡r ❛♣r♦❢✉♥❞❛❞❛ ❛ ❛❧❛r❣❛❞❛✱ ❝♦♠ ♠❛t❡♠át✐❝♦s t❛✐s ❝♦♠♦ ❉✐r✐❝❤❧❡t✱ ❲❡✐❡rstr❛ss ❡ ❘✐❡♠❛♥♥✳ ◆♦ sé❝✉❧♦ ❳❳ ♠✉✐t♦s ♠❛t❡♠át✐❝♦s ❝♦♥t✐♥✉❛r❛♠ ❛ ❞❡❞✐❝❛r✲s❡ ❛ ❡st❛ ❢❛s❝✐✲ ♥❛♥t❡ ár❡❛ ❞❛ ♠❛t❡♠át✐❝❛✱ ♦❜t❡♥❞♦ ✐♠♣♦rt❛♥t❡s ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ❡ ♥♦✈❛s ❛♣❧✐❝❛çõ❡s✳

✶✳✷ ❖ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s

◆❡st❛ s❡çã♦✱ ✈❡r❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ R2_, ♠✉♥✐❞♦ ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡

♠✉❧t✐♣❧✐❝❛çã♦ s❛t✐s❢❛③ ✉♠❛ sér✐❡ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❛s q✉❛✐s ♦ q✉❛❧✐✜❝❛ ❝♦♠♦ ✉♠ ❝♦r♣♦✳ ❊st❡ ❝♦r♣♦ é ✉s✉❛❧♠❡♥t❡ ❞❡♥♦t❛❞♦ ♣♦r C_,❡ ♦s s❡✉s ❡❧❡♠❡♥t♦s sã♦

❝❤❛♠❛❞♦s ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳

❉❛❞♦ ♦ ❝♦♥❥✉♥t♦R2 _{❞❡ t♦❞♦s ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ♣♦❞❡✲} ♠♦s ❞❡✜♥✐r ❞✉❛s ♦♣❡r❛çõ❡s ❜✐♥ár✐❛s ✲ ✉♠❛ ❛❞✐çã♦ ✭❞❡♥♦t❛❞❛ ♣❡❧♦ sí♠❜♦❧♦ ✰✮ ❡ ✉♠❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✭❞❡♥♦t❛❞❛ ♣❡❧♦ sí♠❜♦❧♦ · ✮ ✲ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

(x1, y1) + (x2, y2) = (x1+x2, y1+y2)

(x1, y1). (x2, y2) = (x1x2−y1y2, y1x2+x1y2)

❖❜s❡r✈❡ q✉❡ ❛ ❛❞✐çã♦ ❝♦♠♦ ❢♦✐ ❞❡✜♥✐❞❛ é ❛ ❛❞✐çã♦ ✉s✉❛❧ ❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧

R2_,♣❡❧♦ q✉❡✱ ❝♦♠♦ s❛❜❡♠♦s✱ ❣♦③❛ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❝♦♠✉t❛t✐✈❛ ❡ ❛ss♦❝✐❛t✐✈❛❀

❛❧é♠ ❞✐ss♦✱ ✭✵✱✵✮ é ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ♣❛r❛ ❡st❛ ♦♣❡r❛çã♦ ❡ t♦❞♦ ♣❛r ✭①✱②✮ t❡♠ ✐♥✈❡rs♦ ✭✲①✱✲②✮ t♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ s✐♠étr✐❝♦✳

◗✉❛♥t♦ à ♠✉❧t✐♣❧✐❝❛çã♦✱ ❢❛❝✐❧♠❡♥t❡ s❡ ♠♦str❛ q✉❡✱ t❛♠❜é♠ ❡❧❛✱ é ❝♦♠✉✲ t❛t✐✈❛ ❡ ❛ss♦❝✐❛t✐✈❛✱ q✉❡ ❡①✐st❡ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ✭♦✉ ✐❞❡♥t✐❞❛❞❡✮ ♣❛r❛ ❡st❛ ♦♣❡r❛çã♦ ✭ ♦ ♣❛r ✭✶✱✵✮✮ ❡ q✉❡ t♦❞♦ ♦ ❡❧❡♠❡♥t♦ ✭x, y) ₆= (0,0) t❡♠ ✐♥✈❡rs♦✱

❞❛❞♦ ♣♦r

x x2_+y2,−

y x2 _+y2

.

❋✐♥❛❧♠❡♥t❡✱ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ é ❞✐str✐❜✉t✐✈❛ ❡♠ r❡❧❛çã♦ à ❛❞✐çã♦✳ ❊①❡♠♣❧♦✳ ▼♦str❡ q✉❡ ❛ ♦♣❡r❛çã♦ ❞❡✜♥✐❞❛ ❡♠ C₌R2 _♣♦r

(x₁, y₁).(x₂, y₂) = (x₁x₂₋y₁y₂, x₁y₂+x₂y₁)

é ❝♦♠✉t❛t✐✈❛ ❡ s❛t✐s❢❛③ ✭✶✱✵✮✭①✱②✮ ❂ ✭①✱②✮✱ ♣❛r❛ t♦❞♦ (x, y)_∈R2 _:

❈♦♠✉t❛t✐✈❛✿ (x1, y1).(x2, y2) = (x1x2−y1y2, x1y2 +x2y1) P♦r ♦✉tr♦ ❧❛❞♦✱

(x₂, y₂).(x₁, y₁) = (x₂x₁₋y₂y₁, x₂y₁+x₁y₂).❈♦♠♣❛r❛♥❞♦ ❛s ❡①♣r❡ssõ❡s ❛❝✐♠❛ ♦❜t❡♠♦s ♦ q✉❡ q✉❡rí❛♠♦s ♠♦str❛r✳

❊❧❡♠❡♥t♦ ♥❡✉tr♦✿ ❚❡♠♦s (1,0).(x, y)❂(1x₋0y,1y+ 0x) = (x, y).

