❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✲❯♥❇ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧
♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛
❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❛✳ ■r✐♥❛ ❙✈✐r✐❞♦✈❛
P■✲❡①♣♦❡♥t❡ ❞❡ ➪❧❣❡❜r❛s
❆ss♦❝✐❛t✐✈❛s
❘❡♥❛t❛ ❆❧✈❡s ❞❛ ❙✐❧✈❛
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s ♣♦r ♠❛✐s ❡st❛ ❝♦♥q✉✐st❛✳ ❙❡♠ ❡❧❡✱ ♥❛❞❛ ❞✐ss♦ s❡r✐❛ ♣♦ssí✈❡❧✳
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣❡❧♦ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧✳ ❊s♣❡❝✐❛❧♠❡♥t❡ ❛♦s ♠❡✉s ♣❛✐s✱ ◆❛tá❧✐❛ ❆❧✲ ✈❡s ❞❡ ❙♦✉③❛ ❇r✐t♦ P❡r❡✐r❛ ❡ ◆✐✈❛❧❞♦ P❡r❡✐r❛ ❞❛ ❙✐❧✈❛✱ ♣❡❧♦ ❝❛r✐♥❤♦✱ ♣❡❧♦ ❝✉✐❞❛❞♦ ❡ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❞❡♣♦s✐t❛❞❛ ❡♠ ♠✐♠✳ ❱♦❝ês sã♦ ♦ ♠❡✉ ♠❛✐♦r ❛♠♦r✳
❆♦s ♠❡✉s ✐r♠ã♦s✱ ❚❛✐❧✐♥♥② ❆❧✈❡s ❞❛ ❙✐❧✈❛ ❡ ❘♦❞r✐❣♦ ❆❧✈❡s ❞❛ ❙✐❧✈❛✱ ♣❡❧♦ ❛♠♦r✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❡ r❡s♣❡✐t♦✳
❆♦s ♠❡✉s s♦❜r✐♥❤♦s✱ ❏♦ã♦ ❱✐❝t♦r ❡ ❏♦ã♦ P❡❞r♦✱ ♣❡❧❛ ❛❧❡❣r✐❛ q✉❡ tr♦✉①❡r❛♠ à ♥♦ss❛ ❢❛♠í❧✐❛✳
➚ ♠✐♥❤❛ ❝✉♥❤❛❞❛✱ ❚❛ís✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❝❛r✐♥❤♦✳
❆♦s ♠❡✉s t✐♦s ❡ ♣r✐♠♦s q✉❡ t❛♥t♦ ♠❡ ✐♥❝❡♥t✐✈❛r❛♠ ❡ ♠❡ ❛❥✉❞❛r❛♠✳ ➚ ♠✐♥❤❛ t✐❛ ❏♦s❛✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛✱ ♣❡❧♦ ❛♠♦r ❡ ♣❡❧❛ ❛♠✐③❛❞❡✳
❆♦ ❣r✉♣♦ ❞❡ ♦r❛çã♦ ♣❡❧♦ ❛♣♦✐♦✱ ♣❡❧❛ ❢♦rç❛ ❡ ♣❡❧♦s ♠♦♠❡♥t♦s tã♦ ❛❣r❛❞á✈❡✐s q✉❡ ♣❛s✲ s❛♠♦s ❥✉♥t♦s✳
➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ■r✐♥❛ ❙✈✐r✐❞♦✈❛✱ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ♣r♦✜ss✐♦♥❛❧ í♥t❡❣r❛ ❡ ❝♦♠♣r♦✲ ♠❡t✐❞❛✳ ❆❣r❛❞❡ç♦ ♣♦r r❡s♣❡✐t❛r ❛s ♠✐♥❤❛s ❧✐♠✐t❛çõ❡s✱ ♣♦r t♦❞♦s ♦s ❡♥s✐♥❛♠❡♥t♦s✱ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦✱ ♣❡❧♦ ❝♦♠♣r♦♠❡t✐♠❡♥t♦ ❡ ❞❡❞✐❝❛çã♦ ❛ ❡st❡ tr❛❜❛❧❤♦✳ ▼✉✐t♦ ♦❜r✐❣❛❞❛✦
❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✱ ❆♥❛ ❈r✐st✐♥❛ ❱✐❡✐r❛ ❡ ❏♦sé ❆♥tô♥✐♦ ❖❧✐✈❡✐r❛ ❞❡ ❋r❡✐t❛s✱ ♣❡❧❛ ❛t❡♥çã♦ ♥❛s ❧❡✐t✉r❛s ❡ ♣❡❧❛s ❝♦♥tr✐❜✉✐çõ❡s ❢❡✐t❛s ❛♦ tr❛❜❛❧❤♦✳
❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠♥❡t♦ ❞❡ ▼❛t❡♠át✐❝❛✲❯♥❇ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ❡ ❛♣♦✐♦✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛♦s ♣r♦❢❡ss♦r❡s ❍❡♠❛r ❡ ▼❛r❝♦ P❡❧❧❡❣r✐♥✐✳
❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞❛ ❣r❛❞✉❛çã♦ ♣❡❧♦ ✐♥❝❡♥t✐✈♦✱ ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ♣❡❧♦ ❛♣♦✐♦✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛♦s ♣r♦❢❡ss♦r❡s ❊✉❞❡s ❆♥tô♥✐♦✱ ❆❞r✐❛♥♦ ❘♦❞r✐❣✉❡s✱ ❑❛❧❡❞ ❙✉❧❛✐♠❛♥ ❑❤✐❞✐r✱ ❉✐r❧❡②✱ ●✐s❡❧❡✱ ❲❛❧ér✐❛✱ ●✐❧♠❛r ❡ ■❞❡♠❛r ❱✐s♦❧❧✐✳
❆♦ ♠❡✉ ♥❛♠♦r❛❞♦✱ ❘ô♠✉❧♦✱ ♠❡✉ ❝♦♠♣❛♥❤❡✐r♦ ♥❡st❛ tr❛❥❡tór✐❛✱ q✉❡ s♦✉❜❡ ❝♦♠♣r❡✲ ❡♥❞❡r ❝♦♠♦ ♥✐♥❣✉é♠ ❡st❛ ❢❛s❡ tã♦ ✐♠♣♦rt❛♥t❡✳ ❆❣r❛❞❡ç♦ ♣❡❧♦ ❛♠♦r✱ ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ♣❡❧❛ ♣r❡s❡♥ç❛ ♥♦s ❜♦♥s ❡ ♠❛✉s ♠♦♠❡♥t♦s ❞❛ ♠✐♥❤❛ ✈✐❞❛✳
❆♦s ♠❡✉s ❝♦❧❡❣❛s ❡ ❛♠✐❣♦s ❞❛ Pós✲●r❛❞✉❛çã♦ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✲❯♥❇ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ♣♦r t♦❞♦s ♠♦♠❡♥t♦s ✈✐✈❡♥❝✐❛❞♦s✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛ ❙❛✐❡♥②✱ ▲❛✉r♦✱ ▼❛②r❛✱ ❋á❜✐♦✱ ❏♦sé ❈❛r❧♦s✱ ❖tt♦✱ ❇r✉♥♦ ❚r✐♥❞❛❞❡✱ ■❧❛♥❛✱ ❑❡✐❞♥❛✱ ❚✐❛❣♦ ▲✐♠❛✱ ▲✉✐③ ▼❛t❡✉s✱ ❊♠❡rs♦♥✱ ▼❛②❡r✱ ▼❛r✐♥❛✱ ❆r✐stót❡❧❡s✱ ❏♦❛❜②✱ ❚❤❛②♥❛r❛✱ ❑❛❧✐❛♥❛✱ ❑❡❧❡♠✱ ❇r✉♥♦ ❙♦✉③❛✱ ▲✐♥♥✐❦❡r ▼♦♥t❡✐r♦✱ ❱✐♥í❝✐✉s ▼❛rt✐♥s✱ ❱✐♥í❝✐✉s ❊❧✐❛s✱ ❱❛❧❞✐❡❣♦✱ ▼❛r✐❛ ▲❡✐t❡✱ ❘❛✐♠✉♥❞♦ ❡ ❊❞♠✐❧s♦♥✳
❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛ ❣r❛❞✉❛çã♦ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ♣❡❧♦ ❝❛r✐♥❤♦✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛ ▼❛r✐st❡❧❛✱ ❈❧❡✐❞✐❛♥❡✱ ❏❛❡❧t♦♥ ❡ ❋á❜✐♦✳
♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❡ ♣❡❧❛ ❝♦♥✈✐✈ê♥❝✐❛ ❛❣r❛❞á✈❡❧✳
❆ ❙❛✐❡♥②✱ ▼❛②r❛✱ ▲❛✉r♦✱ ❋á❜✐♦✱ ❏♦sé ❈❛r❧♦s ❡ ❖tt♦✳ ❆ ❛♠✐③❛❞❡ ❞❡ ✈♦❝ês ❢♦✐ ❡ss❡♥❝✐❛❧ ♣❛r❛ ❡st❛ ❝♦♥q✉✐st❛✳
❆ t♦❞♦s ♦s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❈♦❧✐♥❛✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛ ❉♦♥❛ ❘❛✐♠✉♥❞❛✱ ❘♦s❡ ❡ ▼❛r❝❡❧♦✱ ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ♣❡❧❛ ♣r❡st❛t✐✈✐❞❛❞❡✳
❆ t♦❞♦s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✲❯♥❇✳ ❊s♣❡❝✐❛❧♠❡♥t❡ ❛ ❈❧❛ú✲ ❞✐❛✱ ❲✐❧❧✐❛♠✱ ❉♦♥❛ ■r❡♥❡✱ ▼❛rt❛✱ ❇r✉♥❛✱ ❱✐✈✐❛♥✱ ▲✉ís✱ ❊✈❡❧✐♥❡✱ ❋❛❜✐❛♥❛ ❡ ❚❤✐❛❣♦✳
➚ ❈❛♣❡s ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
❘❡s✉♠♦
❙❡❥❛♠ A ✉♠❛ P I✲á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ s♦❜r❡ ✉♠ ❝♦r♣♦ F ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦ ❡
{cn(A)} ❛ s❡q✉ê♥❝✐❛ ❞❡ ❝♦❞✐♠❡♥sõ❡s ❞❡ A✳ ◆❡st❡ tr❛❜❛❧❤♦ ✈❛♠♦s ❡st✉❞❛r ♦ ❝♦♠♣♦rt❛✲
♠❡♥t♦ ❞❡st❛s s❡q✉ê♥❝✐❛s✳ ❘❡❣❡✈ ♠♦str♦✉ q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❝♦❞✐♠❡♥sõ❡s é ❡①♣♦♥❡♥❝✐✲ ❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛✳ ❖ ♥♦ss♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ é ❛♣r❡s❡♥t❛r ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣♦r ❆✳ ●✐❛♠❜r✉♥♦ ❡ ▼✳ ❩❛✐❝❡✈ ❡♠ ❬✹❪✱ ♦♥❞❡ ❞❡♠♦♥str❛♠ q✉❡ ♦ P■✲❡①♣♦❡♥t❡ ❞❡A✱ ❞❡♥♦t❛❞♦ ♣♦r exp(A) = lim
n→∞
n
p
cn(A)✱ s❡♠♣r❡ ❡①✐st❡ ❡ é ✉♠ ✐♥t❡✐r♦✳ ❉❛r❡♠♦s ✉♠❛ ♠❛♥❡✐r❛ ❡①♣❧í❝✐t❛
❆❜str❛❝t
▲❡t A ❜❡ ❛♥ ❛ss♦❝✐❛t✐✈❡ ❛❧❣❡❜r❛ ♦✈❡r ❛ ✜❡❧❞ F ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ ③❡r♦ s❛t✐s❢②✐♥❣ ❛ ♣♦❧②♥♦♠✐❛❧ ✐❞❡♥t✐t② ✭P■✲❛❧❣❡❜r❛✮✱ ❛♥❞{cn(A)}❜❡ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❝♦❞✐♠❡♥s✐♦♥s ♦❢ t❤❡A✳
