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❋✉❞❛♥ ❯♥✐✈❡rs✐t② ❙❤❛♥❣❤❛✐✱ ❈❤✐♥❛ ✐❧✐♠♦♥❝❤❡♥❦♦❅❣♠❛✐❧✳❝♦♠ ❚❤❡ ✹✸r❞ ❙②♠♣♦s✐✉♠ ♦♥ ❚r❛♥s❢♦r♠❛t✐♦♥ ●r♦✉♣s ◆♦✈❡♠❜❡r ✶✽✱ ✷✵✶✻ ❍✐♠❡❥✐✱ ❏❛♣❛♥
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❚❤❡♦r❡♠ ✭▼✳❆t✐②❛❤❀ ❱✳●✉✐❧❧❡♠✐♥✱ ❙✳❙t❡r♥❜❡r❣✬✽✷✮ ▲❡t (M, ω) ❜❡ ❛ ✷d✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❛❝t ❝♦♥♥❡❝t❡❞ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞ ✇✐t❤ ❛ ❤❛♠✐❧t♦♥✐❛♥ ❛❝t✐♦♥ ♦❢ ❛ ❝♦♠♣❛❝t t♦r✉s Tn✳ ❚❤❡♥ t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ♠♦♠❡♥t ♠❛♣ µ : M Rn ✐s ❛ ❝♦♥✈❡① ♣♦❧②t♦♣❡ P ✇❤✐❝❤ ✐s t❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ µ(MT)✳ ■❢ d = n ❛♥❞ t❤❡ t♦r✉s ❛❝t✐♦♥ ✐s ❡✛❡❝t✐✈❡✱ t❤❡♥ (M, ω) ✐s ❛ s②♠♣❧❡❝t✐❝ t♦r✐❝ ♠❛♥✐❢♦❧❞✳ ❆ ♣♦❧②t♦♣❡ ✐♥ ✐s ❝❛❧❧❡❞ ❉❡❧③❛♥t ✐❢ ✐ts ♥♦r♠❛❧ ❢❛♥ ✐s s♠♦♦t❤✳ ❚❤❡♦r❡♠ ✭❚✳❉❡❧③❛♥t✬✽✽✮ ❚❤❡r❡ ✐s ❛ ✶✲✶ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ❝♦♠♣❛❝t s②♠♣❧❡❝t✐❝ t♦r✐❝ ♠❛♥✐❢♦❧❞s ✭✉♣ t♦ ❡q✉✐✈❛r✐❛♥t s②♠♣❧❡❝t♦♠♦r♣❤✐s♠✮ ❛♥❞ ❉❡❧③❛♥t ♣♦❧②t♦♣❡s ✭✉♣ t♦ ❧❛tt✐❝❡ ✐s♦♠♦r♣❤✐s♠✮✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❚❤❡♦r❡♠ ✭▼✳❆t✐②❛❤❀ ❱✳●✉✐❧❧❡♠✐♥✱ ❙✳❙t❡r♥❜❡r❣✬✽✷✮ ▲❡t (M, ω) ❜❡ ❛ ✷d✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❛❝t ❝♦♥♥❡❝t❡❞ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞ ✇✐t❤ ❛ ❤❛♠✐❧t♦♥✐❛♥ ❛❝t✐♦♥ ♦❢ ❛ ❝♦♠♣❛❝t t♦r✉s Tn✳ ❚❤❡♥ t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ♠♦♠❡♥t ♠❛♣ µ : M Rn ✐s ❛ ❝♦♥✈❡① ♣♦❧②t♦♣❡ P ✇❤✐❝❤ ✐s t❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ µ(MT)✳ ■❢ d = n ❛♥❞ t❤❡ t♦r✉s ❛❝t✐♦♥ ✐s ❡✛❡❝t✐✈❡✱ t❤❡♥ (M, ω) ✐s ❛ s②♠♣❧❡❝t✐❝ t♦r✐❝ ♠❛♥✐❢♦❧❞✳ ❆ ♣♦❧②t♦♣❡ P ✐♥ Rn ✐s ❝❛❧❧❡❞ ❉❡❧③❛♥t ✐❢ ✐ts ♥♦r♠❛❧ ❢❛♥ ✐s s♠♦♦t❤✳ ❚❤❡♦r❡♠ ✭❚✳❉❡❧③❛♥t✬✽✽✮ ❚❤❡r❡ ✐s ❛ ✶✲✶ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ❝♦♠♣❛❝t s②♠♣❧❡❝t✐❝ t♦r✐❝ ♠❛♥✐❢♦❧❞s (M, ω, µ) ✭✉♣ t♦ ❡q✉✐✈❛r✐❛♥t s②♠♣❧❡❝t♦♠♦r♣❤✐s♠✮ ❛♥❞ ❉❡❧③❛♥t ♣♦❧②t♦♣❡s µ(M) ✭✉♣ t♦ ❧❛tt✐❝❡ ✐s♦♠♦r♣❤✐s♠✮✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❉❡✜♥✐t✐♦♥ ▲❡t P ❜❡ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ s✐♠♣❧❡ ♣♦❧②t♦♣❡ ♦❢ ❞✐♠❡♥s✐♦♥ n✳ ❆ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞ ♦✈❡r P ✐s ❛ s♠♦♦t❤ ✷n✲❞✐♠❡♥s✐♦♥❛❧ ♠❛♥✐❢♦❧❞ M ✇✐t❤ ❛ s♠♦♦t❤ ❛❝t✐♦♥ ♦❢ t❤❡ t♦r✉s T n s❛t✐s❢②✐♥❣ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s✿ ✭✶✮ t❤❡ ❛❝t✐♦♥ ✐s ❧♦❝❛❧❧② st❛♥❞❛r❞❀ ✭✷✮ t❤❡r❡ ✐s ❛ ❝♦♥t✐♥✉♦✉s ♣r♦❥❡❝t✐♦♥ π : M P ✇❤♦s❡ ✜❜❡rs ❛r❡ T n✲♦r❜✐ts✳ ❘❡♠❛r❦s ✭❛✮ ✐s ❤♦♠❡♦♠♦r♣❤✐❝✱ ❛s ❛ ♠❛♥✐❢♦❧❞ ✇✐t❤ ❝♦r♥❡rs✱ t♦ t❤❡ s✐♠♣❧❡ ♣♦❧②t♦♣❡ ❀ ✭❜✮ ❚❤❡ ❛❝t✐♦♥ ✐s ❢r❡❡ ♦✈❡r t❤❡ ✐♥t❡r✐♦r ♦❢ ✱ t❤❡ ✈❡rt✐❝❡s ♦❢ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ✜①❡❞ ♣♦✐♥ts ♦❢ t❤❡ t♦r✉s ❛❝t✐♦♥ ♦♥ ❀ ✭❝✮ ❆ ♣r♦❥❡❝t✐✈❡ t♦r✐❝ ♠❛♥✐❢♦❧❞ ✐s ❛ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❉❡✜♥✐t✐♦♥ ▲❡t P ❜❡ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ s✐♠♣❧❡ ♣♦❧②t♦♣❡ ♦❢ ❞✐♠❡♥s✐♦♥ n✳ ❆ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞ ♦✈❡r P ✐s ❛ s♠♦♦t❤ ✷n✲❞✐♠❡♥s✐♦♥❛❧ ♠❛♥✐❢♦❧❞ M ✇✐t❤ ❛ s♠♦♦t❤ ❛❝t✐♦♥ ♦❢ t❤❡ t♦r✉s T n s❛t✐s❢②✐♥❣ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s✿ ✭✶✮ t❤❡ ❛❝t✐♦♥ ✐s ❧♦❝❛❧❧② st❛♥❞❛r❞❀ ✭✷✮ t❤❡r❡ ✐s ❛ ❝♦♥t✐♥✉♦✉s ♣r♦❥❡❝t✐♦♥ π : M P ✇❤♦s❡ ✜❜❡rs ❛r❡ T n✲♦r❜✐ts✳ ❘❡♠❛r❦s ✭❛✮ M/T ✐s ❤♦♠❡♦♠♦r♣❤✐❝✱ ❛s ❛ ♠❛♥✐❢♦❧❞ ✇✐t❤ ❝♦r♥❡rs✱ t♦ t❤❡ s✐♠♣❧❡ ♣♦❧②t♦♣❡ P❀ ✭❜✮ ❚❤❡ ❛❝t✐♦♥ ✐s ❢r❡❡ ♦✈❡r t❤❡ ✐♥t❡r✐♦r ♦❢ P✱ t❤❡ ✈❡rt✐❝❡s ♦❢ P ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ✜①❡❞ ♣♦✐♥ts ♦❢ t❤❡ t♦r✉s ❛❝t✐♦♥ ♦♥ M❀ ✭❝✮ ❆ ♣r♦❥❡❝t✐✈❡ t♦r✐❝ ♠❛♥✐❢♦❧❞ ✐s ❛ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