❊①❡♠♣❧♦✳ ▼♦str❡ q✉❡ ❛ ♦♣❡r❛çã♦ ❞❡✜♥✐❞❛ ♥♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r é t❛❧ q✉❡ s❡

(x, y)₆= (0,0),❡♥tã♦ ❡①✐st❡ (u, v)_∈Ct❛❧ q✉❡ ✭①✱②✮✳✭✉✱✈✮ ❂ ✭✶✱✵✮✳

■♥✈❡rs♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦✿ ❙❡ (x, y)₆= (0,0)❡♥tã♦ ♣♦❞❡♠♦s ❞❡✜♥✐r (u, v)❂

x x2_+y2,−

y x2_+y2

❡ ♦❜t❡♠♦s

(x, y) .(u, v) = (x, y).

x x2_+y2,−

y x2_+y2

x2 x2_+y2 −

−y2 x2_+y2,

−xy x2_+y2 +

xy x2 _+y2

= (1,0)

❆s ♣r♦♣r✐❡❞❛❞❡s ❛❝✐♠❛ ♠❡♥❝✐♦♥❛❞❛s ♣❡r♠✐t❡♠✲♥♦s✱ ♣♦rt❛♥t♦✱ ❛✜r♠❛r q✉❡ ♦ ❝♦♥❥✉♥t♦ R2_, ♠✉♥✐❞♦ ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡✜♥✐❞❛s

❛❝✐♠❛✱ ❝♦♥st✐t✉✐ ✉♠ ❝♦r♣♦✳ ❈♦♥s✐❞❡r❡ ❛❣♦r❛ ❛ ❛♣❧✐❝❛çã♦

α :R_−→C

x_7−→(x,0), ❡ ♥♦t❡✲s❡ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r ①✱② ∈R_, s❡ t❡♠

α(x+y) = (x+y,0) = (x,0) + (y,0) =α(x) +α(y)

❡

α(xy) = (xy,0) = (x,0).(y,0) =α(x).α(y).

✶✳✸ ❯♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛ ❡ ❢♦r♠❛ ❛❧❣é❜r✐❝❛

➱ ✉s✉❛❧ ❞❡♥♦t❛r ♦ ❝♦♠♣❧❡①♦ ✭✵✱✶✮ ♣❡❧♦ sí♠❜♦❧♦ ✐✳ ❯t✐❧✐③❛♥❞♦ ❡st❛ ♥♦t❛çã♦ ❡ ❢❛③❡♥❞♦ ❛ ✐❞❡♥t✐✜❝❛çã♦ ❛♥t❡r✐♦r ❞♦s ❝♦♠♣❧❡①♦s ❞❛ ❢♦r♠❛ ✭①✱✵✮ ❝♦♠ ♦s ❝♦rr❡✲ s♣♦♥❞❡♥t❡s ♥ú♠❡r♦s r❡❛✐s ①✱ t❡♠✲s❡

(x, y) = (x,0) + (0, y) = (x,0).(1,0) + (0, y).(0,1) = x.1 +y.i=x+yi. ❉❡ss❛ ❢♦r♠❛✱ t♦❞♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ✭①✱②✮ ♣♦❞❡ t❛♠❜é♠ s❡r ❞❡s✐❣♥❛❞♦ ♣♦r x✰yi.❊st❛ é ❛ ❝❤❛♠❛❞❛ ❢♦r♠❛ ❛❧❣é❜r✐❝❛ ❞❡ ③✱ s❡♥❞♦ ❡st❛ ❛ ♥♦t❛çã♦ q✉❡ ♣❛s✲ s❛r❡♠♦s ❛ ✉s❛r ♣❛r❛ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♥❡ss❡ tr❛❜❛❧❤♦✳ ➱✱ ❡♥tã♦✱ ✐♠❡❞✐❛t♦ r❡❝♦♥❤❡❝❡r q✉❡ ✭❞❡♥♦t❛♥❞♦ ♣♦r z2 ♦ ❝♦♠♣❧❡①♦ ③✳③✮ s❡ t❡♠

i2 =i.i= (0,1).(0,1) =₋1.

❉❛❞♦ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ③ ❂ ① ✰ ②✐ ❝❤❛♠❛♠♦s ❛ ① ❛ ♣❛rt❡ r❡❛❧ ❞❡ ③ ❡ ❛ ② ❞❡ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❡ ③✱ ❡ ❡s❝r❡✈❡♠♦s

Rez:=x Imz:=y,

❈♦♠ ♦ ❛✉①í❧✐♦ ❞❛ ❢♦r♠❛ ❛❧❣é❜r✐❝❛ ❞❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ③ ❡ ❝♦♥✲ s✐❞❡r❛♥❞♦ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s z₁ = (x₁+y₁i) ❡ z₂ = (x₂+y₂i), ♣♦❞❡♠♦s

r❡❡s❝r❡✈❡r ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ C ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

❆❞✐çã♦✿ ❛ s♦♠❛ z1 +z2 é ♦❜t✐❞❛ ♣❡❧❛s s♦♠❛s ❞❛s r❡s♣❡❝t✐✈❛s ♣❛rt❡ r❡❛❧ ❡ ✐♠❛❣✐♥ár✐❛✱

z₁+z₂ = (x₁+x₂) +i(y₁+y₂).

▼✉❧t✐♣❧✐❝❛çã♦✿ ❛♣❧✐❝❛♠♦s ❛ ❞✐str✐❜✉t✐✈✐❞❛❞❡ ❡ ❛❣r✉♣❛♠♦s ❛s ♣❛rt❡s r❡❛❧ ❡ ✐♠❛❣✐♥ár✐❛ ✭❧❡♠❜r❛♥❞♦ q✉❡ i2 =₋1✮

z1z2 = (x1+y1i)(x2+y2i)

=x₁x₂+ix₁y₂+iy₁x₂+i2y₁y₂

= (x1x2−y1y2) +i(x1y2 +y1x2)

❯♠❛ ✈❡③ ❞❡✜♥✐❞❛s ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠C_,♣♦❞❡♠♦s

❞❡✜♥✐r ❛s ♦♣❡r❛çõ❡s ❞❡ s✉❜tr❛çã♦ ❡ ❞✐✈✐sã♦ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❞❛ ♠❛♥❡✐r❛ ✉s✉❛❧✿ s❡ z1, z2 ∈C, t❡♠♦s

z_{1 −}z₂ :=z₁+ (₋z₂) e z1 z2

=z₁z₂−1 sez_{2 6}= 0.

❉❛❞♦s ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s z₁ = (x₁ +y₁i) ❡ z₂ = (x₂ +y₂i), ❞✐③❡♠♦s q✉❡z1 =z2,s❡ s✉❛s r❡s♣❡❝t✐✈❛s ♣❛rt❡ r❡❛❧ ❡ ✐♠❛❣✐♥ár✐❛ sã♦ ✐❣✉❛✐s✱ ♦✉ s❡❥❛✱ s❡ x₁ =x₂ ❡ y₁ =y₂.

❉❛❞♦ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z❂x✰yi, ❝❤❛♠❛♠♦s ❝♦♥❥✉❣❛❞♦ ❞❡ ③ ❡ ❞❡✲ ♥♦t❛♠♦s ♣♦r z ♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦

z :=x₋yi.