■♥ t❤✐s ♣❛♣❡r ✇❡ st✉❞② t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡s❡ s❡q✉❡♥❝❡s✳ ❘❡❣❡✈ s❤♦✇❡❞ t❤❛t ❛ s❡q✉❡♥❝❡ ✐s ❡①♣♦♥❡♥t✐❛❧❧② ❝♦❞✐♠❡♥s✐♦♥s ❧✐♠✐t❡❞✳ ❖✉r ♠❛✐♥ ❣♦❛❧ ✐s t♦ s❤♦✇ t❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❜② ❆✳ ●✐❛♠❜r✉♥♦ ❛♥❞ ▼✳ ❩❛✐❝❡✈ ✐♥ ❬✹❪✱ ✇❤❡r❡ t❤❡② ♣r♦✈❡ t❤❛t t❤❡ P■✲❡①♣♦♥❡♥t ♦❢ A✱ ❞❡♥♦t❡❞ ❜②exp(A) = lim
n→∞
n
p
cn(A)✱ ❡①✐sts ❛♥❞ ✐s ❛♥ ✐♥t❡❣❡r✳ ❲❡ ✇✐❧❧ ❣✐✈❡ ❛♥ ❡①♣❧✐❝✐t ✇❛②
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ ■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✹
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✹
✶✳✶ ➪❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❚✲✐❞❡❛✐s ❡ ❱❛r✐❡❞❛❞❡s ❞❡ ➪❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹ P♦❧✐♥ô♠✐♦s ❍♦♠♦❣ê♥❡♦s ❡ ▼✉❧t✐❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✺ ❚✐♣♦s ❊s♣❡❝✐❛✐s ❞❡ ■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✻ ▼ó❞✉❧♦s✱ ❆♥é✐s ❙❡♠✐ss✐♠♣❧❡s ❡ ❘❛❞✐❝❛❧ ❞❡ ❏❛❝♦❜s♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷ Sn✲❘❡♣r❡s❡♥t❛çõ❡s ✷✼
Sn✲❘❡♣r❡s❡♥t❛çõ❡s ✷✼
✷✳✹ Sn✲❛çã♦ ❡♠ P♦❧✐♥ô♠✐♦s ▼✉❧t✐❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✷✳✺ Sn✲❘❡♣r❡s❡♥t❛çõ❡s ❡ ●❛♥❝❤♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✸ ❙❡q✉ê♥❝✐❛ ❞❡ ❈♦❞✐♠❡♥sõ❡s ❞❡ ➪❧❣❡❜r❛s ❡ ➪❧❣❡❜r❛s ●r❛❞✉❛❞❛s ✺✵
❙❡q✉ê♥❝✐❛ ❞❡ ❈♦❞✐♠❡♥sõ❡s ❞❡ ➪❧❣❡❜r❛s ❡ ➪❧❣❡❜r❛s ●r❛❞✉❛❞❛s ✺✵ ✸✳✶ ❈♦❞✐♠❡♥sõ❡s ❞❡ ✉♠❛ ➪❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✷ ➪❧❣❡❜r❛s G✲●r❛❞✉❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✸ ❙✉♣❡r❛❧❣❡❜r❛ ❡ ❊♥✈♦❧✈❡♥t❡ ❞❡ ●r❛ss♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✹ ❖ P■✲❡①♣♦❡♥t❡ ❞❡ ✉♠❛ á❧❣❡❜r❛ ✻✵
❖ P■✲❡①♣♦❡♥t❡ ❞❡ ✉♠❛ á❧❣❡❜r❛ ✻✵
✹✳✶ P■✲❡①♣♦❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✷ ❯♠ ❝❛♥❞✐❞❛t♦ ♣❛r❛ P■✲❡①♣♦❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✸ ■❞❡♥t✐❞❛❞❡s ❡ ■❞❡♥t✐❞❛❞❡s ●r❛❞✉❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✹✳✹ s✉♣❡r❛❧❣❡❜r❛s s✐♠♣❧❡s ❡ s✉❛s ❡♥✈♦❧✈❡♥t❡s ❞❡ ●r❛ss♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹✳✺ ❈♦❧❛♥❞♦ t❛❜❡❧❛s ❞❡ ❨♦✉♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✹✳✻ ❈❛❧❝✉❧❛♥❞♦ ♦ ❧✐♠✐t❡ ✐♥❢❡r✐♦r ❞❛ cn(G(A))✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶
✹✳✼ ❘❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧✳ ❊①✐stê♥❝✐❛ ❞♦ P■✲❡①♣♦❡♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺
■♥tr♦❞✉çã♦
❙❡❥❛♠F ✉♠ ❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦ ❡ FhXi❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❧✐✈r❡ ❞❡ ♣♦st♦ ❡♥✉♠❡rá✈❡❧ ♥♦ ❝♦♥❥✉♥t♦ X = {x1, x2, . . .}✳ ❈♦♥s✐❞❡r❛r❡♠♦s FhXi ♥ã♦ ✉♥✐tár✐❛ ❡ t♦❞❛s ❛s á❧❣❡❜r❛ ❞♦ t❡①t♦ ❛ss♦❝✐❛t✐✈❛s ❡ ♥ã♦ ❝♦♠✉t❛t✐✈❛s ✭❡①❝❡t♦ ♠❡♥çã♦ ❝♦♥trár✐❛✮ ✳ ❉✐r❡♠♦s q✉❡ ✉♠❛F✲á❧❣❡❜r❛A é ✉♠❛ P■✲á❧❣❡❜r❛ s❡ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ♥✉❧♦f(x1, . . . , xn)∈
FhXit❛❧ q✉❡✱f(a1, . . . , an) = 0✱ ♣❛r❛ q✉❛✐sq✉❡ra1, . . . , an ∈A✳ ❉❡♥♦t❛r❡♠♦s ♣♦rId(A)
♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ A✳ ❈❤❛♠❛r❡♠♦s ✉♠ ✐❞❡❛❧I ❞❡FhXi❞❡ T✲✐❞❡❛❧✱ s❡ I é ✐♥✈❛r✐❛♥t❡ ♣♦r EndF (FhXi)✳ P❛r❛ ❝❛❞❛ T✲✐❞❡❛❧ I✱ ❡①✐st❡ á❧❣❡❜r❛ A t❛❧
q✉❡ I = Id(A)✳ ❉❡♥♦t❛r❡♠♦s ♣♦r var(I) ♦✉ var(A) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s q✉❡ t❡♠ I ❝♦♠♦ ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s✳
❙❡❥❛G✉♠ ❣r✉♣♦ q✉❛❧q✉❡r✳ ❯♠❛ á❧❣❡❜r❛A é ❞✐t❛G✲❣r❛❞✉❛❞❛✱ s❡A ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ s♦♠❛ ❞✐r❡t❛ ❞❡ s✉❜❡s♣❛ç♦s A=+.g∈G A(g) t❛✐s q✉❡ ♣❛r❛ t♦❞♦ g, h∈G✱A(g)A(h) ⊆
A(gh)✳ ❉❡♥tr❡ ❛s á❧❣❡❜r❛sG✲❣r❛❞✉❛❞❛s ❝♦♥s✐❞❡r❛❞❛s ♥♦ t❡①t♦✱ ❡♥❢❛t✐③❛r❡♠♦s ❛s á❧❣❡❜r❛s Z2✲❣r❛❞✉❛❞❛s✱ t❛♠❜é♠ ❝❤❛♠❛❞❛s ❞❡ s✉♣❡r❛❧❣❡❜r❛s✳ ❊♥tã♦✱ ✉♠❛ á❧❣❡❜r❛Aé ❝❤❛♠❛❞❛ ❞❡ s✉♣❡r❛❧❣❡❜r❛ ❝♦♠ ❣r❛❞✉❛çã♦ A(0), A(1)✱ s❡A =A(0) +. A(1)é s♦♠❛ ❞✐r❡t❛ ❞❡ s✉❜❡s♣❛ç♦s A(0), A(1) ❡ s❛t✐s❢❛③✿ A(0)A(1) +. A(1)A(0) ⊆ A(1) ❡ A(0)A(0) +. A(1)A(1) ⊆ A(0)✳ ❖❜s❡r✈❡ q✉❡ t♦❞❛ á❧❣❡❜r❛ ❛❞♠✐t❡ ✉♠❛ ❣r❛❞✉❛çã♦ tr✐✈✐❛❧✱ ♦♥❞❡ A(0) =A, A(1) = 0✳ ❯♠ ❡①❡♠♣❧♦ ✐♠♣♦rt❛♥t❡ ❞❡ s✉♣❡r❛❧❣❡❜r❛ é ❛ á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥ G = G(0) +. G(1) ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛ ❣❡r❛❞❛ ♣❡❧♦ ❝♦♥❥✉♥t♦ {e1, e2, . . .| eiej =−ejei, i, j = 1,2, . . .}✱ ♦♥❞❡ G(0) ❡ G(1)
sã♦ ♦s s✉❜❡s♣❛ç♦s ❣❡r❛❞♦s ♣❡❧♦s ♠♦♥ô♠✐♦s ❡♠ ei ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♣❛r ❡ ♦s ♠♦♥ô♠✐♦s ❞❡
❝♦♠♣r✐♠❡♥t♦ í♠♣❛r✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ s✉♣❡r❛❧❣❡❜r❛G(A) = G(0)⊗A(0) +. G(1)⊗A(1) é ❝❤❛♠❛❞❛ ❡♥✈♦❧✈❡♥t❡ ❞❡ ●r❛ss♠❛♥♥ ❞❛ á❧❣❡❜r❛ ❆✱ ❡♠ q✉❡ A é ✉♠❛ s✉♣❡r❛❧❣❡❜r❛✳
➱ ❜❡♠ ❝♦♥❤❡❝✐❞♦ q✉❡✱ s❡ A é ✉♠❛ s✉♣❡r❛❧❣❡❜r❛ s✐♠♣❧❡s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ s♦❜r❡ ✉♠ ❝♦r♣♦ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦ ✱ ❡♥tã♦ A é ✐s♦♠♦r❢❛ ❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s s✉♣❡r❛❧❣❡❜r❛s✿ Mk(F), Mk(F
.
■♥tr♦❞✉çã♦ ✷
r❛❧❣❡❜r❛ A ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ s♦❜r❡ ✉♠ ❝♦r♣♦ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦ é ❡s❝r✐t❛ ❝♦♠♦A=A1⊕. . .⊕An
.
+J(A)✱ ❡♠ q✉❡J =J(A)é ♦ r❛❞✐❝❛❧ ❞❡ ❏❛❝♦❜s♦♥ ❞❡A ❡ A1, . . . , An sã♦ s✉♣❡r❛❧❣❡❜r❛s s✐♠♣❧❡s✳
❯♠❛ á❧❣❡❜r❛A é ❞✐t❛ ✈❡r❜❛❧♠❡♥t❡ ♣r✐♠❛ s❡ ♣❛r❛ q✉❛✐sq✉❡rT✲✐❞❡❛✐sI1 ❡I2 ❞❡FhXi t❛✐s q✉❡ I1I2 ⊆ I ✐♠♣❧✐❝❛ q✉❡ I1 ⊆ I ♦✉ I2 ⊆ I✱ ♦♥❞❡ I é ✉♠ T✲✐❞❡❛❧ ♣r✐♠♦ ❞❡ FhXi✳ ❆✳ ❘✳ ❑❡♠❡r ❝❛r❛❝t❡r✐③♦✉ t♦❞❛s ❛s á❧❣❡❜r❛s ✈❡r❜❛❧♠❡♥t❡ ♣r✐♠❛s s♦❜r❡ ✉♠ ❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦✿ F✱FhXi✱Mk(F)✱ Mk(G)✱ Mk,l(G) (k ≥l)✱ ❡♠ q✉❡G=G0
.