■♥ t❤❡ ✇♦r❦ ♦❢ ▼✳❉❛✈✐s ❛♥❞ ❚✳❏❛♥✉s③❦✐❡✇✐❝③✬✾✶ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥str✉❝t✐♦♥ ❛♣♣❡❛r❡❞✳ ❉❡✜♥✐t✐♦♥ ❙✉♣♣♦s❡ Pn ✐s ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ s✐♠♣❧❡ ♣♦❧②t♦♣❡ ✇✐t❤ ❢❛❝❡ts F✶, . . . , Fm✳ ❉❡♥♦t❡ ❜② T Fi ❛ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♦r❞✐♥❛t❡ s✉❜❣r♦✉♣ ✐♥ T F ∼ = T m ❢♦r ❡❛❝❤ ✶ ≤ i ≤ m ❛♥❞ T G = T Fi ⊂ T F ❢♦r ❛ ❢❛❝❡ G = ∩ Fi ♦❢ ❛ ♣♦❧②t♦♣❡ Pn✳ ❚❤❡♥ t❤❡ ♠♦♠❡♥t✲❛♥❣❧❡ ♠❛♥✐❢♦❧❞ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ P ✐s ❛ q✉♦t✐❡♥t s♣❛❝❡ ZP = T F × Pn/ ∼, ✇❤❡r❡ (t✶, p) ∼ (t✷, q) ✐✛ p = q ∈ P ❛♥❞ t✶t−✶
✷
∈ T G(p)✱ G(p) ✐s ❛ ♠✐♥✐♠❛❧ ❢❛❝❡ ♦❢ P ✇❤✐❝❤ ❝♦♥t❛✐♥s p = q✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❙✐♠♣❧❡ ♣♦❧②t♦♣❡s ◆♦✇ ❝♦♥s✐❞❡r s✐♠♣❧❡ ❝♦♥✈❡① n✲❞✐♠❡♥s✐♦♥❛❧ ♣♦❧②t♦♣❡s P ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ Rn ✇✐t❤ s❝❛❧❛r ♣r♦❞✉❝t , ✳ ❙✉❝❤ ❛ ♣♦❧②t♦♣❡ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s ❛ ❜♦✉♥❞❡❞ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❤❛❧❢s♣❛❝❡s✿ ① ❛ ① ✵ ❢♦r ✶ ✇❤❡r❡ ❛ ✱ ✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❤②♣❡r♣❧❛♥❡s ❞❡✜♥❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s ❛ ① ✵ ❛r❡ ✐♥ ❣❡♥❡r❛❧ ♣♦s✐t✐♦♥✱ t❤❛t ✐s✱ ❛t ♠♦st ♦❢ t❤❡♠ ♠❡❡t ❛t ❛ s✐♥❣❧❡ ♣♦✐♥t✳ ❲❡ ❛❧s♦ ❛ss✉♠❡ t❤❛t t❤❡r❡ ❛r❡ ♥♦ r❡❞✉♥❞❛♥t ✐♥❡q✉❛❧✐t✐❡s ✐♥ ✱ t❤❛t ✐s✱ ♥♦ ✐♥❡q✉❛❧✐t② ❝❛♥ ❜❡ r❡♠♦✈❡❞ ❢r♦♠ ✇✐t❤♦✉t ❝❤❛♥❣✐♥❣ ✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❙✐♠♣❧❡ ♣♦❧②t♦♣❡s ◆♦✇ ❝♦♥s✐❞❡r s✐♠♣❧❡ ❝♦♥✈❡① n✲❞✐♠❡♥s✐♦♥❛❧ ♣♦❧②t♦♣❡s P ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ Rn ✇✐t❤ s❝❛❧❛r ♣r♦❞✉❝t , ✳ ❙✉❝❤ ❛ ♣♦❧②t♦♣❡ P ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s ❛ ❜♦✉♥❞❡❞ ✐♥t❡rs❡❝t✐♦♥ ♦❢ m ❤❛❧❢s♣❛❝❡s✿ P =