Pr♦♣♦s✐çã♦ ✶✳ ❙❡ z _∈C_, ❡♥tã♦ t❡♠♦s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ✿

✭✐✮ z = 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱z = 0;

✭✐✐✮ z =z✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ z _∈R_;

✭✐✐✐✮ (z+w) =z+w ✭✐✈✮ (z₋w) =z₋w ✭✈✮ z.w =z.w;

✭✈✐✮Re(z) = z+z

2 ❡ Im(z) =

z−z

2i

❉❡♠♦♥str❛çã♦✿ ❙❡♥❞♦ ③ ❂ ① ✰ ②✐ ❡ ✇ ❂ ❝ ✰ ❞✐ t❡♠♦s✿ ✭✐✮ z = 0_⇔x₋yi= 0_⇔x=y= 0 _⇔z = 0.

✭✐✐✮❚❡♠♦s

z =z _⇔x₋yi =x+yi _⇔2y ❂0_⇔ z❂x_∈R_.

✭✐✐✐✮

(z+w) = (x+yi) + (c+di) = (x+c) + (y+d)i= (x+c)₋(y+d)i= (x₋yi) + (c₋di) =z+w ✭✐✈✮

(z₋w) = (x+yi)₋(c+di) = (x₋c) + (y₋d)i= (x₋c)₋(y₋d)i= (x₋yi)₋(c₋di) =z₋w ✭✈✮

z.w = (xc₋yd) + (xd+yc)i= (xc₋yd)₋(xd+yc)i. P♦r ♦✉tr♦ ❧❛❞♦

z.w =xc₋xdi₋yci₋yd= (xc₋yd)₋(xd+yc)i, ♣r♦✈❛♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡✳

✭✈✐✮ ❚❡♠♦s

z+z

2 =

x+yi+x₋yi

2 =x=Re(z).

z+z

2i =

x+yi₋x+yi

2i =y=Im(z).

❉❛❞♦ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ③ ❂ ① ✰ ②✐✱ ❝❤❛♠❛♠♦s ♠ó❞✉❧♦ ♦✉ ✈❛❧♦r ❛❜✲ s♦❧✉t♦ ❞❡ ③ ❡ ❞❡♥♦t❛♠♦s ♣♦r ⑤③⑤ ♦ ♥ú♠❡r♦ ✭r❡❛❧ ♥ã♦ ♥❡❣❛t✐✈♦✮ ❞❛❞♦ ♣♦r

|z_|:=px2 _+y2_;

Pr♦♣♦s✐çã♦ ✷✿ ❙❡ z, w _∈C_, ❡♥tã♦ t❡♠♦s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ✿

✭✐✮z.z =_|z_|2, ♣❛r❛ t♦❞♦z _∈C_;

✭✐✐✮|z_|=_|z_|, ♣❛r❛ t♦❞♦_∈C_;

✭✐✐✐✮ Re(z)_{≤ |}Re(z)_{| ≤ |}z_| ❡Im(z)_{≤ |}Im(z)_{| ≤ |}z_|; (iv)⑤③✳✇⑤ ❂ ⑤③⑤✳⑤✇⑤✱ ♣❛r❛ q✉❛✐sq✉❡r z, w _∈C_.

✭✈✮ |z+w_{| ≤ |}z_|+_|w_|, ♣❛r❛ q✉❛✐sq✉❡r z, w _∈C ✭❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✮

✭✈✐✮ ||z_{| − |}w_{|| ≤ |}z_±w_|,♣❛r❛ q✉❛✐sq✉❡r z, w_∈C

❉❡♠♦♥str❛çã♦✿ ❙❡♥❞♦ ③ ❂ ① ✰ ②✐ ❡ w_∈C t❡♠♦s✿

✭✐✮z.z = (x+yi).(x₋yi) =x2+y2 =_|z_|2. ✭✐✐✮ |z_|=px2_+y2 ₌p_x2_{+ (−y)}2 _=|z|

✭✐✐✐✮ ❙❡♥❞♦ z =x+yi✱ ❡♥tã♦ Re(z) =x_{≤ |}x_|=_|Re(z)_| ❡

x_{≤ |}x_|=p_|x_|2 _≤p_x2_+y2 _=|z|.

✭✐✈✮ ❯s❛♥❞♦ ♣r♦♣♦s✐çã♦ ✷ ✭✐✮✱ ♣r♦♣♦s✐çã♦ ✶ ✭✈✮✱ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛ss♦❝✐❛t✐✈✐✲ ❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡✱ ♥♦✈❛♠❡♥t❡ ✭✐✮ ❞❛ ♣r♦♣♦s✐çã♦ ✷✱ t❡♠♦s✿

|z.w_|2 = (z.w)(z.w) = (z.w).(z.w) = (z.z).(w.w) =_|z_|2._|w_|2 = (_|z_|._|w_|)2.

✭✈✮ ❯s❛♥❞♦ ❛s ♣r♦♣♦s✐çõ❡s ✶ ❡ ✷ ❡ ❛ ❞✐str✐❜✉t✐✈✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❝♦♠ r❡❧❛çã♦ ❛ ❛❞✐çã♦ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ t❡♠♦s✿

|z+w_|2 = (z+w)z+w= (z+w)(z+w)

❂ ③✳z+z.w+w.z+w.w

❂ |z_|2+z.w+w.z+w.z+_|w_|2 ❚❡♠♦s ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✶✭✈✮ ❡ ✉s❛♥❞♦ ❛ ♣r♦♣♦s✐çã♦ ✷ q✉❡✿

z.w+zw = 2Re(zw)_≤2_|zw_|= 2_|z_||w_|=_|z_||w_| ❆ss✐♠✱

|z+w_|2 _{≤ |}z_|2+ 2_|z_||w_|+_|w_|2 = (_|z_|+_|w_|)2

❡ ♣♦rt❛♥t♦✱

|z+w_{| ≤ |}z_|+_|w_|

✭✈✐✮ ❊s❝r❡✈❡♥❞♦ z = (z₋w) +w ❡ w= (w₋z) +z ❡ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ ♦❜t❡♠♦s✿

|w_|=_|(w₋z) +z_{| ≤ |}w₋z_|+_|z_|=_|z₋w_|+_|z_|. ❞❛í ❝♦♠♦

|z_{| − |}w_{| ≤ |}z₋w_|e₋(_|z_{| − |}w_|)_{≤ |}z₋w_| ❡♥tã♦

||z_{| − |}w_{|| ≤ |}z₋w_|.