+G1 é ❛ á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥❀ Mk(F) ❡ Mk(G) sã♦ ❛s á❧❣❡❜r❛s ❞❡ ♠❛tr✐③❡s k×k s♦❜r❡ F ❡
G r❡s♣❡❝t✐✈❛♠❡♥t❡ ❡ Mk,l(G) =
(
P Q R S
!
P ∈Mk(G0), Q∈Mk×l(G1), R∈Ml×k(G1)❡ S ∈Ml(G0) )
✳
❖✉tr♦ ❢❛t♦ ❝♦♥❤❡❝✐❞♦ é q✉❡✱ s♦❜r❡ ❝♦r♣♦s ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦✱ t♦❞♦ ♣♦❧✐♥ô♠✐♦ ♥ã♦✲♥✉❧♦ f ∈ FhXi é ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ♠✉❧t✐❧✐♥❡❛r❡s✳ ❉❡♥♦t❛r❡♠♦s ♣♦rPn ♦F✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞♦s ♣♦❧✐♥ô♠✐♦s ♠✉❧t✐❧✐♥❡❛r❡s ❞❡ ❣r❛✉n✳ ❊①✐st❡
✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ F Sn✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ❡♥tr❡ Pn ❡ ❛ á❧❣❡❜r❛ F Sn✱ ❡♠ q✉❡ Sn é ♦
❣r✉♣♦ s✐♠étr✐❝♦ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ {1, . . . , n}✳ ❆ss✐♠✱ ♦ q✉♦❝✐❡♥t❡ Pn/Pn ∩Id(A) t❡♠
✉♠❛ ❡str✉t✉r❛ ❞❡ F Sn✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛✳ ❉❡✜♥✐♠♦s ❛ n✲és✐♠❛ ❝♦❞✐♠❡♥sã♦ ❞❡ A ♣♦r
cn(A) =dimF(Pn/Pn∩Id(A))✳ ❙❡A❢♦r ✉♠❛ P■✲á❧❣❡❜r❛✱ ❡♥tã♦cn(A)< n!♣❛r❛ ❛❧❣✉♠
n✳ ❙❡ A ❢♦r ♥✐❧♣♦t❡♥t❡✱ ❡♥tã♦cn(A) = 0,❛ ♣❛rt✐r ❞❡ ✉♠ ❝❡rt♦ n0.✳
❊♠ ❬✻❪ ❆✳ ❘❡❣❡✈ ❡♠ ✶✾✼✷✱ ♣r♦✈♦✉ q✉❡ s❡ A é ✉♠❛ P I✲á❧❣❡❜r❛✱ ❡♥tã♦ ❡①✐st❡♠ ❝♦♥s✲ t❛♥t❡s α, β t❛✐s q✉❡ cn(A)≤αβn✱ ♦✉ s❡❥❛✱ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❝♦❞✐♠❡♥sõ❡s ❞❡ A é ❡①♣♦♥❡♥✲
❝✐❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛✳
❙❡A é ✉♠❛ P■✲á❧❣❡❜r❛✱ ♦ s❡✉ P■✲❡①♣♦❡♥t❡ é ❞❡✜♥✐❞♦ ❝♦♠♦ exp(A) = lim
n→∞
n
p
cn(A)✳
◆❛ ❞é❝❛❞❛ ❞❡ ✶✾✽✵✱ ❙✳ ❆✳ ❆♠✐ts✉r ❝♦♥❥❡❝t✉r♦✉ q✉❡ ♦ P■✲❡①♣♦❡♥t❡ ❞❡ ✉♠❛ P■✲á❧❣❡❜r❛ ❡①✐st❡ ❡ é ✉♠ ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦✳
■♥tr♦❞✉çã♦ ✸
s✐♠étr✐❝♦ ❡ ❛P I✲t❡♦r✐❛✳
❖ P■✲❡①♣♦❡♥t❡ ❞❛s á❧❣❡❜r❛s ✈❡r❜❛❧♠❡♥t❡ ♣r✐♠❛s✱ t❡♠ ♦s s❡❣✉✐♥t❡s ✈❛❧♦r❡s✿ exp(Mk(F)) =
k2✱ exp(M
k(G)) = 2k2 ❡ exp(Mk,l(G)) = (k+l)2✳
❊st❡ tr❛❜❛❧❤♦ ❡stá ❡str✉t✉r❛❞♦ ❡♠ q✉❛tr♦ ❈❛♣ít✉❧♦s✳ ◆♦ ❈❛♣ít✉❧♦ ✶ ❛♣r❡s❡♥t❛r❡✲ ♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s✱ ❡①❡♠♣❧♦s ❡ r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s s♦❜r❡ ✉♠ ❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦ ❡ ❞❛ t❡♦r✐❛ ❞❡ ♠ó❞✉❧♦s ❡ ❛♥é✐s q✉❡ s❡r✈✐rã♦ ❝♦♠♦ ❜❛s❡ ♣❛r❛ ♦s ❈❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳ ❋❛❧❛r❡♠♦s ❞❡ ❛♥é✐s✱ á❧❣❡❜r❛s s✐♠♣❧❡s ❡ s❡♠✐ss✐♠♣❧❡s✳
◆♦ ❈❛♣ít✉❧♦ ✷ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s ❞❛ t❡♦r✐❛ ❞❡ r❡♣r❡s❡♥✲ t❛çõ❡s ❞❡ ❣r✉♣♦s ✜♥✐t♦s✳ ❋❛❧❛r❡♠♦s ❞❛ t❡♦r✐❛ ❞❡ ❨♦✉♥❣ ❞❡ r❡♣r❡s❡♥t❛çõ❡s ❞♦ ❣r✉♣♦ s✐♠étr✐❝♦✳
◆♦ ❈❛♣ít✉❧♦ ✸ ❡st✉❞❛r❡♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❝♦❞✐♠❡♥✲ sõ❡s cn(A) ❞❡ ✉♠❛ P■✲á❧❣❡❜r❛ A s♦❜r❡ ✉♠ ❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦✳ ❋❛❧❛r❡♠♦s ❞❡
á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s✱ á❧❣❡❜r❛s ✈❡r❜❛❧♠❡♥t❡ ♣r✐♠❛s✱ ❡♥✈♦❧✈❡♥t❡ ❞❡ ●r❛ss♠❛♥♥ ❡ ❞❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s ❞❡st❛s á❧❣❡❜r❛s✳
1
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛
◆❡st❡ ❈❛♣ít✉❧♦ ❞❛r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❡ r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ❞❛ t❡♦r✐❛ ❞❡ á❧❣❡❜r❛s q✉❡ s❛t✐s❢❛③❡♠ ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ✭P■✲á❧❣❡❜r❛s✮✳ ❋❛❧❛r❡♠♦s ❞❡ T✲ ✐❞❡❛✐s✱ ✈❛r✐❡❞❛❞❡s ❞❡ á❧❣❡❜r❛s✱ ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ❡ ♠✉❧t✐❧✐♥❡❛r❡s✳ ❉❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ❞❡ ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ á❧❣❡❜r❛s s♦❜r❡ ✉♠ ❝♦r♣♦ ❜❛s❡ F✳ ❆♣r❡s❡♥t❛r❡♠♦s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❞❡ ❛♥é✐s ❡ á❧❣❡❜r❛s s✐♠♣❧❡s ❡ s❡♠✐ss✐♠♣❧❡s✳ ❉✉✲ r❛♥t❡ t♦❞♦ ❈❛♣ít✉❧♦✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s ❡ ♥ã♦ ❝♦♠✉t❛t✐✈❛s ✭❡①❝❡t♦ ♠❡♥çã♦ ❝♦♥trár✐❛✮ ❡F ✉♠ ❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦✳
✶✳✶ ➪❧❣❡❜r❛s
❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛♠ F ✉♠ ❝♦r♣♦ ❡ A ✉♠ F✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦ q✉❛❧ é ❞❡✜♥✐❞♦ ✉♠ ♣r♦❞✉t♦ ❜✐❧✐♥❡❛r s♦❜r❡ F✳ ❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ t♦❞♦ c∈F ❡ q✉❛✐sq✉❡r x, y, z ∈A✱ t❡♠✲s❡✿
✶✳ ✭①✰②✮③❂①③ ✰ ②③ ✷✳ ③✭①✰②✮❂③① ✰ ③② ✸✳ ✭❝①✮②❂ ❝✭①②✮❂ ①✭❝②✮✳
❊♥tã♦ A é ✉♠❛ F✲á❧❣❡❜r❛✳
❖❜s❡r✈❛çã♦ ✶✳✷✳ ❚♦❞❛ á❧❣❡❜r❛ ❝♦♠ r❡❧❛çã♦ à s♦♠❛ ❡ ❛♦ ♣r♦❞✉t♦ é ✉♠ ❛♥❡❧✳
❆ ❞✐♠❡♥sã♦ ❞❛ F✲á❧❣❡❜r❛ é ❛ s✉❛ ❞✐♠❡♥sã♦ ❝♦♠♦ F✲❡s♣❛ç♦✳ ❙❡ ❛ ❞✐♠❡♥sã♦ é ✜♥✐t❛✱ A ❝❤❛♠❛✲s❡ á❧❣❡❜r❛ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✳
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✺
❊①❡♠♣❧♦ ✶✳✸✳ ❙❡❥❛ Mn(F) ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❛s ♠❛tr✐③❡s n×n ❝♦♠ ❡♥tr❛❞❛s ❡♠ F✳
▼✉♥✐❞♦ ❞♦ ♣r♦❞✉t♦ ✉s✉❛❧ ❞❡ ♠❛tr✐③❡s Mn(F) é ✉♠❛ á❧❣❡❜r❛ ❝♦♠ ✉♥✐❞❛❞❡✱ q✉❡ é ❡①❛t❛✲
♠❡♥t❡ ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ In×n✳ ❉❡st❛❝❛r❡♠♦s ♥❡st❛ á❧❣❡❜r❛ ❛s ♠❛tr✐③❡s ✉♥✐tár✐❛s Eij✱
❝♦♠ 1≤i, j ≤n✱ ♦♥❞❡ Eij é ❛ ♠❛tr✐③ ❝✉❥❛ ú♥✐❝❛ ❡♥tr❛❞❛ ♥ã♦ ♥✉❧❛ é ♥❛ ✐✲és✐♠❛ ❧✐♥❤❛ ❡
❥✲és✐♠❛ ❝♦❧✉♥❛✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ ❛s ♠❛tr✐③❡s ✉♥✐tár✐❛s ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♣❛r❛ Mn(F)✱ ❞♦♥❞❡ dimMn(F) = n2✳
●❡♥❡r❛❧✐③❛♥❞♦✱ s❡ A é ✉♠❛ F✲á❧❣❡❜r❛ ❡ s❡ ❝♦♥s✐❞❡r❛r♠♦s ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ Mn(A) ❞❡
t♦❞❛s ❛s ♠❛tr✐③❡s n×n ❝♦♠ ❡♥tr❛❞❛s ❡♠ A✱ ❞❡✜♥✐♥❞♦ ✉♠ ♣r♦❞✉t♦ ❡♠ Mn(A) ❛♥á❧♦❣♦
❛♦ ♣r♦❞✉t♦ ✉s✉❛❧ ❡♠ Mn(F)✱ ♦❜t❡♠♦s ❛ss✐♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ F✲á❧❣❡❜r❛ ❡♠ Mn(A)✳
❊①❡♠♣❧♦ ✶✳✹✳ ❙❡❥❛V ✉♠ F✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❊♥tã♦ EndF(V)✱ ♦ ❝♦♥❥✉♥t♦ ❞❛s tr❛♥s❢♦r✲
♠❛çõ❡sF✲❧✐♥❡❛r❡s ❞❡V ♠✉♥✐❞♦ ❞❛ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s✱ é ✉♠❛ F✲á❧❣❡❜r❛ ❝♦♠ ✉♥✐❞❛❞❡ ✭♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡✮✳
❊①❡♠♣❧♦ ✶✳✺✳ ❯♠ ❝♦r♣♦ F ♣♦ss✉✐ ♥❛t✉r❛❧♠❡♥t❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ s✐ ♠❡s♠♦✳ ❙❡❥❛ K ✉♠❛ ❡①t❡♥sã♦ ❞♦ ❝♦r♣♦ F✱ ❡♥tã♦ K t❛♠❜é♠ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ F✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡✱ ✈✐st♦ ❞❡st❛ ❢♦r♠❛✱ F ❡ K sã♦ F✲á❧❣❡❜r❛s ❝♦♠✉t❛t✐✈❛s✱ ❝♦♠ ✉♥✐❞❛❞❡✱ ❝✉❥♦s ♣r♦❞✉t♦s sã♦ ❡①❛t❛♠❡♥t❡ ♦s ♣r♦❞✉t♦s ❞♦s ❝♦r♣♦sF ❡K✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❊①❡♠♣❧♦ ✶✳✻✳ ❙❡❥❛♠G✉♠ ❣r✉♣♦ ❡F ✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦rF G=F[G]♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s s♦♠❛s ❢♦r♠❛✐s ✜♥✐t❛sPg∈Gαgg, αg ∈F✱ F G♣♦ss✉✐ ❛ ❡str✉t✉r❛ ❞❡ ❛♥❡❧ ❝♦♠
r❡s♣❡✐t♦ ❛
+ : (X
g∈G
αgg) + ( X
g∈G
βgg) = X
g∈G
(αg+βg)g
0 = X
g∈G
0g
.: (X
g∈G
αgg).( X
h∈G
βhh) = X
g,h∈G
(αg.βh)gh= X
l∈G
γll
λ(X
g∈G
αgg) = X
g∈G
(λαg)g.
