❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦
❢❛❝t♦r ♣♦❧②♥♦♠✐❛❧s
❜②
❚♦✈♦❤❡r② ❍❛❥❛t✐❛♥❛ ❘❛♥❞r✐❛♥❛r✐s♦❛
❚❤❡s✐s ♣r❡s❡♥t❡❞ ✐♥ ♣❛rt✐❛❧ ❢✉❧✜❧♠❡♥t ♦❢ t❤❡ r❡q✉✐r❡♠❡♥ts ❢♦r
t❤❡ ❞❡❣r❡❡ ♦❢ ▼❛st❡r ♦❢ ❙❝✐❡♥❝❡ ✐♥ ▼❛t❤❡♠❛t✐❝s ✐♥ t❤❡
❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ❛t ❙t❡❧❧❡♥❜♦s❝❤ ❯♥✐✈❡rs✐t②
❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ ❙t❡❧❧❡♥❜♦s❝❤✱ Pr✐✈❛t❡ ❇❛❣ ❳✶✱ ▼❛t✐❡❧❛♥❞ ✼✻✵✷✱ ❙♦✉t❤ ❆❢r✐❝❛✳ ❙✉♣❡r✈✐s♦r✿ Pr♦❢✳ ❋❧♦r✐❛♥ ❇r❡✉❡r❉❡❝❡♠❜❡r ✷✵✶✷
❉❡❝❧❛r❛t✐♦♥
❇② s✉❜♠✐tt✐♥❣ t❤✐s t❤❡s✐s ❡❧❡❝tr♦♥✐❝❛❧❧②✱ ■ ❞❡❝❧❛r❡ t❤❛t t❤❡ ❡♥t✐r❡t② ♦❢ t❤❡ ✇♦r❦ ❝♦♥t❛✐♥❡❞ t❤❡r❡✐♥ ✐s ♠② ♦✇♥✱ ♦r✐❣✐♥❛❧ ✇♦r❦✱ t❤❛t ■ ❛♠ t❤❡ s♦❧❡ ❛✉t❤♦r t❤❡r❡♦❢ ✭s❛✈❡ t♦ t❤❡ ❡①t❡♥t ❡①♣❧✐❝✐t❧② ♦t❤❡r✇✐s❡ st❛t❡❞✮✱ t❤❛t r❡♣r♦❞✉❝t✐♦♥ ❛♥❞ ♣✉❜✲ ❧✐❝❛t✐♦♥ t❤❡r❡♦❢ ❜② ❙t❡❧❧❡♥❜♦s❝❤ ❯♥✐✈❡rs✐t② ✇✐❧❧ ♥♦t ✐♥❢r✐♥❣❡ ❛♥② t❤✐r❞ ♣❛rt② r✐❣❤ts ❛♥❞ t❤❛t ■ ❤❛✈❡ ♥♦t ♣r❡✈✐♦✉s❧② ✐♥ ✐ts ❡♥t✐r❡t② ♦r ✐♥ ♣❛rt s✉❜♠✐tt❡❞ ✐t ❢♦r ♦❜t❛✐♥✐♥❣ ❛♥② q✉❛❧✐✜❝❛t✐♦♥✳ ❙✐❣♥❛t✉r❡✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚✳❍✳ ❘❛♥❞r✐❛♥❛r✐s♦❛ ✷✵✶✷✴✶✷✴✶✷ ❉❛t❡✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈♦♣②r✐❣❤t ➞ ✷✵✶✷ ❙t❡❧❧❡♥❜♦s❝❤ ❯♥✐✈❡rs✐t② ❆❧❧ r✐❣❤ts r❡s❡r✈❡❞✳ ✐❆❜str❛❝t
❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦ ❢❛❝t♦r
♣♦❧②♥♦♠✐❛❧s
❚✳❍✳ ❘❛♥❞r✐❛♥❛r✐s♦❛ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ ❙t❡❧❧❡♥❜♦s❝❤✱ Pr✐✈❛t❡ ❇❛❣ ❳✶✱ ▼❛t✐❡❧❛♥❞ ✼✻✵✷✱ ❙♦✉t❤ ❆❢r✐❝❛✳ ❚❤❡s✐s✿ ▼❙❝ ✭▼❛t❤s✮ ❉❡❝❡♠❜❡r ✷✵✶✷ ▼❛❥♦r ✇♦r❦s ❞♦♥❡ ✐♥ ❋✉♥❝t✐♦♥ ❋✐❡❧❞ ❆r✐t❤♠❡t✐❝ s❤♦✇ ❛ str♦♥❣ ❛♥❛❧♦❣② ❜❡t✇❡❡♥ t❤❡ r✐♥❣ ♦❢ ✐♥t❡❣❡rs ❩ ❛♥❞ t❤❡ r✐♥❣ ♦❢ ♣♦❧②♥♦♠✐❛❧s ♦✈❡r ❛ ✜♥✐t❡ ✜❡❧❞ Fq[T ]✳ ❲❤✐❧❡ ❛♥ ❛❧❣♦r✐t❤♠ ❤❛s ❜❡❡♥ ❞✐s❝♦✈❡r❡❞ t♦ ❢❛❝t♦r ✐♥t❡❣❡rs ✉s✐♥❣ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✱ t❤❡ ❞✐s❝♦✈❡r② ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s✱ ✇❤✐❝❤ ❛r❡ ❛♥❛❧♦❣♦✉s t♦ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✱ ♠❛❞❡ ✐t ♣♦ss✐❜❧❡ t♦ ❡①❤✐❜✐t ❛♥ ❛❧❣♦r✐t❤♠ ❢♦r ❢❛❝t♦r✐s✐♥❣ ♣♦❧②♥♦♠✐❛❧s ✐♥ t❤❡ r✐♥❣ Fq[T ]✳ ■♥ t❤✐s t❤❡s✐s✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s✱ t❤❡♥ ✇❡ ❞❡♠♦♥✲ str❛t❡ t❤❡ ❛♥❛❧♦❣② ❜❡t✇❡❡♥ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ❛♥❞ ❊❧❧✐♣t✐❝ ❝✉r✈❡s✳ ❋✐♥❛❧❧②✱ ✇❡ ♣r❡s❡♥t ❛♥ ❛❧❣♦r✐t❤♠ ❢♦r ❢❛❝t♦r✐♥❣ ♣♦❧②♥♦♠✐❛❧s ♦✈❡r ❛ ✜♥✐t❡ ✜❡❧❞ ✉s✐♥❣ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s✳ ✐✐❯✐ttr❡❦s❡❧
❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ❡♥ ❤✉❧ t♦❡♣❛ss✐♥❣s t♦t ❢❛❦t♦r ♣♦❧✐♥♦♠❡
✭✏❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦ ❢❛❝t♦r ♣♦❧②♥♦♠✐❛❧s✑✮ ❚✳❍✳ ❘❛♥❞r✐❛♥❛r✐s♦❛ ❉❡♣❛rt❡♠❡♥t ▼❛t❤❡♠❛t✐❦✱ ❯♥✐✈❡rs✐t❡✐t ✈❛♥ ❙t❡❧❧❡♥❜♦s❝❤✱ Pr✐✈❛❛ts❛❦ ❳✶✱ ▼❛t✐❡❧❛♥❞ ✼✻✵✷✱ ❙✉✐❞ ❆❢r✐❦❛✳ ❚❡s✐s✿ ▼❙❝ ✭❲✐s❦✮ ❉❡s❡♠❜❡r ✷✵✶✷ ✬♥ ●r♦♦t ❞❡❡❧ ✈❛♥ ❞✐❡ ✇❡r❦ ✇❛t r❡❡❞s ✐♥ ❢✉♥❦s✐❡❧✐❣❣❛❛♠ r❡❦❡♥❦✉♥❞❡ ✈♦❧t♦♦✐ ✐s t♦♦♥ ✬♥ st❡r❦ ✈❡r❜❛♥❞ t✉ss❡♥ ❞✐❡ r✐♥❣ ✈❛♥ ❤❡❡❧❣❡t❛❧❧❡✱ Z, ❡♥ ❞✐❡ r✐♥❣ ✈❛♥ ♣♦❧✐♥♦♠❡ ♦♦r ✬♥ ❡✐♥❞✐❣❡ ❧✐❣❣❛❛♠✱ F[T ]. ❚❡r✇②❧ ❞❛❛r ❛❧r❡❡❞s ✬♥ ❛❧❣♦r✐t♠❡✱ ✇❛t ❣❡❜r✉✐❦ ♠❛❛❦ ✈❛♥ ❡❧❧✐♣t✐❡s❡ ❦✉r✇❡s✱ ♦♥t✇❡r♣ ✐s ♦♠ ❤❡❡❧❣❡t❛❧❧❡ t❡ ❢❛❦t♦r✐s❡❡r✱ ❤❡t ❞✐❡ ♦♥t❞❡❦❦✐♥❣ ✈❛♥ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s✱ ✇❛t ❛♥❛❧♦♦❣ ✐s ❛❛♥ ❡❧❧✐♣t✐❡s❡ ❦✉r✇❡s✱ ❞✐t ♠♦♦♥t❧✐❦ ❣❡♠❛❛❦ ♦♠ ✬♥ ❛❧❣♦r✐t♠❡ t❡ ❦♦♥str✉❡❡r ♦♠ ♣♦❧✐♥♦♠❡ ✐♥ ❞✐❡ r✐♥❣ F[T ] t❡ ❢❛❦t♦r✐s❡❡r✳ ■♥ ❤✐❡r❞✐❡ t❡s✐s ♠❛❛❦ ♦♥s ❞✐❡ ❦♦♥s❡♣ ✈❛♥ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ❜❡❦❡♥❞ ❞❡✉r s❡❦❡r❡ ❛s♣❡❦t❡ ❞❛❛r✈❛♥ t❡ ❜❡st✉❞❡❡r✳ ❖♥s ❣❛❛♥ ✈♦♦rt ❞❡✉r ✬♥ ✈♦♦r❜❡❡❧❞ t❡ ✈♦♦rs✐❡♥ ✇❛t ❞✐❡ ❛♥❛❧♦♦❣ t✉ss❡♥ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ❡♥ ❡❧❧✐♣t✐❡s❡ ❦✉r✇❡s ✐❧❧✉str❡❡r✳ ❯✐t❡✐♥✲ ❞❡❧✐❦✱ ❞❡✉r ❣❡❜r✉✐❦ t❡ ♠❛❛❦ ✈❛♥ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s✱ ❜❡✈❡st✐❣ ♦♥s ❤✐❡r❞✐❡ ❛♥❛❧♦♦❣ ❞❡✉r ❞✐❡ ❛❧❣♦r✐t♠❡ ✈✐r ❞✐❡ ❢❛❦t♦r✐s❡r✐♥❣ ✈❛♥ ♣♦❧✐♥♦♠❡ ♦♦r ❡✐♥❞✐❣❡ ❧✐❣❣❛♠❡ t❡ ✈❡s❦❛❢✳ ✐✐✐❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts
■ ✇♦✉❧❞ ❧✐❦❡ t♦ ❡①♣r❡ss ♠② s✐♥❝❡r❡ ❣r❛t✐t✉❞❡ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❡♦♣❧❡ ❛♥❞ ♦r❣❛♥✲ ✐s❛t✐♦♥s✳ ❲✐t❤♦✉t t❤❡♠ t❤✐s t❤❡s✐s ✇♦✉❧❞ ♥♦t ❤❛✈❡ ❜❡❡♥ ✇r✐tt❡♥✳ ❋✐rst ♦❢ ❛❧❧✱ t❤❡r❡ ✐s Pr♦❢✳ ❋❧♦r✐❛♥ ❇r❡✉❡r✱ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ ❤✐♠ ❢♦r t❤❡ s✉♣♣♦rt✱ ♣❛t✐❡♥❝❡ ❛♥❞ ❣✉✐❞❛♥❝❡ t❤r♦✉❣❤ t❤❡ r❡❛❧✐③❛t✐♦♥ ♦❢ t❤✐s t❤❡s✐s✳ ▼② t❤❛♥❦s ❛❧s♦ t♦ ❆■▼❙ ❛♥❞ t❤❡ ❙t❡❧❧❡♥❜♦s❝❤ ❯♥✐✈❡rs✐t② ❢♦r t❤❡✐r ✜♥❛♥❝✐❛❧ ❛♥❞ ♠❛t❡r✐❛❧ s✉♣♣♦rt✳ ■ ❛❧s♦ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ ♠② ❢❛♠✐❧②✱ ❢r✐❡♥❞s ❢♦r t❤❡✐r s✉♣♣♦rt ❞✉r✐♥❣ t❤❡ ②❡❛rs ♦❢ st✉❞②✳ ❆ s♣❡❝✐❛❧ t❤❛♥❦s t♦ t❤❡♠ ❢♦r t❤❡✐r s✉♣♣♦rt✱ ❝♦♠♠❡♥ts✱ s✉❣❣❡st✐♦♥s ❛♥❞ ❤❡❧♣✳ ❆♠♦♥❣ t❤❡♠✱ t❤❡r❡ ❛r❡ ♠② ♠♦t❤❡r✱ ❢❛t❤❡r✱ ❜r♦t❤❡rs ❛♥❞ s✐st❡r✳ ❆♥❞ ❛❧s♦ s♦♠❡ ❢r✐❡♥❞s✿ ❋r❛♥❝❡s✱ ❆♥❞r②✱ ❘♦♥❛❧❞❛ ❛♥❞ ❉❛r❧✐s♦♥✳ ❋✐♥❛❧❧②✱ ■ t❤❛♥❦ ●❖❉✱ t❤❡ ♦♥❡ ✇❤♦ ♠❛❞❡ ❡✈❡r②t❤✐♥❣ ♣♦ss✐❜❧❡✳ ✐✈❉❡❞✐❝❛t✐♦♥s
❚❤✐s t❤❡s✐s ✐s ❞❡❞✐❝❛t❡❞ t♦ ♠② ♣❛r❡♥ts✳ ✏▲✐✈❡ ❛s ②♦✉ ✇❡r❡ t♦ ❞✐❡ t♦♠♦rr♦✇✳ ▲❡❛r♥ ❛s ✐❢ ②♦✉ ✇❡r❡ t♦ ❧✐✈❡ ❢♦r❡✈❡r✳✑ ▼✳●❤❛♥❞✐ ✈❈♦♥t❡♥ts