❆ ❞❡s✐❧❣✉❛❧❞❛❞❡ ||z_{| − |}w_{|| ≤ |}z+w_|♣♦❞❡ s❡r ♦❜t✐❞❛ ❢❛③❡♥❞♦ ❛s ♠♦❞✐✜❝❛çõ❡s ❝♦♥✈❡♥✐❡♥t❡s ❛❝✐♠❛✳

❊①❡♠♣❧♦✳ ❉❛❞♦s z1 = 2 + 4i ❡ z2 = 4−i, t❡♠♦s ✐✮z1+z2 = (2 + 4i) + (4−i) = 6 + 3i

✐✐✮z₁₋z₂ = (2 + 4i)₋(4₋i) =₋2 + 5i

✐✐✐✮z1z2 = (2 + 4i)(4−i) = 8−2i+ 16i−4i2 = 12 + 14i ✐✈✮ z1

z2 = 2+4i

(4−i) (4−i) (4−i) =

8+2i+16i+4i2 16−i2 = 4

17 + 18 17i

❊①❡♠♣❧♦✳ ❙❡ z =x+yi, z ₆= 0, ❞❡t❡r♠✐♥❡ ❛ ❢♦r♠❛ ❛❧❣é❜r✐❝❛ ❞♦ ✐♥✈❡rs♦ ❞❡ ③✳

❆ r❛③ã♦ 1

z é ♦❜t✐❞❛ ♠✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ♦ ♥✉♠❡r❛❞♦r ❡ ♦ ❞❡♥♦♠✐♥❛❞♦r ♣❡❧♦ ❝♦♥✲

❥✉❣❛❞♦ ❞♦ ❞❡♥♦♠✐♥❛❞♦r✱ ✐st♦ é✱

1

z =

1

z z z =

x₋yi

(x+yi)(x₋yi) =

x₋yi x2 _+y2 =

x x2_+y2 −

yi x2_+y2 ❊①❡♠♣❧♦✳ ❉❛❞♦s z1 =x1+y1i ❡ z2 =x2+y2i, ❞❡t❡r♠✐♥❡ ❛ ❞✐✈✐sã♦ z_z12. ❆ ❞✐✈✐sã♦ é ♦❜t✐❞❛ ♠✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ♦ ♥✉♠❡r❛❞♦r ❡ ♦ ❞❡♥♦♠✐♥❛❞♦r ♣❡❧♦ ❝♦♥✲ ❥✉❣❛❞♦ ❞♦ ❞❡♥♦♠✐♥❛❞♦r✱ ✐st♦ é✱

z₁ z2

= z1z2

z2z2

= (x1 +y1i)(x2 −y2i) (x2 +y2i)(x2 −y2i)

= x1x2+y1y2

x22 +y22

+ix2y1−x1y2 x22+y22

❊①❡♠♣❧♦✳ ❖❜t❡♥❤❛ ♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ③ t❛❧ q✉❡z+ 2z₋i= 6 + 3i.❙❡♥❞♦ z =x+yi,❡♥tã♦ z =x₋yi ❡ ❡♥tã♦ t❡♠♦s✿

x₋yi+ 2(x+yi)₋i+ 6 + 3i

3x+yi₋i= 6 + 3i

3x+ (y₋1)i= 6 + 3i.

▲♦❣♦✱ ♣❡❧❛ ✐❣✉❛❧❞❛❞❡ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❞❡✈❡♠♦s t❡r ✸① ❂ ✻ ❡ ② ✲ ✶ ❂ ✸✳ ❉❛í✱ ① ❂ ✷ ❡ ② ❂ ✹✱ ♣♦rt❛♥t♦ z = 2 + 4i

❊①❡♠♣❧♦✳ ❖❜t❡♥❤❛ ♦ ✈❛❧♦r r❡❛❧ ❞❡ ❛ ♣❛r❛ q✉❡ ♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z = _a2+₊i_i

s❡❥❛ r❡❛❧✳ ❚❡♠♦s q✉❡

2 +i a+i =

2 +i a+i

(a₋i) (a₋i) =

2a₋2i+ai+ 1

a2_{+ 1} =

2a+ 1

a2_{+ 1} +

(₋2 +a)i a2_{+ 1} . ❈♦♠♦ ♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ❞❡✈❡ s❡r r❡❛❧✱ ❞❡✈❡♠♦s t❡r a−2

a2

+1 = 0❡ ❝♦♠♦a 2₊₁₆₌

0, ❡♥tã♦ ❛ ✲ ✷ ❂ ✵✱ ✐st♦ é✱ ❛ ❂ ✷✳

❊①❡♠♣❧♦✳ ❙❡❥❛♠ ❞❛❞♦s z1, z2, ..., zn ∈C,♥ã♦ ♥✉❧♦s✳ ▼♦str❡ q✉❡

|z1+z2+...+zn| ≤ |z1|+|z2|+...+|zn|.

❉❡♠♦♥str❛çã♦✿ ❋❛ç❛♠♦s ✐♥❞✉çã♦ s♦❜r❡ ♥✳ P❛r❛ ♥ ❂ ✶✱ ❞❡ ❢❛t♦✱|z_{1| ≤ |}z_1|. ❙✉♣♦♥❤❛ q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✈❛❧❡ ♣❛r❛ ✉♠ ❝❡rt♦ ♥ ❡ ♣r♦✈❡♠♦s q✉❡ ❝♦♥✲ s❡q✉❡♥t❡♠❡♥t❡ ✈❛❧❡ ♣❛r❛ ♥ ✰ ✶✳ ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ♣❛r❛ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡ ❡♠ s❡❣✉✐❞❛ ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦s q✉❡✿

|z1+z2+...+zn+zn+1| ≤ |z1+z2+...+zn|+|zn+1| ≤ |z1|+|z2|+...+|zn|+|zn+1|. ▲♦❣♦✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✈❛❧❡ ♣❛r❛ t♦❞♦ n_∈N_.