❆ss✐♠ ♠✉♥✐❞♦ ❞❡st❛s ♦♣❡r❛çõ❡s t❡♠♦s q✉❡ F G é ✉♠❛ F✲á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛✱ ✉♥✐tár✐❛ ❝♦♠ ✉♥✐❞❛❞❡ 1F1G✱ ♦♥❞❡ G ✭✈✐st♦ ❡♠ F G✮ ❢♦r♠❛ ✉♠❛ ❜❛s❡ ♣❛r❛ F G✳ ❉✐st♦✱ s❡❣✉❡ q✉❡
dimFF G =|G|✳ ❆ á❧❣❡❜r❛ F G é ❞✐t❛ ❞❡ á❧❣❡❜r❛ ❞❡ ❣r✉♣♦ s♦❜r❡ F✳ ❆ á❧❣❡❜r❛ F G é
❝♦♠✉t❛t✐✈❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ G é ❝♦♠✉t❛t✐✈♦✳
❆ á❧❣❡❜r❛F Sn ❝♦♥str✉í❞❛ à ♣❛rt✐r ❞❡ Sn✱ ♦ ❣r✉♣♦ ❞❛s ♣❡r♠✉t❛çõ❡s ❞❡ n ❡❧❡♠❡♥t♦s✱
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✻
❊①❡♠♣❧♦ ✶✳✼✳ ❖ ❋✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✿
Mk,l(F) =
P Q R S
!
♦♥❞❡ P, Q, R, S sã♦ ♠❛tr✐③❡s k×k, k×l, l×k ❡ l×l✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❝♦♠ k≥l > 0✱ é ✉♠❛ F✲á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛✱ ✉♥✐tár✐❛✱ ♥ã♦ ❝♦♠✉t❛t✐✈❛ ❞❡ ❞✐♠❡♥sã♦(k+l)2✱ ❝♦♠ ♣r♦❞✉t♦ ✉s✉❛❧ ❞❡ ♠❛tr✐③ ❡♠ ❜❧♦❝♦✳
❊①❡♠♣❧♦ ✶✳✽✳ ❙❡❥❛G ✉♠ ❣r✉♣♦ ❞❡ ♦r❞❡♠2 ❣❡r❛❞♦ ♣♦rc✳ ❖F✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ Mn(F +
cF) ❝♦♠ ❡♥tr❛❞❛s ♥❛ á❧❣❡❜r❛ ❞❡ ❣r✉♣♦ F +cF é ✉♠❛ F✲á❧❣❡❜r❛ ❞❡ ❞✐♠❡♥sã♦ 2n2✳ ❉❡✜♥✐çã♦ ✶✳✾✳ ❯♠ s✉❜❡s♣❛ç♦ B ❞❛ á❧❣❡❜r❛ A é ❝❤❛♠❛❞♦ s✉❜á❧❣❡❜r❛ s❡ é ❢❡❝❤❛❞♦ ❝♦♠ r❡s♣❡✐t♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ♦✉ s❡❥❛✱ b1·b2 ∈B ♣❛r❛ q✉❛✐sq✉❡r b1, b2 ∈B✳ ❉✐③❡♠♦s t❛♠❜é♠ q✉❡ ✉♠ s✉❜❡s♣❛ç♦ I ❞❡ A é ✉♠ ✐❞❡❛❧ à ❡sq✉❡r❞❛ ✭à ❞✐r❡✐t❛✮ ❞❡ A✱ s❡ ax ∈ I ✭xa ∈ I✮ ♣❛r❛ q✉❛✐sq✉❡r x ∈ I ❡ a ∈ A r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡ I é ✉♠ ✐❞❡❛❧ à ❡sq✉❡r❞❛ ❡ à ❞✐r❡✐t❛ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ❞✐③❡♠♦s q✉❡ I é ✉♠ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❞❡ A✳
❊①❡♠♣❧♦ ✶✳✶✵✳ ✭❈❡♥tr♦ ❞❡ ✉♠❛ ➪❧❣❡❜r❛✮ ❙❡❥❛ A ✉♠❛ á❧❣❡❜r❛✳ ❖ ❝♦♥❥✉♥t♦ Z(A) =
{a ∈ A | ax = xa,∀x ∈ A} é ✉♠❛ s✉❜á❧❣❡❜r❛ ❞❡ A q✉❡ ❝❤❛♠❛♠♦s ❞❡ ❝❡♥tr♦ ❞❡ A✳ ❙❛❜❡♠♦s ❞❛ á❧❣❡❜r❛ ❧✐♥❡❛r q✉❡✱ ✜①❛❞♦ n∈N ❛s ú♥✐❝❛s ♠❛tr✐③❡s q✉❡ ❝♦♠✉t❛♠ ❝♦♠ t♦❞❛s ❛s ♠❛tr✐③❡s✱ sã♦ ❛s ♠❛tr✐③❡s ❡s❝❛❧❛r❡s✳ ❚❡♠♦s ❡♥tã♦ q✉❡ Z(Mn(F)) ={λIn×n|λ∈F}✳
❉❡✜♥✐çã♦ ✶✳✶✶✳ ❙❡❥❛♠ A1 ❡ A2 ❋✲á❧❣❡❜r❛s✳ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r Φ : A1 →A2 é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ F✲á❧❣❡❜r❛s s❡✿
Φ(ab) = Φ(a)Φ(b), ∀a, b∈A1.
❉✐r❡♠♦s q✉❡ Φ é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ F✲á❧❣❡❜r❛s s❡ Φ é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❜✐❥❡t✐✈♦ ❞❡ F✲á❧❣❡❜r❛s✱ ♥❡st❡ ❝❛s♦✱ t❛♠❜é♠ ❞✐③❡♠♦s q✉❡ A1 ❡ A2 sã♦ á❧❣❡❜r❛s ✐s♦♠♦r❢❛s✳ ❙❡ Φ :A1 →A1 é ✉♠ ❤♦♠♦♠♦r✜s♠♦✱ ❞✐③❡♠♦s q✉❡ Φé ✉♠ ❡♥❞♦♠♦r✜s♠♦✳
❊①❡♠♣❧♦ ✶✳✶✷✳ ❙❡❥❛A ✉♠❛ F✲á❧❣❡❜r❛✳ ❊♥tã♦ Mn(F)⊗A∼=Mn(A) ❝♦♠♦ á❧❣❡❜r❛s✳ ❉❡
❢❛t♦✱ ❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❛ss✐♠ ❞❡✜♥✐❞❛
ϕ:Mn(F)⊗A → Mn(A)
Eij ⊗a 7→ aEij
é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s✱ ♦♥❞❡ {Eij, i, j = 1, . . . , n} sã♦ ❛s ♠❛tr✐③❡s ❡❧❡♠❡♥t❛r❡s
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✼
❖❜s❡r✈❛çã♦ ✶✳✶✸✳ ❆ss✐♠ ❝♦♠♦ ♥❛ t❡♦r✐❛ ❞❡ ❣r✉♣♦s ❡ ❞❡ ❛♥é✐s ♦ t❡♦r❡♠❛ ❞♦ ❤♦♠♦♠♦r✲ ✜s♠♦ t❛♠❜é♠ é ✈á❧✐❞♦ ♣❛r❛ á❧❣❡❜r❛s✱ ♦✉ s❡❥❛✱ ♣❛r❛ ✉♠ ❤♦♠♦♠♦r✜s♠♦ Φ : A1 → A2 t❡♠♦s
Ker(Φ) ={a1 ∈A1,Φ(a1) = 0}, t❡♠♦s q✉❡ ♦ q✉♦❝✐❡♥t❡ A1
Ker(Φ) é ✐s♦♠♦r❢♦ ❛ ✐♠❛❣❡♠ Im(Φ) ={Φ(a)| a∈A1}✳ ◆♦t❡ q✉❡ ❛ á❧❣❡❜r❛ q✉♦❝✐❡♥t❡ A1
Ker(Φ) ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s Ker(Φ) é ✉♠ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❞❡ A1✳
❉❡✜♥✐çã♦ ✶✳✶✹✳ ❙❡❥❛ A ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ F✳
✭✐✮ A é ❛ss♦❝✐❛t✐✈❛ s❡ (ab)c=a(bc) ♣❛r❛ t♦❞♦ a, b ❡ c∈A✳ ✭✐✐✮ A é ❝♦♠✉t❛t✐✈❛ s❡ ab=ba✱ ♣❛r❛ t♦❞♦ a, b∈A✳
✭✐✐✐✮ A é ✉♥✐tár✐❛ s❡ A t❡♠ ✉♥✐❞❛❞❡✱ ♦✉ s❡❥❛✱ s❡ ❡①✐st❡ 1 ∈ A t❛❧ q✉❡ 1a = a1 = a✱
∀a∈A✳
❉❡✜♥✐çã♦ ✶✳✶✺✳ ❙❡❥❛ A ✉♠❛ F✲á❧❣❡❜r❛✳ ❉✐③❡♠♦s q✉❡✿
✭✐✮ ❯♠ ❡❧❡♠❡♥t♦ a∈A ❝❤❛♠❛✲s❡ ♥✐❧ ✭♦✉ ♥✐❧♣♦t❡♥t❡✮ s❡ ❡①✐st❡ n∈N t❛❧ q✉❡ an= 0✳ ❙❡
t♦❞♦s ❡❧❡♠❡♥t♦s ❞❡ A sã♦ ♥✐❧♣♦t❡♥t❡s✱ ❞✐③❡♠♦s q✉❡ A é ♥✐❧✳ ❙❡ ❡①✐st❡ n∈N t❛❧ q✉❡ ♣❛r❛ t♦❞♦ a ∈A✱ t❡♠✲s❡ an = 0✱ ❡♥tã♦ A ❝❤❛♠❛✲s❡ ♥✐❧ ❞❡ ❣r❛✉ ❧✐♠✐t❛❞♦✳ ❖ ♠❡♥♦r ♥❛t✉r❛❧ n
❝♦♠ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❝❤❛♠❛❞♦ ❞❡ ♥✐❧ ❡①♣♦❡♥t❡ ❞❡ A✳
✭✐✐✮ ❯♠❛ á❧❣❡❜r❛A ❝❤❛♠❛✲s❡ ♥✐❧♣♦t❡♥t❡✱ s❡ ❡①✐st❡ n∈N t❛❧ q✉❡ a1. . . an = 0 ♣❛r❛ t♦❞♦
a1, . . . , an ∈ A✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ♦ ♠❡♥♦r ♥❛t✉r❛❧ n ❝♦♠ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ♦
í♥❞✐❝❡ ✭♦✉ ❣r❛✉✮ ❞❡ ♥✐❧♣♦tê♥❝✐❛ ❞❡ A✳