❢♦r i = ✶, . . . , m
(∗) ✇❤❡r❡ ❛i ∈ Rn✱ bi ∈ R✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❤②♣❡r♣❧❛♥❡s ❞❡✜♥❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s ❛i, ① + bi = ✵ ❛r❡ ✐♥ ❣❡♥❡r❛❧ ♣♦s✐t✐♦♥✱ t❤❛t ✐s✱ ❛t ♠♦st n ♦❢ t❤❡♠ ♠❡❡t ❛t ❛ s✐♥❣❧❡ ♣♦✐♥t✳ ❲❡ ❛❧s♦ ❛ss✉♠❡ t❤❛t t❤❡r❡ ❛r❡ ♥♦ r❡❞✉♥❞❛♥t ✐♥❡q✉❛❧✐t✐❡s ✐♥ (∗)✱ t❤❛t ✐s✱ ♥♦ ✐♥❡q✉❛❧✐t② ❝❛♥ ❜❡ r❡♠♦✈❡❞ ❢r♦♠ (∗) ✇✐t❤♦✉t ❝❤❛♥❣✐♥❣ P✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❚❤❡♥ P ❤❛s ❡①❛❝t❧② m ❢❛❝❡ts ❣✐✈❡♥ ❜② Fi =
❢♦r i = ✶, . . . , m. ▲❡t ❜❡ t❤❡ ♠❛tr✐① ♦❢ r♦✇ ✈❡❝t♦rs ❛ ✱ ❛♥❞ ❧❡t ❜ ❜❡ t❤❡ ❝♦❧✉♠♥ ✈❡❝t♦r ♦❢ s❝❛❧❛rs ✳ ❚❤❡♥ ✇❡ ❝❛♥ ✇r✐t❡ ❛s ① ① ❜ ✵ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❛✣♥❡ ♠❛♣ ① ① ❜ ■t ❡♠❜❡❞s ✐♥t♦ ② ✵ ❢♦r ✶
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❚❤❡♥ P ❤❛s ❡①❛❝t❧② m ❢❛❝❡ts ❣✐✈❡♥ ❜② Fi =
❢♦r i = ✶, . . . , m. ▲❡t AP ❜❡ t❤❡ m × n ♠❛tr✐① ♦❢ r♦✇ ✈❡❝t♦rs ❛i✱ ❛♥❞ ❧❡t ❜P ❜❡ t❤❡ ❝♦❧✉♠♥ ✈❡❝t♦r ♦❢ s❝❛❧❛rs bi ∈ R✳ ❚❤❡♥ ✇❡ ❝❛♥ ✇r✐t❡ (∗) ❛s P =
❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❛✣♥❡ ♠❛♣ iP : Rn → Rm, iP(①) = AP① + ❜P. ■t ❡♠❜❡❞s ✐♥t♦ ② ✵ ❢♦r ✶
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❚❤❡♥ P ❤❛s ❡①❛❝t❧② m ❢❛❝❡ts ❣✐✈❡♥ ❜② Fi =
❢♦r i = ✶, . . . , m. ▲❡t AP ❜❡ t❤❡ m × n ♠❛tr✐① ♦❢ r♦✇ ✈❡❝t♦rs ❛i✱ ❛♥❞ ❧❡t ❜P ❜❡ t❤❡ ❝♦❧✉♠♥ ✈❡❝t♦r ♦❢ s❝❛❧❛rs bi ∈ R✳ ❚❤❡♥ ✇❡ ❝❛♥ ✇r✐t❡ (∗) ❛s P =
❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❛✣♥❡ ♠❛♣ iP : Rn → Rm, iP(①) = AP① + ❜P. ■t ❡♠❜❡❞s P ✐♥t♦ Rm
≥ = {② ∈ Rm : yi ≥ ✵
❢♦r i = ✶, . . . , m}.