❉❡❝❧❛r❛t✐♦♥ ✐ ❆❜str❛❝t ✐✐ ❯✐ttr❡❦s❡❧ ✐✐✐ ❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts ✐✈ ❉❡❞✐❝❛t✐♦♥s ✈ ❈♦♥t❡♥ts ✈✐ ✶ ■♥tr♦❞✉❝t✐♦♥ ✶ ✶✳✶ ❊❧❧✐♣t✐❝ ❝✉r✈❡s ❛♥❞ ✐♥t❡❣❡r ❢❛❝t♦r✐s❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✶ ❊❈▼ ❛❧❣♦r✐t❤♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❈❛r❧✐t③ ♠♦❞✉❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷✳✶ ❚❤❡ ❈❛r❧✐t③ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ ❖✉t❧✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ♦✈❡r ✜❡❧❞s ✽ ✷✳✶ ●❡♥❡r❛❧✐s✐♥❣ t❤❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✷ ❚♦rs✐♦♥ ♠♦❞✉❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✸ ❚❤❡ ♥♦t✐♦♥ ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸✳✶ ❚❤❡ ♠♦❞✉❧❡ str✉❝t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸✳✷ ❚❤❡ ❝❛t❡❣♦r② ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✹ ❆♥❛❧②t✐❝ ❝♦♥str✉❝t✐♦♥ ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✹✳✶ ❈♦♠♣❧❡① t❤❡♦r② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✹✳✷ ▲❛tt✐❝❡s ❛ss♦❝✐❛t❡❞ t♦ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸ ❆♥❛❧♦❣② ✇✐t❤ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ✷✼ ✸✳✶ ❚❤❡ ❲❡✐❡rstr❛ss ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✷ ❖♥ t❤❡ s✐❞❡ ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷✳✶ ❚❛t❡ ♠♦❞✉❧❡ ♦♥ ❊❧❧✐♣t✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✸ ❖♥ t❤❡ s✐❞❡ ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✸✳✶ ❚❛t❡ ♠♦❞✉❧❡ ♦♥ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✈✐❈❖◆❚❊◆❚❙ ✈✐✐ ✹ ❋❛❝t♦r✐s❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧s ✸✽ ✹✳✶ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ♦✈❡r r✐♥❣s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✷ ❋❛❝t♦r✐s❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷✳✶ ❆❧❣♦r✐t❤♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✷✳✷ ❈♦♠♣❧❡①✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✺ ❈♦♥❝❧✉s✐♦♥ ✺✺ ❆♣♣❡♥❞✐❝❡s ✺✻ ❆ ❙✐♥❣✉❧❛r ♣r♦❣r❛♠ ✺✼ ❇ ❇✐❣ ❡①❛♠♣❧❡ ✻✺ ▲✐st ♦❢ ❘❡❢❡r❡♥❝❡s ✻✾
❈❤❛♣t❡r ✶
■♥tr♦❞✉❝t✐♦♥
■♥t❡r❡st ❢♦r ❢❛❝t♦r✐♥❣ ♣♦❧②♥♦♠✐❛❧s ❤❛s ✐♥❝r❡❛s❡❞ ❛s ✐t ❤❛s ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s ✐♥ t❤❡ ✜❡❧❞ ♦❢ ❈♦♠♣✉t❡r ❛❧❣❡❜r❛✱ ❈♦❞✐♥❣ t❤❡♦r②✱ ❈r②♣t♦❣r❛♣❤②✳ ❋♦r ❡①❛♠♣❧❡✱ ✐t ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❝♦♠♣✉t❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s✱ ✇❤✐❝❤ ✐s ❛♥ ✐♠♣♦rt❛♥t ♣r♦❜❧❡♠ ✐♥ ♣✉❜❧✐❝✲❦❡② ❝r②♣t♦❣r❛♣❤②✱ ♦✈❡r ✜♥✐t❡ ✜❡❧❞s ♦❢ ♣r✐♠❡✲♣♦✇❡r ♦r❞❡r✳ ❚❤❡r❡ ❛r❡ ❛❧r❡❛❞② ♠❛♥② ❛❧❣♦r✐t❤♠s ❢♦r ❢❛❝t♦r✐♥❣ ♣♦❧②♥♦♠✐❛❧s ♦✈❡r ✜♥✐t❡ ✜❡❧❞ ❜✉t r❡s❡❛r❝❤ st✐❧❧ ❝♦♥t✐♥✉❡s t♦ ❞❡✈❡❧♦♣ ❜❡tt❡r ♠❡t❤♦❞s✳ ❚❤❡ ❇❡r❧❡❦❛♠♣✬s ❛♥❞ t❤❡ ❈❛♥t♦r✲❩❛ss❡♥❤❛✉s✬ ❛❧❣♦r✐t❤♠s ❛r❡ ❡①❛♠♣❧❡s ♦❢ ❛❧❣♦r✐t❤♠s t♦ ❢❛❝t♦r ♣♦❧②✲ ♥♦♠✐❛❧s ♦✈❡r ✜♥✐t❡ ✜❡❧❞✳ ❍♦✇❡✈❡r s✐♥❝❡ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ t❤❡♦r② ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s✱ ✇❤✐❝❤ ✐s ❛ ✏❣❡♥❡r❛❧✐s❛t✐♦♥✑ ♦❢ t❤❡ ♥♦t✐♦♥ ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✱ ❛ ♥❡✇ ❛❧❣♦r✐t❤♠ ✇❛s ❞❡✈❡❧♦♣❡❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❜② ❆✳ P❛♥❝❤✐s❤❦✐♥ ❛♥❞ ■✳ P♦t❡♠✲ ✐♥❡ ✭P❛♥❝❤✐s❤❦✐♥ ❛♥❞ P♦t❡♠✐♥❡✱ ✶✾✽✾✮✱ ❛♥❞ ❛❧s♦ ❜② ✈❛♥ ❞❡r ❍❡✐❞❡♥ ✭✈❛♥ ❞❡r ❍❡✐❞❡♥✱ ✷✵✵✹✮✳✶✳✶ ❊❧❧✐♣t✐❝ ❝✉r✈❡s ❛♥❞ ✐♥t❡❣❡r ❢❛❝t♦r✐s❛t✐♦♥
❆s ✇❡ ❤❛✈❡ s❛✐❞ ❡❛r❧✐❡r✱ t❤❡ ♥♦t✐♦♥ ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ✐s ❛ ✏❣❡♥❡r❛❧✐s❛t✐♦♥✑ ♦❢ t❤❡ ♥♦t✐♦♥ ♦❢ ❊❧❧✐♣t✐❝ ❝✉r✈❡s✳ ❚❤✉s ♦♥❡ ♠✐❣❤t t❤✐♥❦ ✐❢ ❛♥ ❛♥❛❧♦❣♦✉s t❤❡♦r② ❡①✐sts ♦♥ t❤❡ s✐❞❡ ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s✱ ✐❢ ✇❡ ❤❛✈❡ ♦♥❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❡❧❧✐♣t✐❝✳ ■♥❞❡❡❞✱ t❤❡ ❛❧❣♦r✐t❤♠ ✇❡ ✇✐❧❧ ❞❡✈❡❧♦♣ ✐s ❛♥❛❧♦❣♦✉s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣♦r✐t❤♠✱ ❝❛❧❧❡❞ ▲❡♥str❛ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❢❛❝t♦r✐③❛t✐♦♥ ♦r ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❢❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ ✭❊❈▼✮✳ ❙♦ ✐t ✐s ♥❛t✉r❛❧ t♦ ✜rst s❡❡ t❤❛t ❛❧❣♦r✐t❤♠✳✶✳✶✳✶ ❊❈▼ ❛❧❣♦r✐t❤♠
■♥ t❤✐s ❛❧❣♦r✐t❤♠ ✇❡ ✇✐❧❧ ❞❡❛❧ ✇✐t❤ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ E ♦❢ t❤❡ ❣❡♥❡r✐❝ ❢♦r♠ ✐✳❡✳ ✐ts ❡q✉❛t✐♦♥ ✐s ♦❢ t❤❡ ❢♦r♠ y2 = x3+ ax + b. ✭✶✳✶✳✶✮ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ♣♦✐♥ts ♦♥ t❤❡ ❝✉r✈❡ ✇❡ ❛❧s♦ ❤❛✈❡ ❛♥♦t❤❡r ♣♦✐♥t O ❛♥❞ ✇❡ ❝❛♥ ❢♦r♠ ❛♥ ❛❜❡❧✐❛♥ ❣r♦✉♣ ✇✐t❤ ✐❞❡♥t✐t② O✳ ▼♦r❡ ❡①♣❧❛♥❛t✐♦♥ ❛❜♦✉t t❤✐s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❙✐❧✈❡r♠❛♥ ✭✷✵✵✾✮✳ ✶❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✷ ❍❡r❡ ❛r❡ t❤❡ st❡♣s ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✳ ❙✉♣♣♦s❡ n ✐s t❤❡ ✐♥t❡❣❡r t♦ ❜❡ ❢❛❝t♦r❡❞ ❛♥❞ ✇❡ ❛ss✉♠❡ t❤❛t 2 ♦r 3 ❞♦❡s♥✬t ❞✐✈✐❞❡ n✳ ❲❡ ❝❛♥ ❛❧s♦ s✉♣♣♦s❡ t❤❛t n ✐s ♥♦t ❛ ♣❡r❢❡❝t ♣♦✇❡r✳ ✶✳ ❈❤♦♦s❡ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ y2 = x3+ ax + b ( mod n)❛♥❞ ❛ ♣♦✐♥t P = (x 0, y0) ♦♥ t❤❡ ❝✉r✈❡✳ ❲❡ ❝❤♦♦s❡ t❤❡ ✐♥t❡❣❡r a ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t gcd (4a3+ 27c2, n) 6= n✳ ✷✳ ■❢ gcd (4a3+ 27c2, n) 6= 1✱ t❤❡♥ ✇❡ ❣❡t ❛ ♣r♦♣❡r ❢❛❝t♦r ♦❢ n✳ ❖t❤❡r✇✐s❡ ❣♦ t♦ t❤❡ ♥❡①t st❡♣✳ ✸✳ ❈❤♦♦s❡ e ❛s ❛ ♣r♦❞✉❝t ♦❢ ♠❛♥② s♠❛❧❧ ♣r✐♠❡ ♥✉♠❜❡rs ❛♥❞ ❝♦♠♣✉t❡ eP ✇❤✐❝❤ ✐s t❤❡ e t✐♠❡s s✉♠ ♦❢ P ✇✳r✳t t❤❡ ❣r♦✉♣ ❧❛✇✳ ✹✳ eP ✐s ♦❢ t❤❡ ❢♦r♠ p u2, q u3 ❛♥❞ ✇❡ s❡t v = gcd (u, n)✳ ✺✳ ■❢ v 6= 1, n✱ t❤❡♥ ✇❡ ❤❛✈❡ ❛ tr✐✈✐❛❧ ❢❛❝t♦r ♦❢ n✳ ■❢ v = n ✇❡ ❣♦ t♦ st❡♣ ✸ ❜② ❝❤♦♦s✐♥❣ ❛ s♠❛❧❧❡r e✳ ❖t❤❡r✇✐s❡ ❢♦r v = 1✱ ✇❡ ❝❛♥ ❡✐t❤❡r ❝❤♦♦s❡ ❛♥♦t❤❡r ❝✉r✈❡ ✐♥ st❡♣ ✶ ♦r ✐♥❝r❡❛s❡ e ✐♥ st❡♣ ✸✳ ❚❤✐s ❛❧❣♦r✐t❤♠ ✉s❡s t❤❡ tr✐❛❧ ❛♥❞ ❡rr♦r ♠❡t❤♦❞✳ ◆❛♠❡❧②✱ ♦♥❡ ❡①❡❝✉t✐♦♥ ♦❢ t❤✐s ❛❧❣♦r✐t❤♠ ❣✐✈❡s ✉s ❛ ♣r♦♣❡r ❢❛❝t♦r ❢♦r s♦♠❡ ❝❤♦✐❝❡ ♦❢ ❝✉r✈❡ ❛♥❞ ❛❧s♦ ❢♦r s♦♠❡ ❝❤♦✐❝❡ ♦❢ e✳ ■♥❞❡❡❞✱ ❧❡t ✉s ❛ss✉♠❡ t❤❛t Ep ✐s t❤❡ s❡t ♦❢ ♣♦✐♥ts s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✶✳✶✮ ♠♦❞✉❧♦ p✱ ✇❤❡r❡ p ✐s ❛ ♣r♦♣❡r ❢❛❝t♦r ♦❢ n✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ♯Ep ❞✐✈✐❞❡s e✳ ❚❤❡♥✱ ❢♦r ❛ r❛t✐♦♥❛❧ ♣♦✐♥t P ♦♥ t❤❡ ❝✉r✈❡ E✱ eP = O✱ ✇❤❡r❡ P