❉❡✜♥✐çã♦ ✶✳✶✻✳ ❙❡❥❛ ϑ ✉♠❛ ❝❧❛ss❡ ❞❡ F✲á❧❣❡❜r❛s ❡ A ∈ ϑ ✉♠❛ F✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣♦r ✉♠ ❝♦♥❥✉♥t♦X✳ ❆ F✲á❧❣❡❜r❛ A é ❝❤❛♠❛❞❛ ✉♠❛ F✲á❧❣❡❜r❛ ❧✐✈r❡ ♥❛ ❝❧❛ss❡ϑ ❧✐✈r❡♠❡♥t❡ ❣❡r❛❞❛ ♣❡❧♦ ❝♦♥❥✉♥t♦ X✱ s❡ ♣❛r❛ q✉❛❧q✉❡r á❧❣❡❜r❛ B ∈ϑ✱ q✉❛❧q✉❡r ❛♣❧✐❝❛çã♦ ψ :X → B ♣♦❞❡ s❡r ❡st❡♥❞✐❞❛ ❛ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ϕ:A →B✳ ❆ ❝❛r❞✐♥❛❧✐❞❛❞❡ |X| ❞♦ ❝♦♥❥✉♥t♦ X é ❝❤❛♠❛❞♦ ♣♦st♦ ❞❡ A✳
❊①❡♠♣❧♦ ✶✳✶✼✳ P❛r❛ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ X ❛ á❧❣❡❜r❛ ♣♦❧✐♥♦♠✐❛❧ F[X] é ❧✐✈r❡ ♥❛ ❝❧❛ss❡ ❞❡ t♦❞❛s á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✱ ❝♦♠✉t❛t✐✈❛s ❡ ✉♥✐tár✐❛s✳
❊①❡♠♣❧♦ ✶✳✶✽✳ ❙❡❥❛ X = {x1, . . . , xn, . . .} ✉♠ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ ❡♥✉♠❡rá✈❡❧✳ ❈❤❛✲
♠❛r❡♠♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ X ❞❡ ✈❛r✐á✈❡✐s ♦✉ ✐♥❞❡t❡r♠✐♥❛❞❛s✳ ❯♠❛ ♣❛❧❛✈r❛ ❞❡ X é ✉♠❛ s❡q✉ê♥❝✐❛ xi1· · ·xin✱ ❝♦♠ n ∈ N✳ ❆ ♣❛❧❛✈r❛ xi1· · ·xis✱ ❡♠ q✉❡ s = 0✱ é
❛ ♣❛❧❛✈r❛ ✈❛③✐❛ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r 1✳ ●❡r❛❧♠❡♥t❡✱ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r s ≥ 1 ✭s❛❧✈♦ ♠❡♥çã♦ ❝♦♥trár✐❛✮✳ ❈❤❛♠❛r❡♠♦s ❞❡ ♠♦♥ô♠✐♦s ♦ ♣r♦❞✉t♦ ❞❡ ✉♠ ❡s❝❛❧❛r ♣♦r ✉♠❛ ♣❛❧❛✈r❛ ❡♠ ❳✱ ♦✉ s❡❥❛✱ αxi1· · ·xin ♦♥❞❡ α ∈ F ❡ xi1, . . . , xin ∈ X✳ ❉✐r❡♠♦s q✉❡
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✽
it = jt✱ ♣❛r❛ t♦❞♦ t✳ ❖ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ♠♦♥ô♠✐♦s é ❞❛❞♦ ♣♦r ❥✉st❛♣♦s✐çã♦ ❞❡✜♥✐❞❛ ♣♦r
(xi1· · ·xim)(xj1· · ·xjn) =xi1· · ·xim·xj1· · ·xjn, xik, xjl ∈X✳
❖ F✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ FhXi = {P(i)αixi1. . . xin| αi ∈ F, n ≥ 1} é ✉♠❛ F✲á❧❣❡❜r❛✱
♥ã♦ ❝♦♠✉t❛t✐✈❛✱ ❛ss♦❝✐❛t✐✈❛ ❡ s❡♠ ✉♥✐❞❛❞❡✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ FhXi sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ♦✉ s❡❥❛✱ s♦♠❛s ❢♦r♠❛✐s ❞❡ ♠♦♥ô♠✐♦s✳ ❙❡ f ∈ FhXi✱ ❡s❝r❡✈❡r❡♠♦s f = f(x1, . . . , xn) =
n X
i=1
αiwi ♣❛r❛ ✐♥❞✐❝❛r q✉❡ x1, . . . , xn ∈ X sã♦ ❛s ú♥✐❝❛s ✐♥❞❡✲
t❡r♠✐♥❛❞❛s q✉❡ ❛♣❛r❡❝❡♠ ❡♠ f✱ ❡♠ q✉❡ αi ∈ F ❡ wi sã♦ ♣❛❧❛✈r❛s q✉❡ ❞❡♣❡♥❞❡♠ ❞❡
x1, . . . , xn✳
❉❡ ❛❣♦r❛ ❡♠ ❞✐❛♥t❡ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❛ á❧❣❡❜r❛ ❧✐✈r❡ FhXi ❛ss♦❝✐❛t✐✈❛✱ ♥ã♦ ❝♦♠✉✲ t❛t✐✈❛✱ ♥ã♦ ✉♥✐tár✐❛ ❞❡ ♣♦st♦ ❡♥✉♠❡rá✈❡❧ ♥♦ ❝♦♥❥✉♥t♦ X = {x1, x2, . . .} ✭s❛❧✈♦ ♠❡♥çã♦ ❝♦♥trár✐❛✮✳
❖❜s❡r✈❛çã♦ ✶✳✶✾✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ ❝♦♥s✐❞❡r❛r♠♦s ❛ ♣❛❧❛✈r❛ ✈❛③✐❛✱ ❡♥tã♦ ✈❛♠♦s ♦❜t❡r ❛ á❧❣❡❜r❛ ❧✐✈r❡ ❛ss♦❝✐❛t✐✈❛✱ ♥ã♦ ❝♦♠✉t❛t✐✈❛✱ ✉♥✐tár✐❛✱ ❝✉❥❛ ✐❞❡♥t✐❞❛❞❡ é ❛ ♣❛❧❛✈r❛ ✈❛③✐❛ ✭♠♦♥ô♠✐♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♥✉❧♦✮✳
✶✳✷ ■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s
◆❡st❛ s❡çã♦ ❞❛r❡♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ ❡①❡♠♣❧♦s ❞❡ á❧❣❡❜r❛s q✉❡ s❛t✐s❢❛③❡♠ ✐❞❡♥✲ t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s✳
❉❡✜♥✐çã♦ ✶✳✷✵✳ ❙❡❥❛A✉♠❛F✲á❧❣❡❜r❛ ❡f =f(x1, . . . , xn)∈FhXi✳ ❉✐③❡♠♦s q✉❡f ≡0
é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ❞❡ A s❡ f(a1, . . . , an) = 0 ♣❛r❛ t♦❞♦s a1, . . . , an∈A✳
❱❛♠♦s ❞❡♥♦t❛r ♣♦r Id(A) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ✉♠❛ F✲ á❧❣❡❜r❛A✳
❉❡✜♥✐çã♦ ✶✳✷✶✳ ❙❡ A s❛t✐s❢❛③ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♥ã♦ tr✐✈✐❛❧ f ≡ 0✱ ❞✐③❡♠♦s q✉❡ A é ✉♠❛ P■✲á❧❣❡❜r❛✳
❉❡✜♥✐çã♦ ✶✳✷✷✳ ❉✉❛s P■✲á❧❣❡❜r❛s A ❡ B sã♦ ❞✐t❛s P■✲❡q✉✐✈❛❧❡♥t❡s s❡ Id(A) = Id(B)✳
❙❡❥❛♠ A ✉♠❛ á❧❣❡❜r❛ ❡ a, b∈A ❞❡✜♥✐♠♦s [a, b] = ab−ba ♦ ❝♦♠✉t❛❞♦r ❞❡ ▲✐❡ ❞❡ a ❡ b✳ ❉❡ ✉♠ ♠♦❞♦ ❣❡r❛❧✱ ❞❡✜♥✐♠♦s ♦ ❝♦♠✉t❛❞♦r ❞❡ ❝♦♠♣r✐♠❡♥t♦ n [a1, . . . , an−1, an] =
[[a1, . . . , an−1], an]♣❛r❛ ai ∈A✳
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✾
❊①❡♠♣❧♦ ✶✳✷✸✳ ❙❡ A é ✉♠❛ á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛✱ ❡♥tã♦A é ✉♠❛ P■✲á❧❣❡❜r❛ q✉❡ s❛t✐s❢❛③ ❛ ✐❞❡♥t✐❞❛❞❡ [x, y]≡0✳
❊①❡♠♣❧♦ ✶✳✷✹✳ ◗✉❛❧q✉❡r á❧❣❡❜r❛ ♥✐❧♣♦t❡♥t❡ é ✉♠❛ P■✲á❧❣❡❜r❛✳ ❉❡ ❢❛t♦ t❡♠♦s q✉❡An=
0✱ ♣❛r❛ ❛❧❣✉♠ n≥1✱ ❡♥tã♦ x1. . . xn ≡0 é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ❞❡ A✳
❊①❡♠♣❧♦ ✶✳✷✺✳ ❙❡❥❛ A ✉♠❛ á❧❣❡❜r❛ ♥✐❧ ❞❡ ❣r❛✉ ❧✐♠✐t❛❞♦✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ n ≥ 1 t❛❧ q✉❡ an = 0✱ ♣❛r❛ t♦❞♦ a ∈ A✳ ❊♥tã♦ ♦ ♣♦❧✐♥ô♠✐♦ xn ≡ 0 é ✉♠❛ ✐❞❡♥t✐❞❛❞❡
♣♦❧✐♥♦♠✐❛❧ ❞❡ A✳
❊①❡♠♣❧♦ ✶✳✷✻✳ ❙❡❥❛ A ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❡ s❡❥❛ dimA < n✳ ❊♥tã♦ A s❛t✐s❢❛③ ❛ ✐❞❡♥t✐❞❛❞❡ ❙t❛♥❞❛r❞ ❞❡ ♣♦st♦ n
Stn(x1, . . . , xn) = X
σ∈Sn
(sign σ) xσ(1)· · ·xσ(n) = 0
♦♥❞❡ Sn é ♦ ❣r✉♣♦ s✐♠étr✐❝♦ ❞❡ ❣r❛✉ n✳ ❆ á❧❣❡❜r❛ A t❛♠❜é♠ s❛t✐s❢❛③ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡
❈❛♣❡❧❧✐ ❞❡ ♣♦st♦ n
Capn(x1, . . . , xn;y1, . . . , yn+1) = X
σ∈Sn
(sign σ) y1xσ(1)y2. . . ynxσ(n)yn+1 = 0.