✶✳✹ P❧❛♥♦ ❝♦♠♣❧❡①♦

❋✐①❛❞♦ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ♥✉♠ ♣❧❛♥♦✱ ❛ ❝❛❞❛ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z =x+yi ♣♦❞❡rá ❛ss♦❝✐❛r✲s❡✱ ❞❡ ♠♦❞♦ ú♥✐❝♦✱ ♦ ♣♦♥t♦ ❞❡ ❝♦♦r❞❡✲ ♥❛❞❛s ✭①✱②✮ ❂ ✭❘❡ ③✱ ■♠ ③✮ ❞❡ss❡ ♣❧❛♥♦ ✭♦✉ s❡ ♣r❡❢❡r✐r♠♦s✱ ♦ ✈❡t♦r q✉❡ ✉♥❡ ❛ ♦r✐❣❡♠ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛ ❡ss❡ ♣♦♥t♦✮✳ ❊st❛❜❡❧❡❝❡✲s❡✱ ❛ss✐♠✱ ✉♠❛ ❜✐❥❡çã♦ ❡♥tr❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❡ss❡ ♣❧❛♥♦✳ ❖s ♥ú♠❡r♦s r❡❛✐s ✭♦✉ s❡❥❛✱ ♦s ❝♦♠♣❧❡①♦s ❞❡ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ♥✉❧❛✮ ❝♦rr❡s♣♦♥❞❡♠ ❛♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✱ ♦ q✉❛❧ é ❞❡s✐❣♥❛❞♦ ♣♦r ❡✐①♦ r❡❛❧❀ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ♦s ❝♦♠♣❧❡①♦s ❞❛ ❢♦r♠❛ ✐② ✭✐st♦ é✱ ♦s ❝♦♠♣❧❡①♦s ❞❡ ♣❛rt❡ r❡❛❧ ♥✉❧❛✮✲ ❝❤❛♠❛❞♦s ✐♠❛❣✐♥ár✐♦s ♣✉r♦s ✲ ❝♦rr❡s♣♦♥❞❡♠ ❛♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s✱ ♦ q✉❛❧ ❝♦st✉♠❛ s❡r ❞❡s✐❣♥❛❞♦ ♣♦r ❡✐①♦ ✐♠❛❣✐♥ár✐♦✳ ◗✉❛♥❞♦ ♦ ♣❧❛♥♦ ①② é ✉t✐❧✐③❛❞♦✱ ❞❡st❡ ♠♦❞♦✱ ♣❛r❛ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ é ✉s✉❛❧ ❝❤❛♠❛r✲❧❤❡ ♣❧❛♥♦ ❝♦♠✲ ♣❧❡①♦ ♦✉ ♣❧❛♥♦ ❞❡ ❆r❣❛♥❞✲●❛✉ss ▼✉✐t❛s ✈❡③❡s✱ ✐❞❡♥t✐✜❝❛♠♦s ❝♦♠♣❧❡t❛♠❡♥t❡ ♦ ❝♦♥❥✉♥t♦ C ❝♦♠ ❡st❡ ♣❧❛♥♦ ❡ r❡❢❡r✐♠♦✲♥♦s ❛ C ❝♦♠♦ ♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✱

❢❛❧❛♥❞♦ ♥♦ ♣♦♥t♦ z =x+yi.

✷ P♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s

◆❡st❛ s❡çã♦ ✈❡r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ C_, ❛

❞❡✜♥✐çã♦ ❞❡ ❣r❛✉ ❡ ♠❛✐s ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❡♥✈♦❧✈❡♥❞♦ ❡ss❡s ♣♦❧✐♥ô♠✐♦s✳

❯♠ ♣♦❧✐♥ô♠✐♦ ♣✭③✮ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ Cé ✉♠❛ ❡①♣r❡ssã♦ ❢♦r♠❛❧ ❞♦ t✐♣♦

p(z) = a_nzn+a_n−1zn−1+...+a₁z+a₀ =

n

X

j=0 a_jzj

♦♥❞❡ n _∈ N_{, aj ∈} C_, ♣❛r❛ _{0 ≤ j ≤ n} ❡ ❛ ✈❛r✐á✈❡❧ ③ ♣♦❞❡ ❛ss✉♠✐r q✉❛❧q✉❡r

✈❛❧♦r ❝♦♠♣❧❡①♦✳

❖s ❡❧❡♠❡♥t♦saj ♣❛r❛ 0≤ j ≤n sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝♦❡✜❝✐❡♥t❡s ❞❡ ♣✭③✮✱

❛s ♣❛r❝❡❧❛s a_jzj _{❞❡ t❡r♠♦s s❡♥❞♦a}

0 ♦ t❡r♠♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ③ ❡ ♦s t❡r♠♦s a_jzj t❛✐s q✉❡ a_{j 6}= 0,❞❡ t❡r♠♦s ❞❡ ❣r❛✉ ❥ ❞♦ ♣♦❧✐♥ô♠✐♦ ♣✭③✮✳

❈❤❛♠❛♠♦s p(z) =a0, ❝♦♠ a0 ∈C, ❞❡ ♣♦❧✐♥ô♠✐♦ ❝♦♥st❛♥t❡✳ ❙❡ p(z) = 0, ❝❤❛♠❛♠♦s ♣✭③✮ ❞❡ ♣♦❧✐♥ô♠✐♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ ❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r t♠❛❜é♠ p(z)_≡0. ❊st❡ ♣♦❧✐♥ô♠✐♦ ♣♦❞❡rá s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛

p(z) = 0zn+ 0zn−1+...+ 0z+ 0,_∀n _∈N_.

❙❡ ♣✭③✮ ❢♦r ✉♠ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦✱ ♦✉ s❡❥❛✱ p(z)₆= 0, ❡♥✲ tã♦ ❛❧❣✉♠ ❝♦❡✜❝✐❡♥t❡ ❞❡✈❡ s❡r ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ ❡ ❞❛í ❤❛✈❡rá ✉♠ ♠❛✐♦r í♥❞✐❝❡ ♥ t❛❧ q✉❡ a_{n 6}= 0✳ ❉❡✜♥✐♠♦s ♦ ❣r❛✉ ❞❡ ♣✭③✮ ❝♦♠♦ s❡♥❞♦ ❡st❡ ♥ú♠❡r♦ ♥ ❡

♦ ❞❡♥♦t❛♠♦s ♣♦r ❣r✭♣✭③✮✮✳ ◆❡st❡ ❝❛s♦✱ an é ❝❤❛♠❛❞♦ ❞❡ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r

❞❡ ♣✭③✮✳ ❈❤❛♠❛r❡♠♦s ❞❡ ♣♦❧✐♥ô♠✐♦s ♠ô♥✐❝♦s ❛q✉❡❧❡s ❞❡ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r a_n= 1.