✐s t❤❡ r❡❞✉❝t✐♦♥ ♦❢ P ♠♦❞✉❧♦ p✳ ❖♥❡ s❤♦✇s t❤❛t p ❞✐✈✐❞❡s u✳ ❍❡♥❝❡✱ ✇❡ ❣❡t ❛ ♣r♦♣❡r ❢❛❝t♦r gcd (u, n)✱ ❛ss✉♠✐♥❣ t❤❛t✱ ❢♦r ♦✉r ❝❤♦✐❝❡ ♦❢ ❝✉r✈❡ ❛♥❞ e✱ n ✐s ♥♦t ❛ ❞✐✈✐s♦r ♦❢ u✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛ ♣r♦♣❡r ❢❛❝t♦r ❢♦r ❛♣♣r♦♣r✐❛t❡ ❝❤♦✐❝❡s ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡✳ ❋♦r ♠♦r❡ ❞❡t❛✐❧s ♦♥ t❤✐s✱ ✇❡ ❝❛♥ r❡❢❡r t♦ ❙✐❧✈❡r♠❛♥ ❛♥❞ ❚❛t❡ ✭✶✾✾✹✱ ❝❤❛♣✳ ■❱✮✳
✶✳✷ ❈❛r❧✐t③ ♠♦❞✉❧❡
❚❤❡ ♥♦t✐♦♥ ♦❢ ❊❧❧✐♣t✐❝ ♠♦❞✉❧❡s ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ❉r✐♥❢❡❧❞✱ ✐♥ ❤✐s ♣❛♣❡r ❉r✐♥✲ ❢❡❧❞ ✭✶✾✼✹✮✱ ❛s ❛ ✏❣❡♥❡r❛❧✐s❛t✐♦♥✑ ♦❢ t❤❡ ♥♦t✐♦♥ ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✳ ◆♦✇❛❞❛②s✱ t❤✐s ❝♦♥❝❡♣t ✐s ❦♥♦✇♥ ❛s ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s✳ ❆❧t❤♦✉❣❤✱ t❤❡ ❛rt✐❝❧❡ ✇❛s ♣✉❜✲ ❧✐s❤❡❞ ✐♥ ✶✾✼✹✱ ❛ ♣❛rt✐❝✉❧❛r ❝❛s❡ ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ✇❛s ❛❧r❡❛❞② st✉❞✐❡❞ ❜② ❈❛r❧✐t③ ✐♥ t❤❡ ✶✾✸✵✬s ✭❈❛r❧✐t③✱ ✶✾✸✷❛✱ ✶✾✸✺✮✳ ❚❤✐s ✐s ❛ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡ ♦❢ r❛♥❦ 1 ❛♥❞ ✐s ❝❛❧❧❡❞ ❈❛r❧✐t③ ♠♦❞✉❧❡s✳ ❙♦ ❜❡❢♦r❡ ✇❡ st✉❞② t❤❡ ♥♦t✐♦♥ ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s✱ ❧❡t ✉s ❜r✐❡✢② s❡❡ t❤❡ s✐♠♣❧❡st ❝❛s❡ ✇❤✐❝❤ ✐s t❤❡ ❈❛r❧✐t③ ♠♦❞✉❧❡✳ ❉❡✜♥❡ t❤❡ r✐♥❣ A = F [T ] ❛s t❤❡ r✐♥❣ ♦❢ ♣♦❧②♥♦♠✐❛❧s ✐♥ t❤❡ ✈❛r✐❛❜❧❡ T ✇✐t❤ ❝♦♥st❛♥ts ✐♥ F✱ ✇❤❡r❡ F ✐s ❛ ✜♥✐t❡ ✜❡❧❞ ♦❢ ♣r✐♠❡ ❝❤❛r❛❝t❡r✐st✐❝ p ❛♥❞ ❝❛r❞✐♥❛❧ q✳ ▲❡t k = F (T ) ❜❡ t❤❡ ❢r❛❝t✐♦♥ ✜❡❧❞ ♦❢ A✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② ∞ t❤❡ ♣❧❛❝❡ ♦❢ k ❣✐✈❡♥ ❜② t❤❡ ❡❧❡♠❡♥t T−1✳ ◆♦t✐❝❡ t❤❛t✱ t❤❡ r✐♥❣ A ✐s ❡①❛❝t❧②✱ t❤❡ r✐♥❣❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✸ ♦❢ ❡❧❡♠❡♥ts ♦❢ k s✉❝❤ t❤❛t t❤❡ ♦♥❧② ♣♦❧❡s ❛r❡ ❛t ∞✳ ❚❤❡ ♣❧❛❝❡ ∞ ✐♥❞✉❝❡s ❛ t♦♣♦❧♦❣② ♦♥ k ❛♥❞ ❧❡t ✉s ❞❡♥♦t❡ ❜② k∞t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ k ✇✳r✳t✳ t❤✐s t♦♣♦❧♦❣②✳ ❚❤❡ ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡ k∞ ♦❢ k∞ ✐s ♥♦t ❝♦♠♣❧❡t❡ ❜✉t ✐❢ ✇❡ t❛❦❡ t❤❡ ❝♦♠♣❧❡t✐♦♥ C∞ ♦❢ k∞✱ ✇❡ s❡❡ t❤❛t C∞ ✐s ❜♦t❤ ❝♦♠♣❧❡t❡ ❛♥❞ ❛❧❣❡❜r❛✐❝❛❧❧② ❝❧♦s❡❞✳ ❘❡♠❛r❦ ✶✳✷✳✶✳ ❚❤✐s s❡t✉♣ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ ♥✉♠❜❡r ✜❡❧❞ ❛♥❞ ❢✉♥❝t✐♦♥ ✜❡❧❞✿ ◆✉♠❜❡r ✜❡❧❞✿ Z Q R C C ❋✉♥❝t✐♦♥ ✜❡❧❞✿ A k k∞ k∞ C∞.
✶✳✷✳✶ ❚❤❡ ❈❛r❧✐t③ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥
▲❡t j ❜❡ ❛♥ ✐♥t❡❣❡r✳ ❲❡ ❞❡✜♥❡ [j] = Tqj − T ∈ A✳ ▲❡t ✉s ❛❧s♦ ❞❡✜♥❡ π ❜② π = ∞ Y j=1 1 − [j] [j + 1] . ❲❡ ✇✐❧❧ s♦♦♥ s❡❡ t❤❛t t❤✐s ♣r♦❞✉❝t ✐s ✇❡❧❧ ❞❡✜♥❡❞ ✐♥ k∞✳ ▲❡t ✉s ✜rst ❛ss✉♠❡ t❤✐s ❢❛❝t✱ s♦ t❤❛t ✇❡ ❝❛♥ ❞❡✜♥❡ ❛♥ A✲❧❛tt✐❝❡✱ πA✱ ♦❢ C∞✳ ❚❤✐s ❧❛tt✐❝❡ ✐s ♦❢ ❞✐♠❡♥s✐♦♥ 1✱ s♦ t❤❛t t❤❡ ♦❜❥❡❝t ✇❡ ✇✐❧❧ ❝♦♥str✉❝t ✐s ❝❛❧❧❡❞ ♦❢ r❛♥❦ 1✳ ❚♦ ❞♦ t❤❡ ❝♦♥str✉❝t✐♦♥✱ ❧❡t ✉s ✇♦r❦ ♦✉t t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ eA(z) = z Y λ∈A−{0} 1 − z λ . ▲❡t n ❜❡ ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡r ❛♥❞ ❧❡t ✉s ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s ✐♥ A ✇✐t❤ ❞❡❣r❡❡ ❧❡ss t❤❛♥ n ❜② An = {a ∈ A : deg a < n}✳ ❲❡ ❞❡✜♥❡ ❢♦r n ≥ 0✱ e′An(z) = Y a∈An (z − a) . ❉❡✜♥✐t✐♦♥ ✶✳✷✳✷✳ ❲❡ ❞❡✜♥❡ L0 = D0 = 1✱ ❛♥❞ ❢♦r n ≥ 1✱ Ln= n Y j=1 [j] ❛♥❞ Fn= n−1 Y j=0 [n − j]qj. ❋♦r t❤❡ ❢✉♥❝t✐♦♥ e′ An✱ ❈❛r❧✐t③ ❤❛s s❤♦✇♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt② ✭❈❛r❧✐t③✱ ✶✾✸✷❜✮✿ ❚❤❡♦r❡♠ ✶✳✷✳✸✳ ▲❡t n ≥ 0 ❜❡ ❛♥ ✐♥t❡❣❡r✳ ❚❤❡♥✱ e′ An = n X i=0 (−1)n−izqi Fn FiLq i n−i . ❚❛❦✐♥❣ s♦♠❡ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ t❤❡✐r ♣r♦❞✉❝t ✐♥ t❤❡ r✐♥❣ A✱ ✇❡ ❣❡t t❤❡ ♥❡①t r❡s✉❧ts✱ ❛s ❢♦✉♥❞ ✐♥ ●♦ss ✭✶✾✾✼✱ ❝❤❛♣✳ ✸✮✳❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✹ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✹✳ ❼ Y a♠♦♥✐❝ ✐♥ A deg a=n a = Fn✱ ❼ Y a∈A−{0} deg a=n a = (−1)n Fn Ln✱ ❼ Fn= Tqi − TFn−1q ✳ ◆♦✇✱ ❛s a ❛♥❞ −a ❛r❡ ❜♦t❤ ✐♥ An✱ t❤❡♥ ✇❡ ❛❧s♦ ❤❛✈❡ e′An(z) = Y a∈An (z − a) = Y a∈An (z + a) . ✭✶✳✷✳✶✮ ❲❡ ♠✉❧t✐♣❧② t❤❡ ❡q✉❛❧✐t② ✐♥ t❤❡♦r❡♠ ✶✳✷✳✸ ❜② (−1)n Ln Fn✳ ❯s✐♥❣ t❤❡ ♣r♦♣♦s✐t✐♦♥ ✶✳✷✳✹ ❛♥❞ t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✷✳✶✮✱ ✇❡ ❤❛✈❡ z Y a∈A−{0} deg a<n z a + 1 = n X i=0 (−1)izqi Ln FiLq i n−i . ❆❣❛✐♥✱ ✐♥t❡r❝❤❛♥❣✐♥❣ a ❛♥❞ −a✱ ✇❡ ❣❡t z Y a∈A−{0} deg a<n 1 −z a = n X i=0 (−1)izqi Ln FiLq i n−i . ✭✶✳✷✳✷✮ ▼♦r❡♦✈❡r✱ ❛s ✇❡ ✇✐❧❧ s❡❡ ❧❛t❡r ✐♥ t❤❡ ♣r♦♣♦s✐t✐♦♥ ✷✳✹✳✻✱ eAn(z) := z Y a∈A−{0} deg a<n 1 − z a , ❝♦♥✈❡r❣❡s ✐♥ C∞ ✇❤❡♥ n → ∞✳ ❇✉t k∞ ✐s ❝♦♠♣❧❡t❡✱ t❤❡♥ t❤❡ ❧✐♠✐t ♠✉st ❜❡ ✐♥ k∞✳ ❍❡♥❝❡✱ Pni=0(−1) i zqi L n FiLqin−i ❝♦♥✈❡r❣❡s ✐♥ k∞✳ ▲❡♠♠❛ ✶✳✷✳✺✳ ❙✉♣♣♦s❡✱ πi := [1] qi−1 q−1 Li ✱ t❤❡♥✱ πi = i−1 Y j−1 1 − [j] [j + 1] . ❍❡♥❝❡✱ πi ❝♦♥✈❡r❣❡s t♦ π := Q∞j=1 1 − [j+1][j] ✳
❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✺ Pr♦♦❢✳ ❲❡ ❤❛✈❡✱ i−1 Y j=1 1 − [j] [j + 1] = i−1 Y j=1 [j + 1] − [j] [j + 1] = i−1 Y j=1 [1]qj [j + 1] ! = Qi−1 j=0[1] qj Li = [1] qi−1 q−1 Li . ❲❤❛t r❡♠❛✐♥s t♦ s❤♦✇ ✐s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ s✐♥❝❡ t❤❡ ❧✐♠✐t ✇✐❧❧ ❝♦♠❡ ❛✉t♦♠❛t✐✲ ❝❛❧❧②✳ ❇✉t [1]qi−1q−1 Li ❝♦♥✈❡r❣❡s ✐♥ k∞❛s ✐t ✐s ❛ ❈❛✉❝❤② s❡q✉❡♥❝❡ ✐♥ t❤❛t ✜❡❧❞ ✇❤✐❝❤ ✐s ❝♦♠♣❧❡t❡✳ 2 ◆♦✇✱ eAn(z) = Pn i=0(−1) i zqi L n FiLqin−i ✱ t❤❡♥✱ ❛s πi → π✱ ✇❡ ❣❡t eAn(z) = n X i=0 (−1)izqi π qi n−i[1] qn−1 q−1 [1]qn−qiq−1 π nFi = 1 πn n X i=0 (−1)izqiπ qi n−i[1] qi−1 q−1 Fi . ✭✶✳✷✳✸✮ ❖❜✈✐♦✉s❧②✱ lim eAn = eA✳ ❚❤✐s s✉❣❣❡st ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✱ ✇❤✐❝❤ ✇❡ ✇✐❧❧ ♥♦t ♣r♦✈❡ ❤❡r❡✿ ❚❤❡♦r❡♠ ✶✳✷✳✻✳ ❚❤❡ s❡r✐❡s 1 πn n X i=0 (−1)izqiπ qi n−i[1] qi−1 q−1 Fi , ❝♦♥✈❡r❣❡s ❛s n → ∞ ❛♥❞ eA(z) = 1 π ∞ X i=0 (−1)izqiπ qi [1]qi−1q−1 Fi . ❘❡♠❛r❦ ✶✳✷✳✼✳ ❚❤❡ t❤❡♦r❡♠ ✶✳✷✳✻ ❞♦❡s ♥♦t ❢♦❧❧♦✇ ❞✐r❡❝t❧② ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✷✳✸✮✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s ❤❡r❡ t❤❛t t❤❡ ✐♥❞❡① n ✐s ✐♥s✐❞❡ t❤❡ s✉♠♠❛t✐♦♥ ❛s ✇❡❧❧ ❛s ✐t ✐s ❛❧s♦ t❤❡ ♦r❞❡r ♦❢ t❤❡ s✉♠♠❛t✐♦♥✳ ▼♦r❡ ❞❡t❛✐❧s ❛r❡ ✐♥ ●♦ss ✭✶✾✾✼✱ ❝❤❛♣✳ ✸✮
❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✻ ❲❡ ♥♦✇✱ s❡t ξ t♦ ❜❡ ❛ (q − 1)✲t❤ r♦♦t ♦❢ [1]✱ t❤✉s ✇❡ ❣❡t eA(z) = 1 π ∞ X i=0 (−1)izqiπq i [1]qi−1q−1 Fi = 1 π [1]q−11 ∞ X i=0 (−1)izqiπ qi [1]q−1qi Fi = 1 πξ ∞ X i=0 (−1)izqi(πξ) qi Fi . ❋r♦♠ ❛❧❧ ♦❢ t❤❡s❡✱ ✇❡ ♠❛② ♥♦✇ ❞❡✜♥❡ t❤❡ ❈❛r❧✐t③ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ t♦ ❜❡ eC(z) := eπξA(z)✳ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✽✳ ❚❤❡ ❈❛r❧✐t③ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ s❛t✐s✜❡s πξeA(z) = eC(πξz) . ▼♦r❡♦✈❡r ✐t ❤❛s t❤❡ ❝♦♠♣❧❡① ♠✉❧t✐♣❧✐❝❛t✐♦♥ eC(T z) = T eC(z) − eC(z)q. Pr♦♦❢✳ ❲❡ ❤❛✈❡ eA(z) = z Y a∈A−{0} 1 − z a = 1 πξ ∞ X i=0 (−1)izqi(πξ) qi Fi . ◆♦✇✱ eπξA(z) = z Y a∈A−{0} 1 − z πξa = πξ z πξ Y a∈A−{0} 1 − z πξa = πξeA z πξ . ❋♦r t❤❡ s❡❝♦♥❞ ❛ss❡rt✐♦♥✱ ✇❡ ❤❛✈❡✱ ❢r♦♠ ❛❜♦✈❡✱ eC(z) = ∞ X i=0 (−1)i z qi Fi . ❍❡♥❝❡✱ T eC(z) − eC(T z) = T ∞ X i=0 (−1)i z qi Fi − ∞ X i=0 (−1)iTqiz qi Fi = ∞ X i=0 (−1)iT − Tqiz qi Fi
❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✼ ❇② ♣r♦♣♦s✐t✐♦♥ ✶✳✷✳✹✱ ✇❡ ❣❡t T eC(z) − eC(T z) = ∞ X i=1 (−1)i−1 z qi Fi−1q = ∞ X i=0 (−1)iz qiq Fiq = ∞ X i=0 (−1)i z qi Fi !q = eC(z)q. ◆♦t❡ t❤❛t (−1)iq = (−1)i ✐s ♦❜✈✐♦✉s ❢♦r ❛♥ ♦❞❞ ❝❤❛r❛❝t❡r✐st✐❝ p✳ ❋♦r t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ p = 2✱ ✇❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t −1 = 1✳ 2 ❆s ✇❡ ✇✐❧❧ s❡❡ ❧❛t❡r✱ s✉❝❤ ❛♥ ❡❧❧✐♣t✐❝ ♠♦❞✉❧❡ ❣✐✈❡s r✐s❡ t♦ ❛ t✇✐st❡❞ ♣♦❧②♥♦♠✐❛❧s φT = T − τ✳
✶✳✸ ❖✉t❧✐♥❡
❚♦ ❝♦♥❝❧✉❞❡ t❤❡ ✜rst ❝❤❛♣t❡r ❧❡t ✉s ♥♦✇ ❞❡s❝r✐❜❡ ❜r✐❡✢② t❤❡ ❝♦♥t❡♥t ♦❢ t❤✐s t❤❡s✐s✳ ❲❡ ✇✐❧❧ ❣❡♥❡r❛❧✐s❡ t❤❡ t✇♦ ♣r❡✈✐♦✉s s❡❝t✐♦♥s ✇❡ ❤❛✈❡ s❡❡♥ ✐♥ t❤✐s ✜rst ❝❤❛♣t❡r✳ ■♥ ❈❤❛♣t❡r ✷✱ ✇❡ ✇✐❧❧ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ♦✈❡r ❛♥ ❛r❜✐tr❛r② ✜❡❧❞✳ ❚❤❡r❡ ✇❡ ✇✐❧❧ s❡❡ ❤♦✇ ✇❡ ❝❛♥ ❝♦♥str✉❝t ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ♦✈❡r t❤❡ ✜❡❧❞ C∞✳ ❚❤❡ ❛♥❛❧♦❣② ♠❡♥t✐♦♥❡❞ ❡❛r❧✐❡r ✇✐❧❧ ❜❡ st✉❞✐❡❞ ✐♥ ❈❤❛♣t❡r ✸✱ ✇❤❡r❡ ✇❡ ✇✐❧❧ ✐♥✈❡st✐❣❛t❡ ✐t ♠♦r❡ ❝❧♦s❡❧② ❢♦r t❤❡ ❚❛t❡ ♠♦❞✉❧❡s✳ ❚❤❡♥✱ ✐♥ ❈❤❛♣t❡r ✹✱ ✇❡ ✇✐❧❧ ❞❡✈❡❧♦♣ t❤❡ ❛❧❣♦r✐t❤♠ ❢♦r ❢❛❝t♦r✐♥❣ ♣♦❧②♥♦♠✐❛❧s ✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❊❈▼ ✇❡ ❤❛✈❡ s❡❡♥ ❛❜♦✈❡❀ ❜✉t ❜❡❢♦r❡ t❤❛t ✇❡ ❡①♣❧❛✐♥ t❤❡ ♥♦t✐♦♥ ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ♦✈❡r r✐♥❣s✳ ❋✐♥❛❧❧② ✇❡ ✇✐❧❧ ❝♦♥❝❧✉❞❡ ✐♥ ❈❤❛♣t❡r ✺ ❛♥❞ t❤❡♥✱ ✐♥ t❤❡ ❛♣♣❡♥❞✐①✱ ✇❡ ✐♠♣❧❡♠❡♥t t❤✐s ❛❧❣♦r✐t❤♠ ✉s✐♥❣ ❙■◆●❯▲❆❘ ✭❉❡❝❦❡r ❡t ❛❧✳✱ ✷✵✶✶✮✳ ❲❡ ✇✐❧❧ ❛❧s♦ ❣✐✈❡ ♦♥❡ ❡①❛♠♣❧❡ t♦ ❡①♣❧❛✐♥ s♦♠❡ ♣r♦❝❡❞✉r❡s ✐♥ t❤❡ ♣r♦❣r❛♠✳❈❤❛♣t❡r ✷
❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ♦✈❡r ✜❡❧❞s
▲❡t F ❜❡ ❛ ✜♥✐t❡ ✜❡❧❞✱ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ p✱ ✇✐t❤ q = pr ❡❧❡♠❡♥ts ❛♥❞ ❧❡t k/F ❜❡ ❛ ❢✉♥❝t✐♦♥ ✜❡❧❞ ✇✐t❤ ✜❡❧❞ ♦❢ ❝♦♥st❛♥ts F✳ ❲❡ ✜① ❛ ♣❧❛❝❡ ♦❢ k✱ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ ❜② ∞✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ ∞ ✐s ❞❡♥♦t❡❞ ❜② d∞✳ ❲❡ s❡t A t♦ ❜❡ t❤❡ r✐♥❣ ♦❢ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ k ✇✐t❤ t❤❡ ♦♥❧② ♣♦❧❡s ❛t ∞✳ ❆❢t❡r t❤❛t✱ ✇❡ ❛ss✉♠❡ L ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ✜❡❧❞ F✳ ■❢ ✇❡ s❡t τ t♦ ❜❡ t❤❡ q✲❋r♦❡❜❡♥✐✉s ❡♥❞♦♠♦r♣❤✐s♠ ♦✈❡r F✱ t❤❡♥ ❛❧❧ t❤❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ t❤❡ ✈❛r✐❛❜❧❡ τ ❢♦r♠ ❛ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ r✐♥❣✱ t❤❡ s❦❡✇ ♣♦❧②♥♦♠✐❛❧ r✐♥❣✱ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ ❜② L hτi✳ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❧❛✇ ♦❢ t❤❡ ❧❛t❡r r✐♥❣ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇✱ aτm.bτn= abqmτm+n.✷✳✶ ●❡♥❡r❛❧✐s✐♥❣ t❤❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣
●❡♥❡r❛❧❧②✱ ✇❤❡♥ ✇❡ ❞❡✜♥❡ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s✱ ✇❡ ❞♦ ♥♦t r❡str✐❝t ♦✉rs❡❧❢ t♦ ❛ r✐♥❣ A = F [T ]✳ ❖✉r ❝♦♥str✉❝t✐♦♥ ♦❢ A ✐s ♠♦r❡ ❣❡♥❡r❛❧ ❛♥❞ ✇❡ st✐❧❧ ❤❛✈❡ t♦ ❦❡❡♣ s♦♠❡ ♣r♦♣❡rt② ❢♦r t❤❛t r✐♥❣✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✳ Pr♦♣♦s✐t✐♦♥ ✷✳✶✳✶✳ A ✐s ✐♥t❡❣r❛❧❧② ❝❧♦s❡❞ ✐♥ k✳ ❆♥❞ t❤❡r❡❢♦r❡ A ✐s ❛ ❉❡❞❡❦✐♥❞ ❞♦♠❛✐♥✳ Pr♦♦❢✳ ▲❡t R✱ ❜❡ t❤❡ ✐♥t❡❣r❛❧ ❝❧♦s✉r❡ ♦❢ A ✐♥ k✳ ▲❡t x ❜❡ ❛♥ ❡❧❡♠❡♥t ♦❢ R s✉❝❤ t❤❛t t❤❡ ✐♥t❡❣r❛❧ ❞❡♣❡♥❞❡♥❝❡ ❢♦r x ♦✈❡r A ✐s✱ xn+ an−1xn−1+ · · · + a0 = 0. ✭✷✳✶✳✶✮ ❋✐rst✱ ✇❡ ✇❛♥t t♦ s❤♦✇ t❤❛t vP (x) ≥ 0 ❢♦r ❛❧❧ ♣❧❛❝❡s P ♦❢ k ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ∞✳ ❙✉♣♣♦s❡ ✐t ✐s ♥♦t t❤❡ ❝❛s❡ ❢♦r s♦♠❡ ♣❧❛❝❡ P 6= ∞ ♦❢ k✳ ❲❡ ❦♥♦✇ t❤❛t vP(ai) ≥ 0❢♦r ❛❧❧ 0 ≤ i < n✳ ❚❤❡r❡❢♦r❡ (n − i) vP (x) + ivP (x) < vP(ai) + ivP (x) , ❢♦r ❛❧❧ 0 ≤ i < n. ❚❤✉s ❢♦r ❛❧❧ 0 ≤ i < n✱ ✇❡ ❤❛✈❡ vP (aixi) > vP (xn)✳ ❍❡♥❝❡✱ min 0≤i≤n−1vP aix i > v P(xn) . ✭✷✳✶✳✷✮ ✽❈❍❆P❚❊❘ ✷✳ ❉❘■◆❋❊▲❉ ▼❖❉❯▲❊❙ ❖❱❊❘ ❋■❊▲❉❙ ✾ ❋r♦♠ t❤❡ ❡q✉❛t✐♦♥ ✭✷✳✶✳✶✮✱ vP(xn) = vP an−1xn−1+ · · · + a0 . ❇② t❤❡ ♣r♦♣❡rt② ♦❢ ✈❛❧✉❛t✐♦♥✱ vP (xn) ≥ minvP an−1xn−1 , · · · , vP(a0) . ✭✷✳✶✳✸✮ ❆♥❞ ✇❡ s❡❡ t❤❛t ✇❡ ❤❛✈❡ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ❜❡t✇❡❡♥ ✭✷✳✶✳✷✮ ❛♥❞ ✭✷✳✶✳✸✮ ✱ t❤❡r❡❢♦r❡ vP(x) ≥ 0✳ ❚❤✉s✱ ✇❡ ❤❛✈❡ R ⊂ A✳ ❚❤❡ ♦t❤❡r ✐♥❝❧✉s✐♦♥ ✐s ♦❜✈✐♦✉s s♦ t❤❛t ✜♥❛❧❧② ✇❡ ❤❛✈❡ R = A ✐✳❡✳ A ✐s ✐♥t❡❣r❛❧❧② ❝❧♦s❡❞✳ ❋♦r t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠✱ ❙✉♣♣♦s❡ a ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ A ❤❛✈✲ ✐♥❣ ❛ ♣♦❧❡ ♦♥❧② ❛t ∞✳ ❚❤❡♥✱ ❛s ❛ ♠❛tt❡r ♦❢ ❢❛❝t ✭s❡❡ ❩❛r✐s❦✐ ❡t ❛❧✳✱ ✶✾✼✺✱ ❝❤❛♣✳ ❱✱❚❤❡♦r❡♠✳ ✶✾✮✱ t❤❡ ✐♥t❡❣r❛❧ ❝❧♦s✉r❡ B ♦❢ F [a] ✐♥ k ✐s ❛ ❉❡❞❡❦✐♥❞ ❞♦✲ ♠❛✐♥✳ ❲❡ ✇❛♥t t♦ s❤♦✇ t❤❛t B = A✳ ❲❡ ❥✉st s❤♦✇ t❤❛t A ✐s ✐♥t❡❣r❛❧❧② ❝❧♦s❡❞ s♦ t❤❛t B ⊂ A✳ ◆♦t✐❝❡ t❤❛t✱ ❢♦r ❛ ♣❧❛❝❡ P ❞✐✛❡r❡♥t ❢r♦♠ ∞✱ A = TP 6=∞RP s♦ t❤❛t B ⊂ RP✱ ✇❤❡r❡ RP ✐s t❤❡ ✈❛❧✉❛t✐♦♥ r✐♥❣ ♦❢ k ❛t P ✳ ❘❡❝❛❧❧ t❤❛t RP ❤❛s ❛ ✉♥✐q✉❡ ♠❛①✐♠❛❧ ✐❞❡❛❧ P ✳ ❆s P ✐s ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧ ♦❢ RP✱ t❤❡♥ P ✐s ❛ ♣r✐♠❡ ✐❞❡❛❧ ♦❢ RP✳ ❆♥❞ ✇❡ ❤❛✈❡ P ∩ B ✐s ❛ ♥♦♥③❡r♦ ✐❞❡❛❧ ❜❡❝❛✉s❡✱ ✐❢ ✐t ✐s ♥♦t t❤❡ ❝❛s❡✱ t❤❡r❡❢♦r❡✱ s✐♥❝❡ t❤❡ ❢r❛❝t✐♦♥ ✜❡❧❞ ♦❢ R ✐s k✱ ✇❡ ❤❛✈❡ ❛♥ ✐♥❝❧✉s✐♦♥ ♦❢ k ✐♥t♦ RP/P✳ ❇✉t s✐♥❝❡ RP/P ✐s ✜♥✐t❡ ♦✈❡r F✱ t❤✉s ✐t ✇♦✉❧❞ ❜❡ t❤❡ ❝❛s❡ ❢♦r k✱ ✇❤✐❝❤ ✐s ✐♠♣♦ss✐❜❧❡ s✐♥❝❡ k ✐s ♥♦t ❛❧❣❡❜r❛✐❝ ♦✈❡r F✳ ❚❤❡r❡❢♦r❡ P ∩ B ✐s ❛❧s♦ ❛ ♣r✐♠❡ ✐❞❡❛❧ ♦❢ RP ∩ B = B✱ ❜✉t B ✐s ❛ ❉❡❞❡❦✐♥❞ ❞♦♠❛✐♥✱ t❤❡♥ P ∩ B ✐s ♠❛①✐♠❛❧ ✐♥ B✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❧♦❝❛❧✐s❛t✐♦♥✱ BP ∩B✱♦❢ B ❛t P ∩ B ✐s ❛ s✉❜r✐♥❣ ♦❢ RP✳ ❇✉t t❤✐s ❧♦❝❛❧✐s❛t✐♦♥ ✐s ❞✐s❝r❡t❡ ✈❛❧✉❛t✐♦♥ r✐♥❣✱ t❤❡♥ ✐t s❤♦✉❧❞ ❜❡ ♠❛①✐♠❛❧✳ ❚❤✉s BP ∩B = RP✳ ❚❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ A = \ P 6=∞ BP ∩B. ✭✷✳✶✳✹✮ ■♠♣❧✐❝✐t❧②✱ ❢r♦♠ ❛ ♣❧❛❝❡ ♦❢ k ✇❤✐❝❤ ✐s ♥♦t t❤❡ ♣❧❛❝❡ ❛t ✐♥✜♥✐t②✱ ✇❡ ❣❡t ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧ ♦❢ B✳ ◆♦✇ ❧❡t ✉s t❛❦❡ ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧ M ♦❢ B✱ t❤❡♥ BM ✐s ❛ ♣❧❛❝❡ ✭❤❡r❡ ✇❡ r❡❢❡r t♦ t❤❡ ♣❧❛❝❡ ❛s t❤❡ ✈❛❧✉❛t✐♦♥ r✐♥❣ ♥♦t t❤❡ ♠❛①✐♠❛❧ ✐❞❡❛❧✦✮ ♦❢ k✳ ❆♥❞ t❤✐s ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♣❧❛❝❡ ❛t ✐♥✜♥✐t② s✐♥❝❡ ✐t ❝♦♥t❛✐♥s a✳ ❈♦♥s❡q✉❡♥t❧②✱ ✇❡ ❤❛✈❡ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣❧❛❝❡s ♦❢ k ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ✐♥✜♥✐t② ❛♥❞ t❤❡ ♠❛①✐♠❛❧ ✐❞❡❛❧ ♦❢ B✳ ❆♥❞ t❤❡♥✱ t❤❡ ❡q✉❛❧✐t② ✭✷✳✶✳✹✮ ❜❡❝♦♠❡s A = \ M♠❛①✐♠❛❧ ✐♥ B BM. ❆♥❞ ❢r♦♠ ❛ ♣r♦♣❡rt② ♦❢ ❛ ❉❡❞❡❦✐♥❞ ❞♦♠❛✐♥✱ \ M♠❛①✐♠❛❧ ✐♥ B BM = B ❛♥❞ t❤❡r❡✲ ❢♦r❡ A = B✳ 2
❈❍❆P❚❊❘ ✷✳ ❉❘■◆❋❊▲❉ ▼❖❉❯▲❊❙ ❖❱❊❘ ❋■❊▲❉❙ ✶✵ ●❡♥❡r❛❧✐s✐♥❣ t❤❡ r✐♥❣ A ✐♠♣❧✐❡s t❤❛t ✇❡ ❛❧s♦ ❣❡♥❡r❛❧✐s❡ t❤❡ ♥♦t✐♦♥s ❢r♦♠ ♣♦❧②✲ ♥♦♠✐❛❧ r✐♥❣✳ ❍❡♥❝❡✱ ▲❡♠♠❛ ✷✳✶✳✷✳ ■❢ A ✐s ❛ ❉❡❞❡❦✐♥❞ ❞♦♠❛✐♥ ❝♦♥t❛✐♥❡❞ ✐♥ k ❛♥❞ a ∈ A✱ t❤❡♥ ❢♦r ❛ ♣r✐♠❡ ✐❞❡❛❧ I ♦❢ A✱ ✐❢ t❤❡ ❧♦❝❛❧✐③❛t✐♦♥ ♦❢ A ❛t I ❣✐✈❡s ❛ ♣❧❛❝❡ ♦❢ k ✇✐t❤ ♠❛①✐♠❛❧ ✐❞❡❛❧ P ✱ ✇❡ ❤❛✈❡ vP(a) = m✱ ✇❤❡r❡ m ✐s t❤❡ ♣♦✇❡r ♦❢ I ✐♥ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ (a) ❛s ❢❛❝t♦r ♦❢ ♣r✐♠❡ ✐❞❡❛❧s ♦❢ A✳ Pr♦♦❢✳ ❙✉♣♣♦s❡ (a) = ImQ kJ mk k ✐s t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♦❢ t❤❡ ✐❞❡❛❧ ❣❡♥❡r❛t❡❞ ❜② (a)✱ t❤❡♥ ✇❡ ❤❛✈❡ aAI = Im Y k Jmk k AI. ◆♦✇✱ JkAI ✐s ❛♥ ✐❞❡❛❧ ♦❢ AI✳ ❆♥ ❡❧❡♠❡♥t ♦❢ P ✐s ♦❢ t❤❡ ❢♦r♠ si✱ ✇❤❡r❡ i ∈ I ❛♥❞ s /∈ I✳ ❚❛❦✐♥❣ ❛♥ ❡❧❡♠❡♥t j ♦❢ Jk ✇❤✐❝❤ ✐s ♥♦t ✐♥ I✱ ✇❡ ❤❛✈❡ si = jsji ✳ ❆♥❞ t❤❡ ❧❛st ♦♥❡ ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ JAI✳ ❚❤✉s P ⊂ JkAI✳ ❆♥❞ s✐♥❝❡ t❤✐s ✐♥❝❧✉s✐♦♥ ✐s str✐❝t✱ ❜② t❤❡ ♠❛①✐♠❛❧✐t② ♦❢ P ✱ ✇❡ ❤❛✈❡ JkAI = AI✳ ❚❤❡r❡❢♦r❡ (a) AI = ImAI = Pm✳ ❆♥❞ t❤❡♥ vP (a) = m✳ 2 ❉❡✜♥✐t✐♦♥ ✷✳✶✳✸✳ ❋♦r ❛♥ ❡❧❡♠❡♥t a ♦❢ A✱ ✇❡ ❞❡✜♥❡ deg a = −v∞(a) d∞✳
❘❡♠❛r❦ ✷✳✶✳✹✳ ❆ ♣❧❛❝❡ ♦❢ k ✐s ❣✐✈❡♥ ❜② ❛ ❞✐s❝r❡t❡ ✈❛❧✉❛t✐♦♥ r✐♥❣ R ✇✐t❤ ♠❛①✲ ✐♠❛❧ ✐❞❡❛❧ P ✳ ❖♥❡ s❤♦✇ t❤❛t ✇❡ ❛❝t✉❛❧❧② ❤❛✈❡ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ♣❧❛❝❡s ❞✐✛❡r❡♥t ❢r♦♠ ∞ ❛♥❞ t❤❡ ♣r✐♠❡ ✐❞❡❛❧s ♦❢ A ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦rr❡s♣♦♥❞❡♥❝❡✿ P (✐❞❡❛❧ ♦❢ A) ⇔ (AP, P AP) . ❋r♦♠ t❤✐s ❝♦rr❡s♣♦♥❞❡♥❝❡✱ s♦♠❡t✐♠❡s ✇❡ r❡❢❡r t♦ t❤❡ ♣❧❛❝❡ P ❛s t❤❡ ♣r✐♠❡ ✐❞❡❛❧ ♦❢ A✳ ▼♦r❡♦✈❡r✱ t❤❡ ❧❡♠♠❛ ✷✳✶✳✷ ❛♥❞ t❤❡ ❢❛❝t t❤❛t A/Pm ≡ A P/ (P AP)m ❛❧❧♦✇ ✉s t♦ ❞❡✜♥❡ vP ❛♥❞ deg P ✇✐t❤ t❤❡ s❛♠❡ ♥♦t✐♦♥s ✇❤❡♥❡✈❡r ✇❡ ❛r❡ t❤✐♥❦✐♥❣ ♦❢ P ❛s ❛ ♣❧❛❝❡ ♦❢ k✱ ♦r ❛ ♣r✐♠❡ ✐❞❡❛❧ ♦❢ A✳ ❚❤❡♦r❡♠ ✷✳✶✳✺✳ ■❢ a ∈ A✱ t❤❡♥ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ A/ (a) ♦✈❡r F ✐s ❡q✉❛❧ t♦ deg a✳ Pr♦♦❢✳ ■❢ t❤❡ ❢❛❝t♦r✐s❛t✐♦♥ ♦❢ (a) ✐s QPPvP(a)✱ P r✉♥♥✐♥❣ t❤r♦✉❣❤ t❤❡ ♣r✐♠❡ ✭t❤✉s ♠❛①✐♠❛❧✮ ✐❞❡❛❧s ♦❢ A✱ t❤❡♥ t❤❡ ❈❤✐♥❡s❡ r❡♠❛✐♥❞❡r t❤❡♦r❡♠ ❣✐✈❡s ✉s A/ (a) = M P A/PvP(a).
❚❤❡r❡❢♦r❡ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ A/ (a) ✐s ❡q✉❛❧ t♦ PPdimFA/PvP(a)✱ P ✐s r✉♥♥✐♥❣
t❤r♦✉❣❤ ❛❧❧ t❤❡ ♣r✐♠❡ ✐❞❡❛❧s ♦❢ A✳ ◆♦✇✱ s✐♥❝❡ P ✐s ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧ ♦❢ A s♦ t❤❛t A/PvP(a) ✐s ✐s♦♠♦r♣❤✐❝ t♦ AP/ (P AP)vP(a)✱ t❤❡♥ A/ (a) = M P AP/ (P AP)vP(a),
❈❍❆P❚❊❘ ✷✳ ❉❘■◆❋❊▲❉ ▼❖❉❯▲❊❙ ❖❱❊❘ ❋■❊▲❉❙ ✶✶ P ✐s r✉♥♥✐♥❣ t❤r♦✉❣❤ ❛❧❧ t❤❡ ♣r✐♠❡ ✐❞❡❛❧s ♦❢ A✳ ◆♦✇✱ P AP ✐s ❛ ♣r✐♥❝✐♣❛❧ ✐❞❡❛❧✱ t❤❡♥ (P AP)i−1/ (P AP)i ∼ AP/P AP✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ AP/ (P AP)vP(a)−1 ∼ AP/ (P AP)vP(a) /(P AP)vP(a)−1/ (P AP)vP(a) . ❇✉t ❢♦r t✇♦ ✈❡❝t♦r✐❛❧ s♣❛❝❡s W ⊂ V ♦✈❡r F✱ ✇❡ ❤❛✈❡ dimFV /W = dimFV − dimFW✳ ❆♥❞ t❤❡r❡❢♦r❡✱
dimFAP/ (P AP)vP(a) = vP(a) deg (P AP) ,
✭P AP ✐s ❛❝t✉❛❧❧② t❤❡ ♣❧❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ♣r✐♠❡ P ✮✱ ❛♥❞ t❤❡♥ ✇❡ ❤❛✈❡ dimFA/ (a) = X P vP (a) deg (P AP) . ❆s P r✉♥♥✐♥❣ t❤r♦✉❣❤ ❛❧❧ t❤❡ ♣r✐♠❡ ✐❞❡❛❧s ♦❢ A✱ t❤❡♥ P AP ✐s r✉♥♥✐♥❣ t❤r♦✉❣❤ ❛❧❧ t❤❡ ♣❧❛❝❡s ♦❢ k ❞✐✛❡r❡♥t ❢r♦♠ ∞✳ ❆♥❞ t❤❡♥ dimFA/ (a) = X P vP (a) deg P, ✇❤❡r❡ P ✐s ♥♦✇ r✉♥♥✐♥❣ t❤r♦✉❣❤ ❛❧❧ t❤❡ ♣❧❛❝❡s ♦❢ k✱ ❞✐✛❡r❡♥t ❢r♦♠ ∞✳ ❆s t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ ❞✐✈✐s♦r (a) ✐s ❡q✉❛❧ t♦ ③❡r♦✱ t❤❡♥ PPvP (a) deg P =
−v∞(a) d∞✳ ❚❤❡r❡❢♦r❡
dimFA/ (a) = deg a.