❆ s❡❣✉✐r ❞❛r❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ P■✲á❧❣❡❜r❛ q✉❡ s❡rá ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♦s ♥♦ss♦s ❡st✉❞♦s ❛❞✐❛♥t❡✳
❊①❡♠♣❧♦ ✶✳✷✼✳ ❙❡❥❛ G ❛ á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥ ✭♦✉ á❧❣❡❜r❛ ❡①t❡r✐♦r✮ ♥ã♦ ✉♥✐tár✐❛ ❡♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ ❡♥✉♠❡rá✈❡❧ s♦❜r❡ ✉♠ ❝♦r♣♦ ❞❡ charF 6= 2✳ ❆ á❧❣❡❜r❛ G ♣♦❞❡ s❡r ❝♦♥str✉í❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳ ❙❡❥❛ FhXi❛ á❧❣❡❜r❛ ❧✐✈r❡ ❛ss♦❝✐❛t✐✈❛ ❞❡ ♣♦st♦ ❡♥✉♠❡rá✈❡❧ ❡♠ X={x1, x2, . . .}✳ ❙❡ ■ é ✉♠ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❞❡FhXi❣❡r❛❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦❧✐♥ô♠✐♦s {xixj +xjxi|i, j ≥ 1}✱ ❡♥tã♦ G = FhXiI ✳ ❊ ❡s❝r❡✈❡♠♦s ei = xi +I ∈ G
♣❛r❛ ❝❛❞❛ i= 1,2, . . .✳ ❈♦♠♦ charF 6= 2 t❡♠♦s q✉❡ e2
i = 0✳ ❊♥tã♦ G é ❣❡r❛❞♦ ♣♦r✿
G=he1, e2, . . .|eiej =−ejei, ∀ i, j ≥1i.
◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ ♦ ❝♦♥❥✉♥t♦ B = {ei1. . . eik|1 ≤ i1 < · · · < iik} é ✉♠❛
F✲❜❛s❡ ♣❛r❛ ❛ á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥ ♥ã♦ ✉♥✐tár✐❛✳ ➱ ❝♦♥✈❡♥✐❡♥t❡ ❡s❝r❡✈❡rG❝♦♠♦ s♦♠❛ ❞✐r❡t❛ ❞♦s s❡❣✉✐♥t❡s s✉❜❡s♣❛ç♦s ✈❡t♦r✐❛✐s✿
G(0) = span{ei1. . . ei2k|1≤i1 <· · ·< i2k, k > 0}
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✶✵
♦♥❞❡ G(0) é ♦ ❡s♣❛ç♦ ❣❡r❛❞♦ ♣♦r t♦❞♦s ♠♦♥ô♠✐♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♣❛r ❡ G(1) é ♦ ❡s✲ ♣❛ç♦ ❣❡r❛❞♦ ♣♦r t♦❞♦s ♦s ♠♦♥ô♠✐♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ í♠♣❛r✳ ❙❡❣✉❡ ❞❡ eiej = −ejei q✉❡
(ei1. . . eim).(ej1. . . ejk) = (−1)
mk(e
j1. . . ejk).(ei1. . . eim) ♣❛r❛ q✉❛✐sq✉❡r m, k ∈ N✳ P♦❞❡✲
♠♦s ❝♦♥❝❧✉✐r q✉❡ ax = xa ♣❛r❛ q✉❛✐sq✉❡r a ∈ G(0) ❡ x ∈ G ❡ bc =−cb ♣❛r❛ q✉❛✐sq✉❡r b, c∈G(1)✳
❆❣♦r❛ ❢❛r❡♠♦s ❝♦♥s✐❞❡r❛çõ❡s ✐♠♣♦rt❛♥t❡s ❛❝❡r❝❛ ❞❡st❛ á❧❣❡❜r❛✿
✶✳ G(0)G(0)+G(1)G(1) ⊆G(0) ✷✳ G(0)G(1)+G(1)G(0) ⊆G(1) ✸✳ G(0) =Z(G)
✹✳ ❖ ♣r✐♠❡✐r♦ ❡ t❡r❝❡✐r♦ ✐t❡♠✱ ♥♦s ❞ã♦ q✉❡G(0) é ✉♠❛ s✉❜á❧❣❡❜r❛ ❞❡ ●✳
✺✳ ● s❛t✐s❢❛③ ❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ [x, y, z] = [[x, y], z] ≡ 0✳ ❉❡ ❢❛t♦ ❝♦♠♦ G(0) = Z(G)✱ é ❝❧❛r♦ q✉❡ q✉❛❧q✉❡r ❝♦♠✉t❛❞♦r ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s ❞❡ G ♣❡rt❡♥❝❡ ❛ G(0)✱ ♦✉ s❡❥❛✱ é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ♠♦♥ô♠✐♦s eis ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♣❛r✳ ❆ss✐♠
[[G, G], G] = 0✳
◆♦t❡ q✉❡ ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s ❞❡ á❧❣❡❜r❛s sã♦ ❡①♣r❡ss❛s ❡♠ ❧✐♥❣✉❛❣❡♠ ❞❡ ✐❞❡♥✲ t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s✳ ◆♦s ❡①❡♠♣❧♦s ❞❛❞♦s ❛❝✐♠❛ ♣♦❞❡♠♦s ♣❡r❝❡❜❡r ✐st♦✳ ❉❡ ❢❛t♦✱ ✈✐♠♦s ♣♦r ❡①❡♠♣❧♦ q✉❡ ✉♠❛ á❧❣❡❜r❛ A é ♥✐❧ ❞❡ ❣r❛✉ ❧✐♠✐t❛❞♦✱ s❡ ❡①✐st❡ ✉♠ n ∈ N t❛❧ q✉❡ xn≡0 é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣❛r❛ A✳
✶✳✸ ❚✲✐❞❡❛✐s ❡ ❱❛r✐❡❞❛❞❡s ❞❡ ➪❧❣❡❜r❛s
◆❡st❛ s❡çã♦ ✐♥tr♦❞✉③✐♠♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ❚✲✐❞❡❛❧ ❡ ❞❡ ✈❛r✐❡❞❛❞❡ ❞❡ á❧❣❡❜r❛s✳ ❱❛♠♦s ♠♦str❛r q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s Id(A) é ✉♠ ❚✲✐❞❡❛❧✳ ❆❧é♠ ❞✐ss♦ ❡①✐✲ ❜✐♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s ❞❡ á❧❣❡❜r❛s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞❛s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳
❯♠❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ✉♠❛ á❧❣❡❜r❛A ♣♦❞❡ ❛✐♥❞❛ s❡r ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣❛r❛ ♦✉tr❛s á❧❣❡✲ ❜r❛s ❞✐❢❡r❡♥t❡s✳ P♦r ✐ss♦✱ ❞❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦❧✐♥ô♠✐♦s S✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❛ ❝❧❛ss❡ ❞❡ t♦❞❛s á❧❣❡❜r❛s q✉❡ s❛t✐s❢❛③❡♠ ❛s ✐❞❡♥t✐❞❛❞❡s ❞❡S✳ ■st♦ ❧❡✈❛ ❛ ♥♦çã♦ ❞❡ ✈❛r✐❡❞❛❞❡s ❞❡ á❧❣❡❜r❛s✳
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✶✶
❉❡♠♦♥str❛çã♦✿ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ ♦ ❝♦♥❥✉♥t♦ Id(A) é ✉♠ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❞❡ FhXi✳ ❆❧é♠ ❞✐ss♦✱ Id(A)é ❢❡❝❤❛❞♦ s♦❜ ❡♥❞♦♠♦r✜s♠♦s ❞❡ FhXi✳ ❉❡ ❢❛t♦✱ s❡❥❛♠
f =f(x1, . . . , xn)∈Id(A)✱ g1, . . . , gn ∈FhXi ❡ ϕ∈End(FhXi)✱ t❛❧ q✉❡
ϕ :FhXi → FhXi xi 7→ gi.
❙❡❣✉❡ q✉❡
ϕ(f(x1, . . . , xn)) = f(ϕ(x1), . . . , ϕ(xn)) =f(g1, . . . , gn) =
= f(g1(xi1, . . . , xin), . . . , gn(xi1, . . . , xin)).
◆♦t❡ q✉❡ f(g1(xi1, . . . , xim), . . . , gn(xi1, . . . , xim)) = 0 ❡♠ A✱ ♣♦✐s ♣❛r❛ q✉❛✐sq✉❡r
a1, . . . , am ∈ A✱ t❡♠♦s gi(a1, . . . , am) ∈ A, ∀i = 1, . . . , m ❡ ❝♦♠♦ f ∈ Id(A)✱ ❡♥tã♦
f(g1, . . . , gn)∈Id(A)✳ P♦rt❛♥t♦ ϕ(Id(A))⊆Id(A)✳
❚❡♠♦s q✉❡ ✉♠ ✐❞❡❛❧ ❝♦♠ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❝❤❛♠❛❞♦ ❞❡ ❚✲✐❞❡❛❧✳
❉❡✜♥✐çã♦ ✶✳✷✾✳ ❯♠ ✐❞❡❛❧I ❞❡FhXié ✉♠ ❚✲✐❞❡❛❧ s❡ϕ(I)⊆I ♣❛r❛ t♦❞♦ ❡♥❞♦♠♦r✜s♠♦ ϕ ❞❡ FhXi✳
❉✐st♦✱ s❡❣✉❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ Id(A) é ✉♠ ❚✲✐❞❡❛❧ ❞❡ FhXi✳ P♦r ♦✉tr♦ ❧❛❞♦ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ t♦❞♦ ❚✲✐❞❡❛❧ ❞❡ FhXi é ❞❡ ❢❛t♦ ❞❡st❡ t✐♣♦✳
▲❡♠❛ ✶✳ ❙❡❥❛ I ✉♠ ❚✲✐❞❡❛❧✳ ❊♥tã♦ I =Id(FhXI i).