❖❜s❡r✈❛çã♦✿ ◆ã♦ ❞❡✜♥✐♠♦s ❣r❛✉ ❞❡ ♣♦❧✐♥ô♠✐♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦p(z)_≡0. ❊①❡♠♣❧♦✳ ❙❡❥❛ p(z) = z2 ₋5z + 2. ❚❡♠♦s ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s ✶✱ ✲✺ ❡ ✷✳

❊①❡♠♣❧♦✳ ❙❡❥❛ p(z) = z2 ₋2iz ₋i. ❚❡♠♦s ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s ✶✱ ✲✷✐ ❡ ✲✐✳

❊①❡♠♣❧♦✳ ❙❡❥❛ ♦ ♣♦❧✐♥ô♠✐♦ ❝♦♥st❛♥t❡ f(z) = 2, t❡♠♦s q✉❡ ❣r ✭❢✭③✮✮ ❂ ✵✱ ✉♠❛ ✈❡③ q✉❡ ❢ ♥ã♦ é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛✳

❊①❡♠♣❧♦✳ ❙❡❥❛ ♦ ♣♦❧✐♥ô♠✐♦ f(z) = z3 _{−2z+ 4, t❡♠♦s q✉❡ ❢ é ♠ô♥✐❝♦ ❡ ❣r} ✭❢✭③✮✮ ❂ ✸✳

❙❡❥❛ ♦ ♣♦❧✐♥ô♠✐♦ ♣✭③✮ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s t❛❧ q✉❡ n _∈ N_{, aj ∈} C_,

♣❛r❛ 0_≤j _≤n,

p(z) = a_nzn+a_n−1zn−1+...+a₁z+a₀ ❡ s❡❥❛ z0 ∈C. ❉❡✜♥✐♠♦s ❛ ❛✈❛❧✐❛çã♦ ❞❡ ♣✭③✮ ❡♠ z0 ❝♦♠♦ s❡♥❞♦

p(z0) =anz0n+an−1z0n−1+...+a1z0+a0 ∈C. ❙❡ p(z₀) = 0, ❞✐③❡♠♦s q✉❡ z₀ é ✉♠❛ r❛✐③ ❞❡ ♣✭③✮✳

❈♦♥s✐❞❡r❡ ❞♦✐s ♣♦❧✐♥ô♠✐♦s ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s ❢ ❡ ❣✳ ❚❡♠♦s q✉❡ ❢ ❡ ❣ s❡rã♦ ✐❣✉❛✐s ✭♦✉ s❡❥❛✱ sã♦ t❛✐s q✉❡f(z) =g(z)♣❛r❛ t♦❞♦ ③_∈C✮✱ s❡ s✉❛ ❞✐❢❡r✲

❡♥ç❛ ❢ ✲ ❣ ❢♦r ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛✳ ❈♦♥t✉❞♦✱ ✐ss♦ ❛❝♦♥t❡❝❡ s♦♠❡♥t❡ s❡ t♦❞♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❢ ✲ ❣ sã♦ ♥✉❧♦s❀ ❧♦❣♦✱ ❞♦✐s ♣♦❧✐♥ô♠✐♦s ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s ❢ ❡ ❣ sã♦ ✐❣✉❛✐s s❡ ✱ ❡ s♦♠❡♥t❡ s❡✱ ❢ ❡ ❣ tê♠ ❝♦❡✜❝✐❡♥t❡s r❡s♣❡❝t✐✈❛♠❡♥t❡ ✐❣✉❛✐s✳ ❊①❡♠♣❧♦✳ ❙❡❥❛ ♦ ♣♦❧✐♥ô♠✐♦ f(z) = z2 +z + 1₋ i. ❖❜s❡r✈❡ q✉❡ f(i) =

i2_{+i+ 1−i=−1 +i+ 1−i= 0. ▲♦❣♦✱ ✐ é r❛✐③ ❞❡ ❢✳}

❊①❡♠♣❧♦✳ ❙❡❥❛ f(z) = z4+ 2z2+ 1. ❙❡ ❢ é ✉♠ ♣♦❧✐♥ô♠✐♦ t❛❧ q✉❡ z _∈R_,∀z,

❡♥tã♦✱ f(z) = (z2_{+ 1)}2 _{>0 ∀z ∈R, ♦ q✉❡ ♠♦str❛ q✉❡ ❢ ♥ã♦ t❡♠ r❛í③❡s r❡❛✐s✳} ❈♦♥t✉❞♦✱ ❢ é ♣♦❧✐♥ô♠✐♦ ❞❡✜♥✐❞♦ ❡♠C❡ ❡♠ ♣❛rt✐❝✉❧❛r_{f(±i) = [(±i)}2_{+ 1]}2 ₌

[₋1 + 1]2 _{= 0✳ P♦rt❛♥t♦✱ ±isã♦ r❛í③❡s ❝♦♠♣❧❡①❛s ❞♦ ♣♦❧✐♥ô♠✐♦ ❢✳}

❊①❡♠♣❧♦✳ ❖s ♣♦❧✐♥ô♠✐♦s f(z) = 3z4₋2z2 +z₋4 +i ❡ g(z) = z₋2z2+ 3z4₋4 +isã♦ ✐❣✉❛✐s✱ ✈✐st♦ q✉❡ ♦s s❡✉s ❝♦❡✜❝✐❡♥t❡sai ❞❛s ✐✲és✐♠❛s ♣♦tê♥❝✐❛s

❞❡ zi _{sã♦ ✿ a}

0 =i, a1 =−4, a2 = 1, a3 =−2, a4 = 3.

❊①❡♠♣❧♦✳ ❖s ♣♦❧✐♥ô♠✐♦s f(z) = z3 ₋3z+ 1 ❡ g(z) = z3+z₋3 sã♦ ❞✐❢❡r✲

❡♥t❡s✱ ✉♠❛ ✈❡③ q✉❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❢✭③✮ sã♦ a₃ = 1, a₁ =₋3 ❡ a₀ = 1 ❡ ♦s

❝♦❡✜❝✐❡♥t❡s ❞❡ ❣✭③✮ sã♦ b3 = 1, b1 = 1 ❡ b0 = −3 ❡ ❞❛í ♥ã♦ ♦❝♦rr❡ aj = bj,

♣❛r❛ 0_≤j _≤3.

✷✳✶ P♦❧✐♥ô♠✐♦s ❡ ❖♣❡r❛çõ❡s

✷✳✶✳✶ ❆❞✐çã♦ ❞❡ P♦❧✐♥ô♠✐♦s

◆♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ❛ ♣❛rt✐r ❞❛ ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦ ❞❡ C_.