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❉❡✜♥✐t✐♦♥✿ ❱✳❇✉❝❤st❛❜❡r ❛♥❞ ❚✳P❛♥♦✈ ✭✶✾✾✽✮ ❲❡ ❞❡✜♥❡ t❤❡ s♣❛❝❡ ZP ❢r♦♠ t❤❡ ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠ ZP
iZ
− − − − → Cm
µ P
iP
− − − − → Rm
≥
✇❤❡r❡ µ(z✶, . . . , zm) = (|z✶|✷, . . . , |zm|✷)✳ ❚❤❡ ❧❛tt❡r ♠❛♣ ♠❛② ❜❡ t❤♦✉❣❤t ♦❢ ❛s t❤❡ q✉♦t✐❡♥t ♠❛♣ ❢♦r t❤❡ ❝♦♦r❞✐♥❛t❡✇✐s❡ ❛❝t✐♦♥ ♦❢ t❤❡ st❛♥❞❛r❞ t♦r✉s Tm = {③ ∈ Cm : |zi| = ✶ ❢♦r i = ✶, . . . , m} ♦♥ Cm✳ ❚❤❡r❡❢♦r❡✱ Tm ❛❝ts ♦♥ ZP ✇✐t❤ q✉♦t✐❡♥t P✱ ❛♥❞ iZ ✐s ❛ Tm✲❡q✉✐✈❛r✐❛♥t ❡♠❜❡❞❞✐♥❣✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❘❡♠❛r❦s ■❢ P✶ ❛♥❞ P✷ ❛r❡ ❝♦♠❜✐♥❛t♦r✐❛❧❧② ❡q✉✐✈❛❧❡♥t✱ ✐✳❡✳ t❤❡✐r ❢❛❝❡ ❧❛tt✐❝❡s ❛r❡ ✐s♦♠♦r♣❤✐❝✱ t❤❡♥ ZP✶ ❛♥❞ ZP✷ ❛r❡ ❤♦♠❡♦♠♦r♣❤✐❝✳ ❚❤❡ ♦♣♣♦s✐t❡ st❛t❡♠❡♥t ✐s ♥♦t tr✉❡ ✭tr✉♥❝❛t✐♦♥ ♣♦❧②t♦♣❡s✮❀ ❋♦r ❛♥② q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞
✷
♦✈❡r ❛ s✐♠♣❧❡ ♣♦❧②t♦♣❡ t❤❡r❡ ✐s ❛ ♣r✐♥❝✐♣❛❧ ✲❜✉♥❞❧❡
✷ ✱ s✳t✳ t❤❡
❝♦♠♣♦s✐t✐♦♥
✷
✐s ❛ ♣r♦❥❡❝t✐♦♥ ♦♥t♦ t❤❡ ♦r❜✐t s♣❛❝❡ ♦❢ t❤❡ ✲❛❝t✐♦♥ ♦♥ ✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❘❡♠❛r❦s ■❢ P✶ ❛♥❞ P✷ ❛r❡ ❝♦♠❜✐♥❛t♦r✐❛❧❧② ❡q✉✐✈❛❧❡♥t✱ ✐✳❡✳ t❤❡✐r ❢❛❝❡ ❧❛tt✐❝❡s ❛r❡ ✐s♦♠♦r♣❤✐❝✱ t❤❡♥ ZP✶ ❛♥❞ ZP✷ ❛r❡ ❤♦♠❡♦♠♦r♣❤✐❝✳ ❚❤❡ ♦♣♣♦s✐t❡ st❛t❡♠❡♥t ✐s ♥♦t tr✉❡ ✭tr✉♥❝❛t✐♦♥ ♣♦❧②t♦♣❡s✮❀ ❋♦r ❛♥② q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞ M✷n P ♦✈❡r ❛ s✐♠♣❧❡ ♣♦❧②t♦♣❡ P t❤❡r❡ ✐s ❛ ♣r✐♥❝✐♣❛❧ T m−n✲❜✉♥❞❧❡ ZP M✷n✱ s✳t✳ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ZP M✷n P ✐s ❛ ♣r♦❥❡❝t✐♦♥ ♦♥t♦ t❤❡ ♦r❜✐t s♣❛❝❡ ♦❢ t❤❡ T m✲❛❝t✐♦♥ ♦♥ ZP✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❊①❛♠♣❧❡s ✶✮ ■❢ P = ∆n t❤❡♥ ZP = S✷n+✶❀ ✷✮ ■❢ P = P✶ × P✷ t❤❡♥ ZP = ZP✶ × ZP✷ ❈♦♥s✐❞❡r ❛ ♣r✐s♠
✸ ✶ ✸
✸
✸ ✺ ❛♥❞ ❝✉t ❛
✈❡rt✐❝❛❧ ❡❞❣❡✳ ❲❡ ❣❡t ❛ ✸✲❝✉❜❡ ✱ ❢♦r ✇❤✐❝❤
✸ ✸ ✸✳
■❢ ✇❡ ♣❡r❢♦r♠ ❛♥ ❡❞❣❡ tr✉♥❝❛t✐♦♥ ♦❢ ✇❡ ❣❡t ❛ ✺✲❣♦♥❛❧ ♣r✐s♠
✺ ❛♥❞
✺
✸ ✹ ✺ ✸✳
❈♦♥s✐❞❡r ❛ ✸✲♣♦❧②t♦♣❡
✶
✳ ❚❤❡♥ ✐s ♥♦t ❤♦♠♦t♦♣② ❡q✉✐✈❛❧❡♥t t♦ ❛ ❝♦♥♥❡❝t❡❞ s✉♠ ♦❢ ♣r♦❞✉❝ts ♦❢ s♣❤❡r❡s✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❊①❛♠♣❧❡s ✶✮ ■❢ P = ∆n t❤❡♥ ZP = S✷n+✶❀ ✷✮ ■❢ P = P✶ × P✷ t❤❡♥ ZP = ZP✶ × ZP✷ ❈♦♥s✐❞❡r ❛ ♣r✐s♠ Pr✸ = vc✶(∆✸), ZPr✸ = S✸ × S✺ ❛♥❞ ❝✉t ❛ ✈❡rt✐❝❛❧ ❡❞❣❡✳ ❲❡ ❣❡t ❛ ✸✲❝✉❜❡ C✱ ❢♦r ✇❤✐❝❤ ZC = S✸ × S✸ × S✸✳ ■❢ ✇❡ ♣❡r❢♦r♠ ❛♥ ❡❞❣❡ tr✉♥❝❛t✐♦♥ ♦❢ ✇❡ ❣❡t ❛ ✺✲❣♦♥❛❧ ♣r✐s♠
✺ ❛♥❞
✺
✸ ✹ ✺ ✸✳
❈♦♥s✐❞❡r ❛ ✸✲♣♦❧②t♦♣❡
✶