❉❡♠♦♥str❛çã♦✿✭⊆✮ ❙❡❥❛f(x1, . . . , xn)∈I✳ ❈♦♥s✐❞❡r❡n❡❧❡♠❡♥t♦s q✉❛✐sq✉❡rg1,¯ g2, . . . ,¯ g¯n ∈ FhXi
I ✳ ❚❡♠♦s f( ¯g1,g2, . . . ,¯ g¯n) = f(g1, . . . , gn) = ¯0✱ ♣♦✐s f(g1, . . . , gn) ∈ I✳ ▲♦❣♦ f ∈
Id(FhXiI )✳
✭⊇✮ ❙❡❥❛ f(x1, . . . , xn)∈ Id(FhXI i)✳ ❈♦♠♦ x1, . . . ,¯ x¯n ∈ FhXiI ✱ t❡♠♦s ¯0 =f( ¯x1, . . . ,x¯n) =
f(x1, . . . , xn)✳ ❊♥tã♦ f(x1, . . . , xn)∈I✳
❉❡✜♥✐çã♦ ✶✳✸✵✳ ❙❡❥❛A ✉♠❛ á❧❣❡❜r❛ ❡ S ⊆A✳ ❖ ✐❞❡❛❧ ❣❡r❛❞♦ ♣♦r S é ♦ ♠❡♥♦r ✐❞❡❛❧ ❞❡ A q✉❡ ❝♦♥té♠ S✳
❖❜s❡r✈❡ q✉❡ ♦ ✐❞❡❛❧ ❣❡r❛❞♦ ♣♦r S ❡①✐st❡✱ ♣♦✐s é ❛ ✐♥t❡rs❡çã♦ ❞❡ t♦❞♦s ♦s ✐❞❡❛✐s ❞❡ A q✉❡ ❝♦♥té♠ S✳
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✶✷
❉❡✜♥✐çã♦ ✶✳✸✷✳ ❙❡❥❛ S ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❡♠ FhXi ❡ f ∈ FhXi✳ ❉✐③❡♠♦s q✉❡ f é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦s ♣♦❧✐♥ô♠✐♦s ❡♠ S✱ s❡ f ∈ hSiT✱ ♦✉ s❡❥❛✱ f ♣❡rt❡♥❝❡ ❛♦
❚✲✐❞❡❛❧ ❣❡r❛❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ S✳
❊①❡♠♣❧♦ ✶✳✸✸✳ ❙❡A é ✉♠❛F✲á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛ ❡ ✉♥✐tár✐❛✱ ❡♥tã♦Id(A) = h[x1, x2]iT✳
❊①❡♠♣❧♦ ✶✳✸✹✳ ❙❡❥❛ G❛ á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥ s♦❜r❡ ✉♠ ❝♦r♣♦ F ❞❡ char6= 2✳ ❊♥tã♦ Id(G) =h[x1, x2, x3]iT✳ ❱❡r ❬✼❪✳
❉❡✜♥✐çã♦ ✶✳✸✺✳ ❉♦✐s ❝♦♥❥✉♥t♦s ❞❡ ♣♦❧✐♥ô♠✐♦s sã♦ ❡q✉✐✈❛❧❡♥t❡s s❡ ❡❧❡s ❣❡r❛♠ ♦ ♠❡s♠♦ ❚✲✐❞❡❛❧✳
❱✐♠♦s q✉❡ q✉❛❧q✉❡r á❧❣❡❜r❛A❞❡t❡r♠✐♥❛ ✉♠ ❚✲✐❞❡❛❧ ❞❡FhXi✳ P♦r ♦✉tr♦ ❧❛❞♦ ✈ár✐❛s á❧❣❡❜r❛s ♣♦❞❡♠ ❝♦rr❡s♣♦♥❞❡r ❛ ✉♠ ♠❡s♠♦ ❚✲✐❞❡❛❧✱ ❝♦♠♦ ♦ ✐❞❡❛❧ ❞❡ s✉❛s ✐❞❡♥t✐❞❛❞❡s✳ ❉❡✜♥✐çã♦ ✶✳✸✻✳ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ S ⊆FhXi✱ ❛ ❝❧❛ss❡ ❞❡ t♦❞❛s ❛s á❧❣❡❜r❛s A t❛✐s q✉❡f ≡0 ❡♠ A ♣❛r❛ t♦❞♦f ∈S é ❝❤❛♠❛❞❛ ❞❡ ✈❛r✐❡❞❛❞❡ ν=ν(S) ❞❡t❡r♠✐♥❛❞❛ ♣♦r S✳
❉✐r❡♠♦s q✉❡νé ✉♠❛ ✈❛r✐❡❞❛❞❡ tr✐✈✐❛❧ s❡Id(ν) = (0)✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ν é ❝❤❛♠❛❞❛ ♥ã♦ tr✐✈✐❛❧ s❡S 6= 0 ❡ν é ❞✐t❛ ♣ró♣r✐❛ s❡ ❡❧❛ ♥ã♦ é tr✐✈✐❛❧ ❡ ❝♦♥té♠ ✉♠❛ á❧❣❡❜r❛ ♥ã♦✲ ♥✉❧❛✳
◆♦t❡ q✉❡ s❡νé ❛ ✈❛r✐❡❞❛❞❡ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ ❝♦♥❥✉♥t♦ S❡hSiT é ♦ ❚✲✐❞❡❛❧ ❞❡FhXi
❣❡r❛❞♦ ♣♦r S✱ ❡♥tã♦ ν(S) = ν(hSiT) ❡ hSiT = ∩
A∈νId(A)✳ ❙❡♥❞♦ ❛ss✐♠ ❡s❝r❡✈❡♠♦s
hSiT = Id(ν)✳ ◗✉❛♥❞♦ ❡①✐st❡ ✉♠❛ á❧❣❡❜r❛ A t❛❧ q✉❡ Id(A) = Id(ν)✱ ❞❡♥♦t❛r❡♠♦s ♣♦r
ν =var(A)✱ ❡ ❞✐r❡♠♦s q✉❡ν é ❛ ✈❛r✐❡❞❛❞❡ ❣❡r❛❞❛ ♣❡❧❛ á❧❣❡❜r❛ A✳ ❉❡✜♥✐çã♦ ✶✳✸✼✳ ❆ ✈❛r✐❡❞❛❞❡ ν˜ é ❝❤❛♠❛❞❛ ❞❡ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ ν s❡ ν˜⊆ν✳
❊①❡♠♣❧♦ ✶✳✸✽✳ ❆ ❝❧❛ss❡ ❞❡ t♦❞❛s ❛s á❧❣❡❜r❛s ❝♦♠✉t❛t✐✈❛s ❢♦r♠❛♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣ró✲ ♣r✐❛✱ ❝♦♠ S ={[x, y]}✳
❊①❡♠♣❧♦ ✶✳✸✾✳ ❆ ❝❧❛ss❡ ❞❡ t♦❞❛s ❛s á❧❣❡❜r❛s ♥✐❧ ❞❡ ❣r❛✉ ❧✐♠✐t❛❞♦ ♣♦r n✱ ❢♦r♠❛♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣ró♣r✐❛ ❝♦♠ S ={xn}✳
❊①❡♠♣❧♦ ✶✳✹✵✳ ❆ ❝❧❛ss❡ ❞❡ t♦❞❛s ❛s á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s é t❛♠❜é♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡✜♥✐❞❛ ♣❡❧♦ ♣♦❧✐♥ô♠✐♦ ♥✉❧♦✳
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✶✸
❚❡♦r❡♠❛ ✶✳✹✶✳ ❊①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐❥❡t✐✈❛ ϕ ❡♥tr❡ ❚✲✐❞❡❛✐s ❞❡ FhXi ❡ ✈❛r✐❡✲ ❞❛❞❡s ❞❡ á❧❣❡❜r❛s✳ P❛r❛ q✉❛✐sq✉❡r ❞♦✐s ❚✲✐❞❡❛✐s I1, I2 ❛ ✐♥❝❧✉sã♦ I1 ⊂I2 é ❡q✉✐✈❛❧❡♥t❡ ❛ ϕ(I1)⊃ϕ(I2)✳
❉❡♠♦♥str❛çã♦✿ P❛r❛ ❝❛❞❛ ❚✲✐❞❡❛❧ I✱ s❡❥❛ V = ϕ(I) ❛ ✈❛r✐❡❞❛❞❡ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦❧✐♥ô♠✐♦s I✳ ➱ ó❜✈✐♦ q✉❡ ϕ é s♦❜r❡❥❡t✐✈❛✱ ♣❡❧❛ ♣ró♣r✐❛ ❞❡✜♥✐çã♦✳ ❆❣♦r❛ s❡❥❛♠ I1 6= I2 ❡ ϕ(Ii) = Vi, i = 1,2✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô♠✐♦ f = f(x1, . . . , xn) q✉❡
❡stá ❡♠ I1\I2 ✭♦✉ ❡♠ I2\I1✮✳ ❙✉♣♦♥❤❛ q✉❡ f ∈ I1\I2✱ ❡♥tã♦ f(x1, . . . , xn) ≡ 0 é ✉♠❛
✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛V1 ❡ ♥ã♦ é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣❛r❛ ❛ á❧❣❡❜r❛ r❡❧❛t✐✈❛♠❡♥t❡ ❧✐✈r❡
FhXi
I2 ∈V2✱ ♣♦✐sId(
FhXi
I2 ) = I2✳ s❡❣✉❡ q✉❡V1 6=V2✳ ▲♦❣♦ϕé ✐♥❥❡t✐✈❛✳ ❊ ❛ss✐♠ ♠♦str❛♠♦s
q✉❡ ϕ é ❜✐❥❡t✐✈❛✳ ❆❣♦r❛ ✈❡❥❛ q✉❡✱ V1 ⊃V2 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t♦❞❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ❞❡V1 é t❛♠❜é♠ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣❛r❛ V2✱ ✐st♦ é✱ Id(V1)⊂Id(V2)✳
✶✳✹ P♦❧✐♥ô♠✐♦s ❍♦♠♦❣ê♥❡♦s ❡ ▼✉❧t✐❧✐♥❡❛r❡s
❖ ❡st✉❞♦ ❞❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ✉♠❛ ❞❛❞❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❝♦r♣♦ ❜❛s❡ F ✐♥✜♥✐t♦✱ ♣♦❞❡ s❡r r❡❞✉③✐❞♦ ❛♦ ❡st✉❞♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ♦✉ ♣♦❧✐♥ô♠✐♦s ♠✉❧t✐❧✐✲ ♥❡❛r❡s ♥♦ ❝❛s♦ ❡♠ q✉❡ ♦ ❝♦r♣♦ ❜❛s❡ é ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦✳
❙❡❥❛Fn=Fhx1, . . . , xni✉♠❛ á❧❣❡❜r❛ ❧✐✈r❡ ❞❡ ♣♦st♦n ≥1s♦❜r❡F✳ ❚❛❧ á❧❣❡❜r❛ ♣♦❞❡
s❡r ❞❡❝♦♠♣♦st❛ ❝♦♠♦✿
Fn =Fn(1)⊕Fn(2)⊕. . .
♦♥❞❡✱ ♣❛r❛ t♦❞♦ k ≥ 1✱ Fn(k) é ♦ s✉❜s❡s♣❛ç♦ ❣❡r❛❞♦ ♣♦r t♦❞♦s ♠♦♥ô♠✐♦s ❞❡ ❣r❛✉ k✳
❉❡s❞❡ ❞❡ q✉❡Fn(i)Fn(j) ⊆Fn(i+j)✱ ♣❛r❛ t♦❞♦ i, j ≥1✱ ❞✐③❡♠♦s q✉❡Fn é ❣r❛❞✉❛❞❛ ♣❡❧♦ ❣r❛✉
♦✉ q✉❡ ❡❧❛ t❡♠ ❡str✉t✉r❛ ❞❡ á❧❣❡❜r❛ ❣r❛❞✉❛❞❛✳ ❆s Fn(i) sã♦ ❝❤❛♠❛❞❛s ❞❡ ❝♦♠♣♦♥❡♥t❡s
❤♦♠♦❣ê♥❡❛s ❞❡ Fn✳ ❙✐♠✐❧❛r♠❡♥t❡✱ ✐♥tr♦❞✉③✐♠♦s ✉♠❛ ♠✉❧t✐❣r❛❞✉❛çã♦ ❡♠ Fn✱ s❡❣✉❡ q✉❡
♣❛r❛ t♦❞♦k ≥1 ❡s❝r❡✈❡♠♦s✿
Fn(k)=⊕i1+···+in=kF
(i1,...,in)
n
♦♥❞❡ F(i1,...,in)
n é ♦ s✉❜❡s♣❛ç♦ ❣❡r❛❞♦ ♣❡❧♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❣r❛✉ i1 ❡♠ x1, . . . , in ❡♠ xn✳
❆❧é♠ ❞✐ss♦F(i1,...,in)
n Fn(j1,...,jn)⊆Fn(i1+j1,...,in+jn) ❡ ❞✐③❡♠♦s q✉❡ Fn é ♠✉❧t✐❣r❛❞✉❛❞❛✳
❉❡✜♥✐çã♦ ✶✳✹✷✳ ❯♠ ♣♦❧✐♥ô♠✐♦ f ♣❡rt❡♥❝❡♥t❡ ❛ Fn(k) ♣❛r❛ ❛❧❣✉♠ k ≥ 1✱ é ❝❤❛♠❛❞♦
❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉k✳ ❙❡f ♣❡rt❡♥❝❡ à ❛❧❣✉♠F(i1,...,in)
n ❡❧❡ é ❝❤❛♠❛❞♦ ♠✉❧t✐❤♦♠♦❣ê♥❡♦
❞❡ ♠✉❧t✐❣r❛✉ (i1, . . . , in)✳ ❚❛♠❜é♠ ❞✐③❡♠♦s q✉❡ ♦ ♣♦❧✐♥ô♠✐♦ f é ❤♦♠♦❣ê♥❡♦ ♥❛ ✈❛r✐á✈❡❧
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✶✹
❊①❡♠♣❧♦ ✶✳✹✸✳ ❖ ♣♦❧✐♥ô♠✐♦
f(x1, x2, x3) = x1x22x3+x22x1x3+x3x1x22
é ♠✉❧t✐❤♦♠♦❣ê♥❡♦ ❞❡ ♠✉❧t✐❣r❛✉ ✭✶✱✷✱✶✮✳ ❊①❡♠♣❧♦ ✶✳✹✹✳ ❖ ♣♦❧✐♥ô♠✐♦
f(x1, x2) = x1x22+x2x1
é ❤♦♠♦❣ê♥❡♦ ❡♠ x1 ❝♦♠ degx1 = 1✳
❙❡f ∈FhXi✱ ♣♦❞❡♠♦s s❡♠♣r❡ ❡s❝r❡✈❡r
f =Pi1≥0,...,in≥0f(i1,...,in)
♦♥❞❡ f(i1,...,in) ∈ F(i1,...,in)
n é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ t♦❞♦s ♠♦♥ô♠✐♦s ❡♠ f ♦♥❞❡ x1
❛♣❛r❡❝❡ ❝♦♠ ❣r❛✉ i1, . . . , xn ❛♣❛r❡❝❡ ❝♦♠ ❣r❛✉ in✳ ❖s ♣♦❧✐♥ô♠✐♦s f(i1,...,in) q✉❡ sã♦ ♥ã♦
♥✉❧♦s✱ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝♦♠♣♦♥❡♥t❡s ♠✉❧t✐❤♦♠♦❣ê♥❡❛s ❞❡f ❝♦♠ ♠✉❧t✐❣r❛✉(i1, . . . , in)✳
❊①❡♠♣❧♦ ✶✳✹✺✳
f(x1, x2) = x1x2+x22x1+ 5x2x1 = x1x2+ 5x2x1
| {z }+x22x1
|{z}
= f(1,1)+f(1,2).