❉❛❞♦s ❞♦✐s ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s ❢✭③✮ ❡ ❣✭③✮ ❝♦♠ ♥ ❃ ♠✳

f(z) =

n

X

i=0

a_izi e g(z) =

m

X

i=0 b_izi

❆♣ós r❡❡s❝r❡✈❡r ❢✭③✮ ❡ ❣✭③✮ ❝♦♠ ❛s ♠❡s♠❛s ♣♦tê♥❝✐❛s ❞❡ ③✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ ♠ ❂ ♥ ❡ ❛ss✐♠✱ ❞❡✜♥✐♠♦s ❛ ❛❞✐çã♦ ❞❡ss❡s ♣♦❧✐♥ô♠✐♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

f(z) +g(z) = (a_n+b_n)zn+...+ (a₂+b₂)z2+ (a₁+b₁)z+ (a₀+b₀)

♦✉ s❡❥❛

f(z) +g(z) =

n

X

i=0

c_izi, onde c_i =a_i+b_i, para 0_≤i_≤n, ❝✉❥♦ r❡s✉❧t❛❞♦ ❝❤❛♠❛✲s❡ s♦♠❛ ❞❡ ❢ ❝♦♠ ❣✳

❊①❡♠♣❧♦✳ ❙♦♠❛r f(z) = z2+ 3z+ 4 ❡g(z) = z4+ 3z2+ 5.❚❡♠♦s

(f +g)(z) = (0 + 1)z4+ (0 + 0)z3+ (1 + 3)z2+ (3 + 0)z+ (4 + 5)

P❛r❛ ❛ ♦♣❡r❛çã♦ ❛❞✐çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ❣r❛✉✿ s❡ f(z)₆= 0, g(z)= 0₆ , f(z) +g(z)₆= 0,

❡♥tã♦

gr(f(z) +g(z))_≤max_{gr(f(z)), g(z))_}, ✈❛❧❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡♠♣r❡ q✉❡ gr(f(z))₆=gr(g(z))

❉❡ ❢❛t♦✱ ❝♦♠♦ ♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r é ♦ ❝♦❡✜❝✐❡♥t❡ ❞♦ t❡r♠♦ ❞❡ ♠❛✐s ❛❧t♦ ❣r❛✉ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦✱ ❡♥tã♦✱ s✉♣♦♥❞♦ ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ ❣r ❢ ❁ ❣r ❣✱ t❡♠♦s q✉❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡ ❢ ✰ ❣ é ♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡ ❣✱ ♦ q✉❡ ♠♦str❛ q✉❡ s❡ gr f ₆=gr g, ❡♥tã♦

gr(f(z) +g(z)) =max_{gr(f(z)), g(z))_},

❙❡ ❣r ❢ ❂ ❣r ❣✱ ♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡ ❢ ✰ ❣ é ❛ s♦♠❛ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❧í❞❡r❡s ❞❡ ❢ ❡ ❣✳ ❈♦♠♦ ❡ss❛ s♦♠❛ ♣♦❞❡ s❡r ♥✉❧❛ ✭❜❛st❛ q✉❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡ ❢ s❡❥❛ ♦♣♦st♦ ❛♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡ ❣✮✱ ♦ ❣r❛✉ ❞❡ ❢ ✰ ❣ ♣♦❞❡ ❝❛✐r ❡ ❞❛í t❡♠♦s q✉❡

gr(f(z) +g(z))_≤max_{gr(f(z)), g(z))_}.

❆ ❛❞✐çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s t❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦✲ ♣r✐❡❞❛❞❡s✱ ♣❛r❛ q✉❛✐sq✉❡r ❢✭③✮✱ ❣✭③✮ ❡ ❤✭③✮✿

• ✭❆ss♦❝✐❛t✐✈❛✮ ((f(z) +g(z)) +h(z) = (f(z) + (g(z) +h(z)); • ✭❈♦♠✉t❛t✐✈❛✮ f(z) +g(z) =g(z) +f(z);

• ✭❊①✐stê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❛❞✐t✐✈♦✮ ❖ ♣♦❧✐♥ô♠✐♦ ♥✉❧♦ é t❛❧

q✉❡ ❢✭③✮ ❂ ✵ ✰ ❢✭③✮ ♣❛r❛ t♦❞♦ ❢✭③✮ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s✳

• ✭❊①✐stê♥❝✐❛ ❞❡ s✐♠étr✐❝♦✮ ❉❛❞♦f(z) = a0+a1z+...anzn,♦ s✐♠étr✐❝♦

❞❡ ❢✭③✮ é ♦ ♣♦❧✐♥ô♠✐♦

−f(z) = (₋a₀) + (₋a₁)z+...(₋a_n)zn.

❊①❡♠♣❧♦✳ ❉❡♠♦♥str❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❛ss♦❝✐❛t✐✈❛ ❞❛ ❛❞✐çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ C_.

❈♦♥s✐❞❡r❡ ♦s ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ C_,

f(z) =

n

X

i=0

aizi, g(z) = m

X

i=0

bizi e h(z) = l

X

i=0 cizi

❯♠❛ ✈❡③ q✉❡ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❢✭③✮✱ ❣✭③✮ ❡ ❤✭③✮ ❝♦♠ ❛s ♠❡s♠❛s ♣♦tê♥✲ ❝✐❛s ❞❡ ③✱ ❡♥tã♦ ♣♦❞❡♠♦s s✉♣♦r q✉❡ ♥ ❂ ♠ ❂ ❧ ❡ t❡♠♦s

((f(z) +g(z)) +h(z) =

n

X

i=0

(a_i+b_i)zi+

n

X

i=0 c_izi

=

n

X

i=0

((ai+bi) +ci)zi

=

n

X

i=0

(ai+ (bi+ci))zi

=

n

X

i=0

aizi+ n

X

i=0

(bi+ci)zi

= (f(z) + (g(z) +h(z)). ❆s ❞❡♠❛✐s ❞❡♠♦♥str❛çõ❡s sã♦ ❝♦rr✐q✉❡✐r❛s✳

✷✳✶✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ P♦❧✐♥ô♠✐♦s

❈♦♥s✐❞❡r❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ♦♣❡r❛çã♦ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ❛ ♣❛rt✐r ❞❛ ♦♣❡r❛çã♦ ❞❡ ♠✉❧✲ t✐♣❧✐❝❛çã♦ ❞❡ C_.