2 Pr♦♣♦s✐t✐♦♥ ✷✳✶✳✻✳ ▲❡t ClA ❞❡♥♦t❡s t❤❡ ❝❧❛ss ❣r♦✉♣ ♦❢ A ❛s ❛ ❉❡❞❡❦✐♥❞ ❞♦✲ ♠❛✐♥❀ D✱ t❤❡ ❣r♦✉♣ ♦❢ ❞✐✈✐s♦rs ♦❢ k❀ P✱ t❤❡ ❣r♦✉♣ ♦❢ ♣r✐♥❝✐♣❛❧ ❞✐✈✐s♦rs❀ D0✱ t❤❡ ❣r♦✉♣ ♦❢ ❞✐✈✐s♦rs ♦❢ ❞❡❣r❡❡ ③❡r♦✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡ ✐s ❡①❛❝t✱ (0) //D 0/P //ClA //Z/ (d∞) //(0) . ❆♥❞ t❤✉s✱ ✐❢ ♯ ClA = hA✱ t❤❡♥ hA ✐s ✜♥✐t❡ ✇✐t❤ hA = d∞hk✱ ✇❤❡r❡ hk ✐s t❤❡ ❝❧❛ss ♥✉♠❜❡r ♦❢ k✳ Pr♦♦❢✳ ❲❡ s❛✇✱ ✐♥ r❡♠❛r❦ ✷✳✶✳✹✱ t❤❛t t❤❡r❡ ✐s ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣r✐♠❡ ✐❞❡❛❧s ♦❢ A ❛♥❞ t❤❡ ♣❧❛❝❡s ♦❢ k ❞✐✛❡r❡♥t ❢r♦♠ ∞✳ ❚❤✉s✱ ✇❡ ❝❛♥ r❡❣❛r❞ ClA ❛s t❤❡ s✉❜❣r♦✉♣ ♦❢ D✱ ❣❡♥❡r❛t❡❞ ❜② ♣❧❛❝❡s ❞✐✛❡r❡♥t ❢r♦♠ ∞✱ ♠♦❞✉❧♦ t❤❡ s✉❜❣r♦✉♣ ♦❢ ♣r✐♥❝✐♣❛❧ ❞✐✈✐s♦rs ✇✐t❤♦✉t ∞ ✭✐✳❡✳ t❤❡ ❡❧❡♠❡♥ts Q P 6=∞PvP(a), a ∈ k∗✮✳ ❙♦ ✇❡ ❝♦♥str✉❝t t❤❡ s❡❝♦♥❞ ♠♦r♣❤✐s♠ ✐♥ t❤❡ s❡q✉❡♥❝❡ ❛s D = QP 6=∞PvP(D)∞v∞(D) 7→Q P 6=∞PvP(D)✳ ❆♥❞ t❤✐s ✐s ♦❜✈✐♦✉s❧② ✐♥❥❡❝t✐✈❡✳ ❚❤❡ t❤✐r❞ ♠♦r♣❤✐s♠ ✐s ♥♦t❤✐♥❣ ❡❧s❡ t❤❛♥ t❤❡ ❞❡❣r❡❡ ♦❢ ❛ ❞✐✈✐s♦r ♠♦❞✉❧♦ d∞✳ ❙✐♥❝❡ t❤❡ ❞❡❣r❡❡ ♦❢ ❛ ♣r✐♥❝✐♣❛❧ ❞✐✈✐s♦r ✐s ❡q✉❛❧ t♦ ③❡r♦✱ t❤✐s ♠♦r♣❤✐s♠ ✐s ✇❡❧❧ ❞❡✜♥❡❞✳ ❆♥❞ ✐t ✐s ✐♥❥❡❝t✐✈❡ s✐♥❝❡✱ ❢♦r m ∈ Z/ (d∞)✱ ✇❡ t❛❦❡ D = Pm✱ ✇❤❡r❡
❈❍❆P❚❊❘ ✷✳ ❉❘■◆❋❊▲❉ ▼❖❉❯▲❊❙ ❖❱❊❘ ❋■❊▲❉❙ ✶✷ P ✐s ❛ ♣❧❛❝❡ ♦❢ ❞❡❣r❡❡ 1 ✭❚❤✐s ❡①✐st ❜② ❙❝❤♠✐❞t✬s ❚❤❡♦r❡♠ ✭❙❝❤♠✐❞t✱ ✶✾✸✶✮✮✳ ❚♦ ❝♦♠♣❧❡t❡ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❡①❛❝t♥❡ss ♦❢ t❤❡ s❡q✉❡♥❝❡✱ ✇❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t t❤❡ ✐♠❛❣❡ Im ♦❢ t❤❡ s❡❝♦♥❞ ♠♦r♣❤✐s♠ ✐s ❡q✉❛❧ t♦ t❤❡ ❦❡r♥❡❧ Ker ♦❢ t❤❡ t❤✐r❞ ♠♦r♣❤✐s♠✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❡❧❡♠❡♥t PP 6=∞PvP(D)∞v ∞(D) ∈ D 0/P ❜❡✐♥❣ ❡q✉❛❧ t♦ ③❡r♦✱ ❤❡♥❝❡ Im ⊂ Ker✳ ■❢ deg QP 6=∞PvP(D) = md∞✱ t❤❡♥ ✇❡ ❥✉st ❝♦♠♣❧❡t❡ ✐t ❜② ∞m t♦ ❣❡t t❤❡ ♦t❤❡r ✐♥❝❧✉s✐♦♥✳ ❚❤❡ r❡♠❛✐♥✐♥❣ ♣❛rt ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥ ❢♦❧❧♦✇s ❛s hk ✐s ✜♥✐t❡✳ 2
✷✳✷ ❚♦rs✐♦♥ ♠♦❞✉❧❡s
■t ✐s ♥♦r♠❛❧ t❤❛t ✇❡ st✉❞② t❤❡ t♦rs✐♦♥ ♠♦❞✉❧❡s ♦✈❡r ❛ ❉❡❞❡❦✐♥❞ ❞♦♠❛✐♥✳ ▲❡t A✱ ❜❡ ❛ ❉❡❞❡❦✐♥❞ ❞♦♠❛✐♥ ❛♥❞ ❛ss✉♠❡ M ✐s ❛♥ A✲♠♦❞✉❧❡✳ ❘❡❝❛❧❧ t❤❛t ❛ t♦rs✐♦♥ s✉❜♠♦❞✉❧❡ ♦❢ M ✐s✱ ❢♦r ❛ ♥♦♥✲③❡r♦ ✐❞❡❛❧ I ♦❢ A✱ ❞❡✜♥❡❞ ❜② M [I] = {m ∈ M : mx = 0, ∀x ∈ I} . ❋♦r t✇♦ r❡❧❛t✐✈❡❧② ♣r✐♠❡ ✐❞❡❛❧s ♦❢ A✱ I1 ❛♥❞ I2✱ t❤❡r❡ ❛r❡ s♦♠❡ ❡❧❡♠❡♥ts x ∈ I1 ❛♥❞ y ∈ I2 s✉❝❤ t❤❛t x + y = 1✳ ❚❤✉s ❛♥② ❡❧❡♠❡♥ts m ♦❢ M [I1I2] ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s mx + my = m✳ ❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ t♦rs✐♦♥ ♠♦❞✉❧❡s✱ ✇❡ s❡❡ t❤❛t t❤✐s s✉♠ ✐s ❛❝t✉❛❧❧② ❞✐r❡❝t✱ ✐✳❡✳ M [I1I2] = M [I1] ⊕ M [I2] . ✭✷✳✷✳✶✮ ❇② ✐♥❞✉❝t✐♦♥✱ t❤✐s ✐♠♣❧✐❡s MIk1 1 Ik22· · · Iknn = M I1k1 ⊕ M Ik22 ⊕ · · · ⊕ M Iknn , ✭✷✳✷✳✷✮ ✇❤❡r❡ t❤❡ Ii✬s ❛r❡ r❡❧❛t✐✈❡❧② ♣r✐♠❡✳ ❉❡✜♥✐t✐♦♥ ✷✳✷✳✶✳ ▲❡t I ❜❡ ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧ ♦❢ A✳ ❲❡ ❞❡✜♥❡ t❤❡ I✲♣r✐♠❛r② ❝♦♠♣♦♥❡♥t ♦❢ ❛ ♠♦❞✉❧❡ M✱ M [I∞]✱ t♦ ❜❡ t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ t♦rs✐♦♥ s✉❜♠♦❞✉❧❡s ♦❢ M ❣✐✈❡♥ ❜② t❤❡ ♣♦✇❡rs ♦❢ I ✐✳❡✳ M [I∞] = ∪∞l=1MIl . Pr♦♣♦s✐t✐♦♥ ✷✳✷✳✷✳ ■❢ t❤❡ ♠♦❞✉❧❡ M ✐s ❛❧s♦ ❛ t♦rs✐♦♥ ♠♦❞✉❧❡✱ t❤❡♥ ✐t ✐s ❛ ❞✐r❡❝t s✉♠ ♦❢ ❛❧❧ t❤❡ M [I∞]✱ I ♠❛①✐♠❛❧ ✐❞❡❛❧s ♦❢ A ✐✳❡✳ ✱ ✐❢ M ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♠❛①✐♠❛❧ ✐❞❡❛❧s ♦❢ A✱ t❤❡♥✱ M =M I∈M M [I∞] . ✭✷✳✷✳✸✮ Pr♦♦❢✳ ❋♦r ❛♥ ❡❧❡♠❡♥t m ∈ M✱ ✇❡ ❝❛♥ ✜♥❞ ❛♥ ❡❧❡♠❡♥t x ∈ A✱ s✉❝❤ t❤❛t xm = 0✳ ❚❛❦✐♥❣ t❤❡ t♦rs✐♦♥ s✉❜♠♦❞✉❧❡ ❣✐✈❡♥ ❜② t❤❡ ♣r✐♥❝✐♣❛❧ ✐❞❡❛❧ (x)✱ ✇❡ ❤❛✈❡ ❢r♦♠ t❤❡ ✐❞❡♥t✐t② ✐♥ ✭✷✳✷✳✷✮✱ M [(x)] = MφIk11 ⊕ MφI2k2 ⊕ · · · ⊕ MφIknn ,❈❍❆P❚❊❘ ✷✳ ❉❘■◆❋❊▲❉ ▼❖❉❯▲❊❙ ❖❱❊❘ ❋■❊▲❉❙ ✶✸ ✇❤❡r❡ Ik1Ik2· · · Ikn ✐s t❤❡ ♣r✐♠❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ (x)✳ ❆s m ❜❡❧♦♥❣s t♦ M [(x)] ❛♥❞ ❛ ♣r✐♠❡ ✐❞❡❛❧ ✐s ♠❛①✐♠❛❧ ✐♥ ❛ ❉❡❞❡❦✐♥❞ ❞♦♠❛✐♥✱ t❤❡♥ ✇❡ s❡❡ t❤❛t m ∈ P I∈MM [I∞]✳ ❚❤✉s ✇❡ ❤❛✈❡ ✇❡ s✉♠ M = X I∈M M [I∞] . ◆♦✇ ✐❢ 0 = Pr l=1ml✱ ✇✐t❤ ml ❜❡❧♦♥❣✐♥❣ t♦ s♦♠❡ ♣r✐♠❛r② ❝♦♠♣♦♥❡♥t M [I∞l ]✱ t❤❡♥ ✇❡ ♠✉st ❤❛✈❡ ml∈ M [Irl]✳ ❚❤✉s✱ ❜② ✭✷✳✷✳✷✮✱ ✇❡ ♠✉st ❤❛✈❡ ml= 0 ❢♦r ❛❧❧ 1 ≤ l ≤ r✳ ❚❤❡r❡❢♦r❡ t❤❡ s✉♠ ✐s ❞✐r❡❝t✳ 2 ◆♦✇ ❧❡t ✉s ❤❛✈❡ ❛♥ ❡①❛❝t s❡q✉❡♥❝❡ ♦❢ t♦rs✐♦♥ A✲♠♦❞✉❧❡s (0) //M1 //M2 //M3 //(0) ■❢ I ✐s ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧ ♦❢ A✱ t❤❡♥ t❤✐s s❡q✉❡♥❝❡ ✐♥❞✉❝❡s ❛♥ ❡①❛❝t s❡q✉❡♥❝❡ ♦♥ t❤❡ I✲♣r✐♠❛r② ❝♦♠♣♦♥❡♥ts ✐✳❡✳ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡ ✐s ❡①❛❝t (0) //M1[I∞] //M2[I∞] //M3[I∞] //(0) ✭✷✳✷✳✹✮ ■❢ ✇❡ ❤❛✈❡ ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧ I ♦❢ A✱ t❤❡♥ ❧❡t ✉s t❛❦❡ ❛♥ ✉♥✐❢♦r♠✐③❡r π ♦❢ I✳ ❆s πl = IlP✱ ❢♦r s♦♠❡ ♣r✐♠❡ P r❡❧❛t✐✈❡❧② ♣r✐♠❡ t♦ I✱ ✇❡ ❤❛✈❡ ❜② ✭✷✳✷✳✶✮✱ Mπl = M Il ⊕ M [P] . ❚❤❡♥✱ ✇❡ ❤❛✈❡ Mπl [I∞] = M Il [I∞] ⊕ M [P] [I∞] . ❇✉t (M [P]) [I∞] = (0)✱ t❤✉s ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✿ Pr♦♣♦s✐t✐♦♥ ✷✳✷✳✸✳ ❋♦r ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧ I ♦❢ A✱ ❛♥❞ ❛ ✉♥✐❢♦r♠✐③❡r π ♦❢ I✱ MIl = M πl [I∞] . ◆❡①t✱ ❧❡t ✉s ❞❡✜♥❡ t❤❡ s❡q✉❡♥❝❡✱ (0) //M [π] //Mπl f //Mπl−1 //(0) , ✇❤❡r❡ f ✐s ❣✐✈❡♥ ❜② t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ ✉♥✐❢♦r♠✐③❡r π ♦❢ ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧ I✳ ■❢ M ✐s ❞✐✈✐s✐❜❧❡✱ t❤❡♥ t❤✐s s❡q✉❡♥❝❡ ✐s ❡①❛❝t✳ ❚❤✉s✱ ❜② ✭✷✳✷✳✹✮ ❛♥❞ t❤❡ ♣r♦♣♦✲ s✐t✐♦♥ ✷✳✷✳✸✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✱ ✇❤✐❝❤ ✐s t❤❡ r❡s✉❧t ✇❡ ♥❡❡❞ ❧❛t❡r ✇❤❡♥ ✇❡ ✇♦r❦ ✇✐t❤ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s✿ ❚❤❡♦r❡♠ ✷✳✷✳✹✳ ❋♦r ❛ ❞✐✈✐s✐❜❧❡ A✲♠♦❞✉❧❡ M ❛♥❞ ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧ I ♦❢ A✱ ✇❡ ❤❛✈❡ ❛♥ ❡①❛❝t s❡q✉❡♥❝❡ (0) //M [I] //MIl f //MIl−1 //(0) .