❖❜s❡r✈❛çã♦ ✶✳✹✻✳ ❯♠❛ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣♦rt❛♥t❡ ❞❡ ❚✲✐❞❡❛✐s é q✉❡ s❡ F é ✉♠ ❝♦r♣♦ ✐♥✜♥✐t♦✱ ❡❧❡s sã♦ ❤♦♠♦❣ê♥❡♦s ❝♦♠ r❡s♣❡✐t♦ ❛ ♠✉❧t✐❣r❛❞✉❛çã♦ ❛❝✐♠❛✳
❚❡♦r❡♠❛ ✶✳✹✼✳ ❙❡❥❛ F ✉♠ ❝♦r♣♦ ✐♥✜♥✐t♦✳ ❙❡ f ≡ 0 é ✉♠ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ ✉♠❛ á❧❣❡❜r❛ A✱ ❡♥tã♦ t♦❞❛ ❝♦♠♣♦♥❡♥t❡ ♠✉❧t✐❤♦♠♦❣ê♥❡❛ ❞❡ f é ❛✐♥❞❛ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ A✳
❉❡♠♦♥str❛çã♦✿ ◆♦t❡ q✉❡ ♣❛r❛ t♦❞❛ ✈❛r✐á✈❡❧ xt✱ 1 ≤ t ≤ n✱ ♣♦❞❡♠♦s ❞❡❝♦♠♣♦r f = Pm
i=0fi✱ ♦♥❞❡ fi é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ t♦❞♦s ♠♦♥ô♠✐♦s ❞❡ f ❡♠ q✉❡ x1 ❛♣❛r❡❝❡ ❝♦♠ ❣r❛✉ i ❡m =degx1f✳
❈♦♠♦ F é ✉♠ ❝♦r♣♦ ✐♥✜♥✐t♦✱ ❡①✐st❡♠ m+ 1 ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ❡♠ F✿ α0, . . . , αm✳ P♦r
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✶✺
F✳ ❊♥tã♦ f(αx1, . . . , xn) = m X
i=0
αifi(x1, . . . , xn)✳
❈♦♠♦f é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣❛r❛ A✱ t❡♠♦s q✉❡
f(αjx1, . . . , xn) = m X
i=0
αijfi(x1, . . . , xn)≡0 ✭✶✳✶✮
❡♠ A✱ ♣❛r❛ t♦❞♦ i= 0, . . . , m ❡j = 1, . . . , n+ 1✳
❆ss✐♠ ♦❜t❡♠♦s ✉♠ s✐st❡♠❛ ❤♦♠♦❣ê♥❡♦ ❝♦♠m+ 1✈❛r✐á✈❡✐s f0, . . . , fm✳ ❆❣♦r❛ ✈❛♠♦s
❡s❝r❡✈❡r ❛ ♠❛tr✐③ ❞❡ ❱❛♥❞❡r♠♦♥❞❡ ❛ss♦❝✐❛❞❛ ❛ ❡st❡ s✐st❡♠❛
∆ =
1 1. . . 1 α0 α1. . . αm
✳✳✳ ✳✳✳ ✳✳✳ αm
0 αm1 . . . αmn
.
❱❛♠♦s ❞❡♥♦t❛r fi(a1, . . . , an) = ¯fi ♣❛r❛ t♦❞♦a1, . . . , an ∈A✱ ❡♥tã♦ ❞❡ ✶✳✶ t❡♠♦s q✉❡
( ¯f0. . .f¯m)∆ = 0.
❈♦♠♦ ♦ ❞❡t❡r♠✐♥❛♥t❡det(∆) =Q0≤i<j≤m(αj −αi) é ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ ♣♦✐s ❡s❝♦❧❤❡♠♦s
t♦❞♦sα′s❞✐st✐♥t♦s✱ ❡♥tã♦∆❛❞♠✐t❡ ✐♥✈❡rs❛✳ ❙❡❣✉❡ q✉❡f0 ≡0, . . . , f
m ≡0sã♦ ✐❞❡♥t✐❞❛❞❡s
❞❡A✳
P♦r ✉♠ ❛r❣✉♠❡♥t♦ ❞❡ ✐♥❞✉çã♦ s♦❜r❡ t✱ ♠♦str❛♠♦s q✉❡ ♣❛r❛ t♦❞❛ ✈❛r✐á✈❡❧ xt✱ fi ≡
0 ♣❛r❛ t♦❞♦ i ≥ 0✳ P♦rt❛♥t♦ t♦❞❛ ❝♦♠♣♦♥❡♥t❡ ♠✉❧t✐❤♦♠♦❣ê♥❡❛ ❞❡ f é ❛✐♥❞❛ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ A✳
❈♦r♦❧ár✐♦ ✶✳✹✽✳ ❙♦❜r❡ ✉♠ ❝♦r♣♦ ✐♥✜♥✐t♦ t♦❞♦ ❚✲✐❞❡❛❧ é ❣❡r❛❞♦ ♣♦r s❡✉s ♣♦❧✐♥ô♠✐♦s ♠✉❧t✐❤♦♠♦❣ê♥❡♦s✳
❉❡✜♥✐çã♦ ✶✳✹✾✳ ❙❡ ✉♠ ♣♦❧✐♥ô♠✐♦f(x1, . . . , xn)é ♠✉❧t✐❤♦♠♦❣ê♥❡♦ ❝♦♠ ♠✉❧t✐❣r❛✉(1, . . . ,1)
❞✐③❡♠♦s q✉❡ f é ♠✉❧t✐❧✐♥❡❛r ❞❡ ❣r❛✉ n ♥❛s ✈❛r✐á✈❡✐s x1, . . . , xn✳
❊①❡♠♣❧♦ ✶✳✺✵✳ ❖s ♣♦❧✐♥ô♠✐♦s ❙t❛♥❞❛r❞ ❡ ❈❛♣❡❧❧✐ ❞❡ ♣♦st♦ n
Stn(x1, . . . , xn) =Pσ∈Sn(−1)
σ x
σ(1)· · ·xσ(n)
Capn(x1, . . . , xn;y1, . . . , yn+1) = X
σ∈Sn
(−1)σ y1xσ(1)y2· · ·ynxσ(n)yn+1
■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ❡ P■✲➪❧❣❡❜r❛ ✶✻
❉❡♥♦t❡ ♣♦r Pn ♦ ❡s♣❛ç♦ ❞❡ t♦❞♦s ♣♦❧✐♥ô♠✐♦s ♠✉❧t✐❧✐♥❡❛r❡s ❞❡ ❣r❛✉ n ♥❛s ✈❛r✐á✈❡✐s
x1, . . . , xn✳ ❯♠❛ ❜❛s❡ ♣❛r❛ Pn é ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❛❧❛✈r❛s
{xσ(1)xσ(2). . . xσ(n) |σ ∈Sn}.
▲♦❣♦✱ ❛ ❞✐♠❡♥sã♦ ❞❡ Pn én!✳
❉❡s❞❡ q✉❡ ❡♠ ✉♠ ♣♦❧✐♥ô♠✐♦ ♠✉❧t✐❧✐♥❡❛rf(x1, . . . , xn)∈Pn❝❛❞❛ ✈❛r✐á✈❡❧ ❛♣❛r❡❝❡ ❡♠
❝❛❞❛ ♠♦♥ô♠✐♦ ❝♦♠ ❣r❛✉ 1✱ ✜❝❛ ❝❧❛r♦ q✉❡ t❛❧ ♣♦❧✐♥ô♠✐♦ é s❡♠♣r❡ ❞❛ ❢♦r♠❛
f(x1, . . . , xn) = X
σ∈Sn
ασxσ(1). . . xσ(n)
✱ ♦♥❞❡ ασ ∈F ❡Sn é ♦ ❣r✉♣♦ s✐♠étr✐❝♦ ❡♠ {1, . . . , n}✳
❖❜s❡r✈❛çã♦ ✶✳✺✶✳ ❙❡ f(x1, . . . , xn) é ❧✐♥❡❛r ❡♠ ✉♠❛ ✈❛r✐á✈❡❧✱ ❞✐❣❛♠♦s x1✱ ❡♥tã♦
f(
k X
i=1
αiy1, x2, . . . , xn) = k X
i=1
αif(yi, x2, . . . , xn)
♦♥❞❡ αi ∈F✱ yi, xi ∈X✳
Pr♦♣♦s✐çã♦ ✶✳✺✷✳ ❙❡❥❛ A ✉♠❛ F✲á❧❣❡❜r❛ ❣❡r❛❞❛ ❝♦♠♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♣❡❧♦ ❝♦♥❥✉♥t♦ B s♦❜r❡F✱ ❡ s❡❥❛ f ∈Pn✳ ❊♥tã♦✱f é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ❞❡As❡✱ ❡ s♦♠❡♥t❡ s❡✱f(b1, . . . , bn) =
0 ♣❛r❛ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦s b1, . . . , bn ∈B✳
❉❡♠♦♥str❛çã♦✿✭⇒✮ ❈♦♠♦ f é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ A✱ ❡♥tã♦ f(a1, . . . , an) = 0 ♣❛r❛
q✉❛✐sq✉❡r a1, . . . , an∈A✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ ♦s ❡❧❡♠❡♥t♦s ❞❡ B✳
✭⇐✮ ❚❡♠♦s q✉❡ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ❞❡ A ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ✜♥✐t❛ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ B✳ ❙❡❥❛♠ ai = Ptjii=1αj(ii)uji ❡❧❡♠❡♥t♦s ❞❡ A✱ ♦♥❞❡ α
(i)
ji ∈ F✱ uji ∈ B
❝♦♠ i = 1, . . . , n✳ ❊♥tã♦✱ ❞❡s❞❡ q✉❡ f = f(x1, . . . , xn) é ❧✐♥❡❛r ❡♠ ❝❛❞❛ ✈❛r✐á✈❡❧✱ ♣❡❧❛
❖❜s❡r✈❛çã♦ ✶✳✺✶
f(a1, . . . , an) = f( t1
X
j1=1
α(1)j1 uj1, . . . ,
tn
X
jn=1
α(jnn)ujn)
=
t1
X
j1=1
α(1)j1 · · ·
tn
X
jn=1
α(jnn)f(uj1, . . . , ujn)
=
t1X,...,tn
j1,...,jn=1
αj1· · ·αjn0 = 0