❉❛❞♦s ❞♦✐s ♣♦❧✐♥ô♠✐♦s ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s

f(z) =

n

X

i=0

aizi e g(z) = m

X

i=0 bizi

❉❡✜♥✐♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ss❡s ♣♦❧✐♥ô♠✐♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

f(z).g(z) = (a₀+a₁z+a₂z2+...+a_nzn).(b₀+b₁z+b₂z2+...+b_mzm) = =a0b0 + (a0b1+a1b0)z+ (a2b0 +a1b1+a0b2)z2+...+anbmzm+n

✐st♦ é✱

f(z).g(z) =

n+m

X

i=0

cizi, onde

c0 =a0b0

c₁ =a₀b₁+a₁b₀

c2 =a2b0+a1b1+a0b2 . . .

cj =a0bj+a1bj−1+...+ajb0 =

X

j+k=i

aj.bk

. . .

c_m+n=a_n.b_m

◆♦t❡♠♦s ❛✐♥❞❛ q✉❡ ❢❣ ♣♦❞❡ s❡r ♦❜t✐❞♦ ♠✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ❝❛❞❛ t❡r♠♦ aizi

❞❡ ❢ ♣♦r ❝❛❞❛ t❡r♠♦ b_jzj ❞❡ ❣✱ s❡❣✉♥❞♦ ❛ r❡❣r❛ (a_izi).(b_jzj) = a_ib_jzi+j✱ ❡ s♦♠❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✳

P❛r❛ ❛ ♦♣❡r❛çã♦ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ✈❛❧❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ♠✉❧t✐✲ ♣❧✐❝❛t✐✈❛ ❞♦ ❣r❛✉✿ s❡ f(z)₆= 0, g(z) ₆= 0 ❡♥tã♦ f(z).g(z) ₆= 0, ❡ s❡♥❞♦ ❣r ❢✭③✮ ❂ ♥ ❡ ❣r ❣✭③✮ ❂ ♠✱ ❡♥tã♦ ✈❛❧❡

gr(f(z).g(z))❂gr(f(z))✰gr(g(z))

❉❡ ❢❛t♦✱ t❡♠♦s

c_m+n=a_n.b_{m 6}= 0

ck = 0,∀k > m+n

❡♥tã♦

gr(f.g) ❂m+n❂gr(f)✰gr(g)

❆ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s t❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✱ ♣❛r❛ q✉❛✐sq✉❡r ❢✭③✮✱ ❣✭③✮ ❡ ❤✭③✮✿

• ✭❆ss♦❝✐❛t✐✈❛✮ (f(z).g(z)).h(z) =f(z).(g(z).h(z)) • ✭❈♦♠✉t❛t✐✈❛✮ f(z).g(z) = g(z).f(z)

• ✭❉✐str✐❜✉t✐✈❛✮ f(z)(g(z) +h(z)) =f(z)g(z) +f(z)h(z);

• ✭❊①✐stê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✮ ❖ ♣♦❧✐♥ô♠✐♦ ❝♦♥st❛♥t❡ ✶ é t❛❧ q✉❡

✶✳❢✭③✮ ❂ ❢✭③✮✱ ♣❛r❛ t♦❞♦ ❢✭③✮ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s✳ ❊①❡♠♣❧♦✳ ▼✉❧t✐♣❧✐❝❛r f(z) = z3 _{+ 2z}2_{+z ♣♦r g(z) = 5z}2_{+ 4z−1.} ❯s❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞✐str✐❜✉t✐✈❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ t❡♠♦s✿

f(z)g(z) = (z3+ 2z2 +z)(5z2+ 4z₋1)

=z3(5z2+ 4z₋1) + 2z2(5z2+ 4z₋1) +z(5z2 + 4z₋1) = (5z5+ 4z4₋z3) + (10z4+ 8z3₋2z2) + (5z3+ 4z2 ₋z) = 5z5+ 14z4+ 12z3+ 2z2₋z

❊①❡♠♣❧♦✳ ❉❡♠♦♥str❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♠✉t❛t✐✈❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ C_.

❈♦♥s✐❞❡r❡ ♦s ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ C_,

f(z) =

n

X

i=0

aizi e g(z) = m

X

i=0 bizi.

❚❡♠♦s q✉❡

f(z).g(z) =

n+m

X

i=0

X

j+k=i

a_j.b_k

!

zi

=

n+m

X

i=0

X

j+k=i

b_k.a_j

!

zi

=g(z).f(z)

♣♦✐s✱ ❡♠ C_, t❡♠♦s _aj.bk=bk.aj, ♣❛r❛ q✉❛✐sq✉❡r ❥ ❡ ❦✳

❊①❡♠♣❧♦✳ ❉❡♠♦♥str❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞✐str✐❜✉t✐✈❛ ❞❡ ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜✲ ❝✐❡♥t❡s ❡♠ C_.

❈♦♥s✐❞❡r❡ ♦s ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ C_,

f(z) =

n

X

i=0

aizi, g(z) = m

X

i=0

bizi e h(z) = l

X

i=0 cizi

❘❡❡s❝r❡✈❡♥❞♦ ❣✭③✮ ❡ ❤✭③✮ ❝♦♠ ❛s ♠❡s♠❛s ♣♦tê♥❝✐❛s ❞❡ ③✱ ♣♦❞❡♠♦s s✉♣♦r ❧ ❂ ♠ ❡ ❡♥tã♦✱

f(z).(g(z) +h(z)) =

n

X

i=0 aizi

!

.

m

X

i=0

(bi +ci)zi

!

=

n+m

X

i=0

X

j+k=i

a_j.(b_k+c_k)

!

zi

=

n+m

X

i=0

X

j+k=i

(aj.bk+aj.ck)

!

zi

=

n+m

X

i=0

X

j+k=i

aj.bk

!

zi+

n+m

X

i=0

X

j+k=i

aj.ck

!

zi

=f(z)g(z) +f(z)h(z), ❆s ❞❡♠❛✐s ❞❡♠♦♥str❛çõ❡s sã♦ ❝♦rr✐q✉❡✐r❛s✳

✷✳✶✳✸ ❉✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s

❈♦♥s✐❞❡r❡ ♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s ❢✭③✮ ❡ ❣✭③✮✳ ❙❡ g(z) ₆= 0, ❞✐③❡♠♦s q✉❡ ❣✭③✮ ❞✐✈✐❞❡ ❢✭③✮ ♦✉ ❢✭③✮ é ❞✐✈✐sí✈❡❧ ♣♦r ❣✭③✮✱ q✉❛♥❞♦ ❡①✐st❡ ♦ ♣♦❧✐♥ô♠✐♦ ❤✭③✮ ❡♠ C_,t❛❧ q✉❡ ❢✭③✮ ❂ ❣✭③✮✳❤✭③✮ ❡ ❞✐③❡♠♦s✱ ♥❡ss❡ ❝❛s♦✱ q✉❡ ❢✭③✮

é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ❣✭③✮✳

❊①❡♠♣❧♦✿ ❚❡♠♦s q✉❡ p(z) =z2₋4z+ 8 ❞✐✈✐❞❡ q(z) =z4+ 64

❉❡ ❢❛t♦✱

q(z) = (z2)2+ 82 = (z2+ 8)2₋2z28 = (z2+ 8)2₋16z2

= (z2+ 8)2₋(4z)2 = (z2+ 8₋4z)(z2+ 8 + 4z).