❈❍❆P❚❊❘ ✷✳ ❉❘■◆❋❊▲❉ ▼❖❉❯▲❊❙ ❖❱❊❘ ❋■❊▲❉❙ ✶✹
✷✳✸ ❚❤❡ ♥♦t✐♦♥ ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s
❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✳ ▲❡t L ❜❡ ❛ ✜❡❧❞✳ ❆ ✜❡❧❞ ♦✈❡r A ♦r s✐♠♣❧② ❛♥ A✲✜❡❧❞ ✐s ❛♥ F✲❛❧❣❡❜r❛ ♠♦r♣❤✐s♠ δ : A → L✳ ❲❡ ❛❧s♦ s❛② t❤❛t L ✐s ❛♥ A✲✜❡❧❞✳ ❚❤❡ A✲✜❡❧❞ δ ✐♥❞✉❝❡s ❛ ♥❛t✉r❛❧ A✲♠♦❞✉❧❡ str✉❝t✉r❡ ♦♥ L✳ ■♥ ♣r❛❝t✐❝❡ t❤✐s ♠♦r♣❤✐s♠ ✐s s❡t t♦ ❜❡ t❤❡ ✐♥❝❧✉s✐♦♥ ♠❛♣ ♦r ❛ r❡❞✉❝t✐♦♥ ♠♦❞✉❧♦ ❛ ♣r✐♠❡ ✐❞❡❛❧ ♦❢ A✳ ❉❡✜♥✐t✐♦♥ ✷✳✸✳✷✳ ❈♦♥s✐❞❡r✐♥❣ L ❛s ❛♥ A✲♠♦❞✉❧❡ ✈✐❛ δ✱ t❤❡ A✲❝❤❛r❛❝t❡r✐st✐❝ ♦❢ L ✐s t❤❡ ❦❡r♥❡❧✱ ker δ✱ ♦❢ t❤❡ ♠❛♣ δ✳ ker δ ✐s ❛ ♣r✐♠❡ ✐❞❡❛❧ ♦❢ t❤❡ r✐♥❣ A✳ ❉❡✜♥✐t✐♦♥ ✷✳✸✳✸ ✭❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡s✮✳ ▲❡t δ ❜❡ ❛♥ A✲✜❡❧❞ ❛♥❞ s✉♣♣♦s❡ D : L hτ i → L,P lnτn7→ l0 ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ❛t ③❡r♦✳ ❆ ❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡ φ ♦✈❡r t❤❡ ✜❡❧❞ L ✐s ❛ F✲❛❧❣❡❜r❛ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ A t♦ t❤❡ r✐♥❣ ♦❢ t✇✐st❡❞ ♣♦❧②♥♦♠✐❛❧s L hτi s✉❝❤ t❤❛t D ◦ φ = δ✱ ❛♥❞ φ(A) * L✳ ❋♦r s✐♠♣❧✐✜❝❛t✐♦♥ ✇❡ ✇✐❧❧ ❞❡♥♦t❡ t❤❡ ✐♠❛❣❡ ♦❢ a ∈ A ❜② φa✐♥st❡❛❞ ♦❢ φ (a) ❛♥❞ ✇❡ ❞❡✜♥❡ t❤❡ ❞❡❣r❡❡ deg φa ❛s t❤❡ ❞❡❣r❡❡ ♦❢ φa t❤♦✉❣❤t ❛s ❛ ♣♦❧②♥♦♠✐❛❧ ✐♥ τ✳ ❉❡✜♥✐t✐♦♥ ✷✳✸✳✹✳ ▲❡t φ ❜❡ ❛ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡ ❛♥❞ s✉♣♣♦s❡ δ ✐s t❤❡ ❝♦rr❡✲ s♣♦♥❞✐♥❣ A✲✜❡❧❞✱ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ char φ ♦❢ t❤❡ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s φ ✐s t❤❡ A✲❝❤❛r❛❝t❡r✐st✐❝ ♦❢ L ✈✐❛ δ✳✷✳✸✳✶ ❚❤❡ ♠♦❞✉❧❡ str✉❝t✉r❡
■t ✐s ♥♦t ❝❧❡❛r ✇❤② ✇❡ ❛r❡ ❝❛❧❧✐♥❣ t❤❡ ♠❛♣ φ ❛s ❛ ♠♦❞✉❧❡✳ ❚❤✐s ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ✇❡ ❝❛♥ ❝♦♥str✉❝t ❛ ♥❡✇ A✲♠♦❞✉❧❡ str✉❝t✉r❡ ♦♥ ❛♥② L✲❛❧❣❡❜r❛ M✱ ❜② ❞❡✜♥✐♥❣ t❤❡ ❡①t❡r♥❛❧ ♣r♦❞✉❝t ❛s a.u = φa(u) , ❢♦r ❛❧❧ a ∈ A, u ∈ M. ❯s✉❛❧❧②✱ ✇❡ ❞❡♥♦t❡ t❤❡ A✲♠♦❞✉❧❡ ❛s Mφ✱ ✐❢ t❤❡ ♠♦❞✉❧❡ str✉❝t✉r❡ ❝♦♠❡s ❢r♦♠ φ✳ ▲✐❦❡ ♠❛♥② str✉❝t✉r❡ ✐♥ ❛❧❣❡❜r❛✱ ✇❡ ❝❛♥ t❤❡♥ ❞❡✜♥❡ ❛ t♦rs✐♦♥ s✉❜♠♦❞✉❧❡ ❛s Mφ[a] = {u ∈ Mφ: φa(u) = 0} . ●❡♥❡r❛❧❧②✱ ✇❡ ❝❛♥ ❞❡✜♥❡✱ ❢♦r ❛♥ ✐❞❡❛❧ I ♦❢ A✱Mφ[I] = {u ∈ Mφ: φa(u) = 0, ∀a ∈ I} .
✷✳✸✳✷ ❚❤❡ ❝❛t❡❣♦r② ♦❢ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s
■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❧♦♦❦ ❛ ❜✐t ✐♥ t❤❡ ❝❛t❡❣♦r② ❢♦r♠❡❞ ❜② ❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡s ♦✈❡r L✱ ✇❤❡r❡ t❤❡ ♠♦r♣❤✐s♠s ❛r❡ ✐s♦❣❡♥✐❡s✳ ▲❡t ✉s ❞❡♥♦t❡ t❤✐s ❝❛t❡❣♦r②✱ ❢♦r ❛ ✜①❡❞ δ✱ ❜② DrinL(A)✳
❈❍❆P❚❊❘ ✷✳ ❉❘■◆❋❊▲❉ ▼❖❉❯▲❊❙ ❖❱❊❘ ❋■❊▲❉❙ ✶✺ Pr♦♣♦s✐t✐♦♥ ✷✳✸✳✺✳ ❙✉♣♣♦s❡ φ ✐s ❛ ❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡✳ ❚❤❡♥✱ ❢♦r s♦♠❡ ♣♦s✐✲ t✐✈❡ r❛t✐♦♥❛❧ ♥✉♠❜❡r rφ✱ deg φa= −rφv∞(a) d∞✱ ❢♦r ❛❧❧ a ✐♥ A✳ ■♥ ♦t❤❡r ✇♦r❞✱
deg φa = rφ deg a✳
Pr♦♦❢✳ ■❢ v (a) = − deg φa✱ v ❞❡✜♥❡s ❛ ✈❛❧✉❛t✐♦♥ ♦♥ A✳ ■♥❞❡❡❞✱
❼ ✇❡ ❝❛♥ ❛ss✉♠❡ v (0) = ∞❀ ❼ v (ab) = v (a) + v (b)❀ ❼ ❛♥❞ ✜♥❛❧❧② v (a + b) ≥ min {v (a) , v (b)}✳ ◆♦✇✱ t❤✐s ✈❛❧✉❛t✐♦♥ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ✈❛❧✉❛t✐♦♥ ♦♥ k ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♣❧❛❝❡ ∞✱ s✐♥❝❡ ♦♥❧② t❤❡ ✈❛❧✉❛t✐♦♥s ❢r♦♠ t❤✐s ♣❧❛❝❡ ❛r❡ ♥❡❣❛t✐✈❡ ♦♥ A✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ ✈❛❧✉❛t✐♦♥s ②✐❡❧❞s✱ ❢♦r s♦♠❡ ♣♦s✐t✐✈❡ r❛t✐♦♥❛❧ rφ ❛♥❞ ❛❧❧ a ∈ A✱ v(a) = rφd∞v∞(a) . 2 ❉❡✜♥✐t✐♦♥ ✷✳✸✳✻✳ ❚❤❡ ♥✉♠❜❡r ❞❡✜♥❡❞ ✐♥ t❤❡ ♣r♦♣♦s✐t✐♦♥ ✷✳✸✳✺ ✐s ❝❛❧❧❡❞ t❤❡ r❛♥❦ ♦❢ ❛ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡✳ ◆♦✇ ❧❡t ✉s ❝♦♥t✐♥✉❡ t♦ t❤❡ ♥♦t✐♦♥ ♦❢ ❤❡✐❣❤t ♦❢ ❛ ❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡✳ ❲❡ ❛ss✉♠❡ t❤❛t φ ✐s ❛ ❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡ ✇✐t❤ ♥♦♥③❡r♦ ❝❤❛r❛❝t❡r✐st✐❝ Q✳ ❉❡✜♥✐t✐♦♥ ✷✳✸✳✼✳ ❲❡ ❞❡✜♥❡ t❤❡ ♠❛♣ ω✱ s✉❝❤ t❤❛t ω (a) ✐s t❤❡ ✐♥❞❡① ♦❢ t❤❡ s♠❛❧❧❡st ♣♦✇❡r ♦❢ τ ✇✐t❤ ♥♦♥③❡r♦ ❝♦❡✣❝✐❡♥t ✐♥ φa ✭✇❡ ❞❡✜♥❡ ω(0) = ∞✮✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✿ Pr♦♣♦s✐t✐♦♥ ✷✳✸✳✽✳ ❚❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ r❛t✐♦♥❛❧ ♥✉♠❜❡r hφ✱ s✉❝❤ t❤❛t
ω (a) = hφvQ(a) deg Q, ∀a ∈ A.
Pr♦♦❢✳ ❚❤❡ ♠❛♣ ω ❞❡✜♥❡s ❛ ✈❛❧✉❛t✐♦♥ ♦♥ A✱ t❤✉s ✐t ❡①t❡♥❞s t♦ ❛ ✈❛❧✉❛t✐♦♥ ♦♥ k✳ ❚❤❡ ✈❛❧✉❛t✐♦♥ r✐♥❣s ❣✐✈❡♥ ❜② t❤✐s ✈❛❧✉❛t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ Q✳ ❚❤✉s t❤❡ t✇♦ ✈❛❧✉❛t✐♦♥s ω ❛♥❞ vQ ❛r❡ ❡q✉✐✈❛❧❡♥ts✳ ❚❤❡ r❡s✉❧t ❢♦❧❧♦✇s ✐♠♠❡❞✐❛t❡❧②✳ 2 ❉❡✜♥✐t✐♦♥ ✷✳✸✳✾✳ ❋♦r ❛ ❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡ φ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ Q✱ ✐❢ Q 6= (0)✱ ✇❡ ❞❡✜♥❡ t❤❡ ❤❡✐❣❤t ❛s t❤❡ ✉♥✐q✉❡ ♣♦s✐t✐✈❡ r❛t✐♦♥❛❧ ♥✉♠❜❡r hφ ✐♥ t❤❡ t❤❡♦r❡♠ ✷✳✸✳✽✳ ■❢ Q 6= (0)✱ t❤❡♥ ✇❡ s❡t hφ= 0✳ ❖♥❡ ♠❛② ❛s❦ ✇❤✐❝❤ ♠♦r♣❤✐s♠ ❝❛♥ ✇❡ ❞❡✜♥❡ ❢♦r ✉s t♦ ❤❛✈❡ ❛ ❝❛t❡❣♦r②✳ ❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✵✳ ■❢ φ, ψ ❛r❡ t✇♦ ❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡s✱ t❤❡♥ ✇❡ ❞❡✜♥❡ ❛ ♠♦r✲ ♣❤✐s♠ ❢r♦♠ φ t♦ ψ ❛s ❛♥ ❡❧❡♠❡♥t f ∈ L hτi s✉❝❤ t❤❛t fφa = ψaf ❢♦r ❛❧❧ ❡❧❡♠❡♥ts a ∈ A✳ ❚❤❡ s❡t ♦❢ ♠♦r♣❤✐s♠s ❢r♦♠ φ t♦ ψ ✐s ❞❡♥♦t❡❞ ❜② homL(φ, ψ)✳
❈❍❆P❚❊❘ ✷✳ ❉❘■◆❋❊▲❉ ▼❖❉❯▲❊❙ ❖❱❊❘ ❋■❊▲❉❙ ✶✻ ■♥ ❢❛❝t✱ ✇❤❡♥ ✇❡ t❛❦❡ ❛♥ ❛❧❣❡❜r❛✐❝❛❧❧② ❝❧♦s❡❞ ✜❡❧❞ ❡①t❡♥s✐♦♥ M ♦❢ L✱ t❤❡♥ ❛s A✲♠♦❞✉❧❡s✱ f ✐s ❛♥ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ Mφ t♦ Mψ✳ ❍❡♥❝❡✱ ❧✐❦❡ ✐♥ t❤❡ t❤❡♦r② ♦❢ ❊❧❧✐♣t✐❝ ❝✉r✈❡s✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t✐♦♥✳ ❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✶✳ ❆ ♥♦♥✲③❡r♦ ♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ t✇♦ ❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡s ψ ❛♥❞ ψ ✐s ❝❛❧❧❡❞ ❛♥ ✐s♦❣❡♥②✳ ❚❤✉s✱ t✇♦ ❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡s ❛r❡ ❝❛❧❧❡❞ ✐s♦❣❡♥♦✉s ✐❢ homL(φ, ψ) ❤❛s ❛ ♥♦♥✲③❡r♦ ❡❧❡♠❡♥t✳ ❚❤❡ ✜rst ♣r♦♣❡rt② ✇❡ ❤❛✈❡ ❢r♦♠ t✇♦ ✐s♦❣❡♥♦✉s ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s ✐s ❛❜♦✉t t❤❡✐r r❛♥❦ ❛♥❞ ❤❡✐❣❤t✿ Pr♦♣♦s✐t✐♦♥ ✷✳✸✳✶✷✳ ❚✇♦ ✐s♦❣❡♥♦✉s ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s φ, ψ ❤❛✈❡ t❤❡ s❛♠❡ r❛♥❦ ❛♥❞ ❤❡✐❣❤t✳ Pr♦♦❢✳ ■❢ φ ❛♥❞ ψ ❛r❡ ✐s♦❣❡♥♦✉s✱ t❤❡♥ ❢♦r s♦♠❡ ♥♦♥✲③❡r♦ f ∈ L hτi✱ ❢♦r ❛❧❧ a ∈ A✱ fφa = ψaf✳ ❚❤❡♥ deg fφa = deg ψaf s♦ t❤❛t deg φa = deg ψa✳ ❍❡♥❝❡✱
❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ r❛♥❦ ✇❡ ❤❛✈❡✱ rφv∞(a) d∞= rψv∞(a) d∞✳ ❙✐♠♣❧✐❢②✐♥❣✱ ✇❡ ❣❡t t❤❡ r❡s✉❧t✳ ❋♦r t❤❡ ❤❡✐❣❤t✱ fφa = ψaf ❛❧s♦ ❣✐✈❡s ✉s hφvQ(a) deg Q = hφvQ′(a) deg Q′✳ ❲❤❡r❡ Q ❛♥❞ Q′ ❛r❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ φ ❛♥❞ ψ✳ ■❢ ✇❡ ❦♥❡✇ t❤❛t Q = Q′✱ t❤❡♥ ✇❡ ❛r❡ ❞♦♥❡✳ ❙♦ ❧❡t ✉s s❤♦✇ t❤❛t t❤❡s❡ ❝❤❛r❛❝t❡r✐st✐❝s ❛r❡ t❤❡ s❛♠❡✳ ■❢ t❤❡ ❝♦♥st❛♥t t❡r♠ ♦❢ f ✐s 0✱ t❤❡♥ ✇❡ ❝❛♥ r❡♠♦✈❡ s♦♠❡ ❢❛❝t♦r ♣♦✇❡r ♦❢ p✱ s♦ t❤❛t f1φ′a = ψ′af2✱ ✇❤❡r❡ ❜♦t❤ f1, f2 ❤❛✈❡ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts ❞✐✛❡r❡♥t ❢r♦♠ 0✳ ❲❤❛t ✇❡ s❤♦✉❧❞ ♥♦t✐❝❡ ✐s t❤❛t t❤❡ ❝♦♥st❛♥t t❡r♠ ♦❢ φa✭r❡s♣✳ ψa✮ ❡q✉❛❧s t♦ ③❡r♦ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❝♦♥st❛♥t t❡r♠ ♦❢ φ′a✭r❡s♣✳ ψ′ a✮ ❡q✉❛❧s t♦ ③❡r♦✳ ❆♥❞ t❤❡ ❡q✉❛❧✐t② Q = Q′ ❢♦❧❧♦✇s ✐♠♠❡❞✐❛t❡❧②✳ 2
■❢ φ ∈ DrinL(A)✱ t❤❡♥ ❢♦r ❛♥ ✐❞❡❛❧ I ♦❢ A✱ t❤❡ ❧❡❢t ✐❞❡❛❧ ♦❢ L hτi ❣❡♥❡r❛t❡❞
❜② t❤❡ ✐♠❛❣❡ ♦❢ I ❜② φ ✐s ♣r✐♥❝✐♣❛❧✳ ❲❡ t❛❦❡ t❤✐s r❡s✉❧t ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡ ❧❡❢t ✐❞❡❛❧s ♦❢ L hτi ❛r❡ ♣r✐♥❝✐♣❛❧ ✭s❡❡ ●♦ss✱ ✶✾✾✼✱ ❝❤❛♣✳ ✶✮✳ ❑❡❡♣✐♥❣ t❤❡s❡ ♥♦t❛t✐♦♥s✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥✿ ❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✸✳ ❚❤❡ s❦❡✇ ♣♦❧②♥♦♠✐❛❧ φI✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ✉♥✐q✉❡ ♠♦♥✐❝ ❣❡♥❡r❛t✐♥❣ t❤❡ ❧❡❢t ♣r✐♥❝✐♣❛❧ ✐❞❡❛❧ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ✐♠❛❣❡ ♦❢ ❛♥ ✐❞❡❛❧ I ♦❢ A ❜② t❤❡ ❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡ φ✳ ❘❡♠❛r❦ ✷✳✸✳✶✹✳ ■❢ I ✐s ❛♥ ✐❞❡❛❧ ♦❢ A✱ t❤❡♥ φI ✐s ❛ ✜♥✐t❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ φai✱ ✇❤❡r❡ ai ∈ I✳ ❚❤❡ s❛♠❡ ❢♦r φa✱ a ∈ I✱ ✐t ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ φI✳ ❚❤✉s φI ✈❛♥✐s❤❡s ✐❢ ❛♥❞ ♦♥❧② ✐❢ φa ✈❛♥✐s❤❡s ❢♦r ❛♥② a ✐♥ I✳ ❚❤❡r❡❢♦r❡✱ ❢♦r ❛♥ L✲❛❧❣❡❜r❛ M✱ ✇❡ ❛❧s♦ ❤❛✈❡ Mφ[I] = {u ∈ Mφ: φI(u) = 0} . ❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✿
Pr♦♣♦s✐t✐♦♥ ✷✳✸✳✶✺✳ ■❢ φ ∈ DrinL(A)✱ t❤❡♥ ❢♦r ❛ ♥♦♥③❡r♦ ✐❞❡❛❧ I ♦❢ A✱ φI ✐s
❈❍❆P❚❊❘ ✷✳ ❉❘■◆❋❊▲❉ ▼❖❉❯▲❊❙ ❖❱❊❘ ❋■❊▲❉❙ ✶✼ Pr♦♦❢✳ hφIi φa ⊂ hφIi ❢♦r ❛♥② ❡❧❡♠❡♥t a ♦❢ A✳ ❚❤✉s ❢♦r a ∈ A✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ s❦❡✇ ♣♦❧②♥♦♠✐❛❧ ψa s✉❝❤ t❤❛t φIψa = ψaφI✳ ❚❤✐s ❣✐✈❡s ❛ F✲❛❧❣❡❜r❛ ψ : A −→ L hτ i ❛♥❞ t❤✐s ✐s ❛ ❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡✳ ❲❤❛t ✇❡ ❞✐❞♥✬t ♣r♦✈❡ ✐s t❤❛t t❤❡ F ✲❛❧❣❡❜r❛ ❤♦♠♦♠♦♣❤✐s♠ D ◦ ψ : A −→ L ✐s ❡q✉❛❧ t♦ δ ✐✳❡✳ φ ❛♥❞ ψ ❤❛s t❤❡ s❛♠❡ ✜❡❧❞ ♦✈❡r A✳ ❲❡ ✇✐❧❧ s❡❡ t❤✐s ✐♥ t❤❡ ❝♦r♦❧❧❛r② ✷✳✸✳✷✵✳ 2 ❲❡ ❞❡♥♦t❡ t❤❡ ❉r✐♥❢❡❧❞ A✲♠♦❞✉❧❡ ψ ✐♥ t❤❡ ♣r♦♣♦s✐t✐♦♥ ✷✳✸✳✶✺ ❜② I ∗ φ✳ ■♥ ❢❛❝t✱ ❛❧t❤♦✉❣❤ ✇❡ ❤❛✈❡♥✬t ②❡t ♣r♦✈❡❞ t❤❛t I ∗ φ ✐s ❛❝t✉❛❧❧② ✐♥ DrinL(A)✱ ✇❡ st✐❧❧ ❝❛♥ ❤❛✈❡ t❤✐s ❞❡✜♥✐t✐♦♥✳ ❍❡r❡ ❛r❡ s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤✐s ♥♦t✐♦♥✿
Pr♦♣♦s✐t✐♦♥ ✷✳✸✳✶✻✳ ■❢ φ ∈ DrinL(A)✱ ❛♥❞ I1, I2 ❛r❡ ✐❞❡❛❧s ♦❢ A✱ t❤❡♥✱
✭❛✮ φI1I2 = (I1∗ φ)I2φI1❀ ✭❜✮ I1 ∗ (I2∗ φ) = I1I2∗ φ❀ ✭❝✮ φ(a)= l−1φa✱ ❢♦r a ∈ A✱ ✇❤❡r❡ l ✐s t❤❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥t ♦❢ φb✳ Pr♦♦❢✳ ✭❛✮ ❆s (I1∗ φ)I2φI1 ✐s ❛ ♠♦♥✐❝✱ t❤❡♥ t♦ ♣r♦✈❡ ✜rst ❛ss❡rt✐♦♥✱ ✇❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t (I1∗ φ)I2φI1 ❛❧s♦ ❣❡♥❡r❛t❡s t❤❡ ❧❡❢t ✐❞❡❛❧ ❣❡♥❡r❛t❡❞ ❜② {φI1I2}✳ ■♥✲ ❞❡❡❞✱ (I1∗ φ)I2φI1 = X x (I1∗ φ)xφI1, ❢♦r s♦♠❡ ✜♥✐t❡ x ∈ I2 =X x φI1φx, ❜② ❞❡✜♥✐t✐♦♥ ♦❢ ✏∗✑ =X x X y φyφx, ❢♦r s♦♠❡ ✜♥✐t❡ y ∈ I1 =X x,y φyx. ❚❤✉s (I1∗ φ)I2φI1 ❜❡❧♦♥❣s t♦ t❤❡ ✐❞❡❛❧ ❣❡♥❡r❛t❡❞ ❜② φI1I2✳ ❈♦♥✈❡rs❡❧②✱ ❧❡t ✉s ♣r♦✈❡ t❤❛t φI1I2 ❜❡❧♦♥❣s t♦ t❤❡ ✐❞❡❛❧ ❣❡♥❡r❛t❡❞ ❜② (I1 ∗ φ)I2φI1✳ ❲❡
❈❍❆P❚❊❘ ✷✳ ❉❘■◆❋❊▲❉ ▼❖❉❯▲❊❙ ❖❱❊❘ ❋■❊▲❉❙ ✶✽ ❤❛✈❡✱ φI1I2 = X a φa, ❢♦r s♦♠❡ ✜♥✐t❡ a ∈ I1I2 = φPa, ❜✉t X a ∈ I1I2✱ t❤✉s = φPyx, ❢♦r s♦♠❡ ✜♥✐t❡ y ∈ I1 ❛♥❞ x ∈ I2 =X y,x φyφx =X y,x fyφI1φx, ❢♦r s♦♠❡ t✇✐st❡❞ ♣♦❧②♥♦♠✐❛❧ fy =X y,x fy(I1∗ φ)xφI1. ❆s (I1∗ φ)x✱ ❢♦r x ∈ I2✱ ❜❡❧♦♥❣s t♦ t❤❡ ✐❞❡❛❧ ❣❡♥❡r❛t❡❞ ❜② (I1∗ φ)I2✱ t❤❡ r❡s✉❧t ❢♦❧❧♦✇s✳ ✭❜✮ ❋♦r t❤❡ s❡❝♦♥❞ ♣♦✐♥t✱ ❜② t❤❡ ✜rst ♣r♦♣❡rt②✱ ✇❡ ❤❛✈❡ φI1I2φa= (I1∗ φ)I2φI1φa = (I1∗ φ)I2(I1∗ φ)aφI1 = (I2∗ (I1∗ φ))a(I1∗ φ)I2φI1 = (I2∗ (I1∗ φ))aφI1I2. ❇✉t I1I2∗ φ✐s t❤❡ ✉♥✐q✉❡ t✇✐st❡❞ ♣♦❧②♥♦♠✐❛❧ s❛t✐s❢②✐♥❣ t❤✐s r❡❧❛t✐♦♥✱ t❤❡♥ ✇❡ ❤❛✈❡ ♦✉r r❡s✉❧t✳ ✭❝✮ ❚❤❡ r❡s✉❧t ✐s tr✐✈✐❛❧✳ 2 ❲❡ ♠❛② ♥♦t✐❝❡ t❤❛t t❤❡ ✜rst ♣❛rt ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥ ✷✳✸✳✶✻ ✐s✱ s♦♠❡❤♦✇✱ ❛ ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ♥♦t✐♦♥ ♦❢ ∗ ❢r♦♠ φa, a ∈ A t♦ φI✱ I ✐❞❡❛❧ ♦❢ A✳ ▲✐❦❡ t❤✐s✱ ❧❡t ✉s ❣✐✈❡ ❛ ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ♠❛♣ ω ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ✷✳✸✳✼✳ ❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✼✳ ❲❡ ❞❡✜♥❡ ❛ ♠❛♣ ω : L hτi −→ Z s✉❝❤ t❤❛t✱ ❢♦r f ∈ L hτi✱ ω (f ) ✐s t❤❡ ✐♥❞❡① ♦❢ t❤❡ s♠❛❧❧❡st ♣♦✇❡r ♦❢ τ ✇✐t❤ ♥♦♥③❡r♦ ❝♦❡✣❝✐❡♥t ✐♥ f✳ ❆s✱ ❢♦r ❛♥ ✐❞❡❛❧ I ♦❢ A✱ φI ✐s ✉♥✐q✉❡✱ t❤❡♥ ✇❡ ❝❛♥ ❞❡✜♥❡ ω (I) = ω (φI)✳ ❘❡♠❛r❦ ✷✳✸✳✶✽✳ ◆♦✇✱ ✇❡ ❤❛✈❡ t❤r❡❡ ❞✐✛❡r❡♥t ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡ ♠❛♣ ω✳ ❚❤❡ ❝♦♥t❡①t ❛❧❧♦✇s ✉s t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ♦❢ t❤❡s❡ ❞❡✜♥✐t✐♦♥s ✇❡ ❛r❡ t❛❧❦✐♥❣ ❛❜♦✉t✳ ❋✉rt❤❡r♠♦r❡✱ ✐❢ ✇❡ ❛r❡ ✇♦r❦✐♥❣ ✇✐t❤ ❛ ✜①❡❞ ❉r✐♥❢❡❧❞ ♠♦❞✉❧❡s φ✱ t❤❡♥ ω (a) = ω (φa)✳ ❆♥❞ t❤✐s ω ❤❛s ❛♥ ❛❞❞✐t✐✈❡ ♣r♦♣❡rt②✱ ♠♦r❡ ♣r❡❝✐s❡❧② ω (φψ) = ω (φ) + ω (ψ)✳