✳ ❚❤❡♥ ✐s ♥♦t ❤♦♠♦t♦♣② ❡q✉✐✈❛❧❡♥t t♦ ❛ ❝♦♥♥❡❝t❡❞ s✉♠ ♦❢ ♣r♦❞✉❝ts ♦❢ s♣❤❡r❡s✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❊①❛♠♣❧❡s ✶✮ ■❢ P = ∆n t❤❡♥ ZP = S✷n+✶❀ ✷✮ ■❢ P = P✶ × P✷ t❤❡♥ ZP = ZP✶ × ZP✷ ❈♦♥s✐❞❡r ❛ ♣r✐s♠ Pr✸ = vc✶(∆✸), ZPr✸ = S✸ × S✺ ❛♥❞ ❝✉t ❛ ✈❡rt✐❝❛❧ ❡❞❣❡✳ ❲❡ ❣❡t ❛ ✸✲❝✉❜❡ C✱ ❢♦r ✇❤✐❝❤ ZC = S✸ × S✸ × S✸✳ ■❢ ✇❡ ♣❡r❢♦r♠ ❛♥ ❡❞❣❡ tr✉♥❝❛t✐♦♥ ♦❢ C ✇❡ ❣❡t ❛ ✺✲❣♦♥❛❧ ♣r✐s♠ Pr✺ ❛♥❞ ZPr✺ = (S✸ × S✹)#✺ × S✸✳ ❈♦♥s✐❞❡r ❛ ✸✲♣♦❧②t♦♣❡
✶
✳ ❚❤❡♥ ✐s ♥♦t ❤♦♠♦t♦♣② ❡q✉✐✈❛❧❡♥t t♦ ❛ ❝♦♥♥❡❝t❡❞ s✉♠ ♦❢ ♣r♦❞✉❝ts ♦❢ s♣❤❡r❡s✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❊①❛♠♣❧❡s ✶✮ ■❢ P = ∆n t❤❡♥ ZP = S✷n+✶❀ ✷✮ ■❢ P = P✶ × P✷ t❤❡♥ ZP = ZP✶ × ZP✷ ❈♦♥s✐❞❡r ❛ ♣r✐s♠ Pr✸ = vc✶(∆✸), ZPr✸ = S✸ × S✺ ❛♥❞ ❝✉t ❛ ✈❡rt✐❝❛❧ ❡❞❣❡✳ ❲❡ ❣❡t ❛ ✸✲❝✉❜❡ C✱ ❢♦r ✇❤✐❝❤ ZC = S✸ × S✸ × S✸✳ ■❢ ✇❡ ♣❡r❢♦r♠ ❛♥ ❡❞❣❡ tr✉♥❝❛t✐♦♥ ♦❢ C ✇❡ ❣❡t ❛ ✺✲❣♦♥❛❧ ♣r✐s♠ Pr✺ ❛♥❞ ZPr✺ = (S✸ × S✹)#✺ × S✸✳ ❈♦♥s✐❞❡r ❛ ✸✲♣♦❧②t♦♣❡ P = vc✶(C)✳ ❚❤❡♥ ZP ✐s ♥♦t ❤♦♠♦t♦♣② ❡q✉✐✈❛❧❡♥t t♦ ❛ ❝♦♥♥❡❝t❡❞ s✉♠ ♦❢ ♣r♦❞✉❝ts ♦❢ s♣❤❡r❡s✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❚❤❡ ♠♦♠❡♥t✲❛♥❣❧❡ ❢✉♥❝t♦r Z r❡♣r❡s❡♥ts t❤❡ ❤♦♠♦t♦♣② t②♣❡ ♦❢ ZP ❛♥❞ t❤❡ r✐♥❣ str✉❝t✉r❡ ♦❢ H∗(ZP; k) ❛s ✐♥✈❛r✐❛♥ts ♦❢ t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ t②♣❡ ✭❢❛❝❡ ❧❛tt✐❝❡ ❡q✉✐✈❛❧❡♥❝❡✮ ♦❢ P✳ ❍❡r❡ ✇❡ ❛r❡ ♠❛✐♥❧② ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ❋♦r♠❛❧✐t② ❛♥❞ ❤✐❣❤❡r ▼❛ss❡② ♣r♦❞✉❝ts ❢♦r ❉❡t❡r♠✐♥❡ t❤❡ ✇✐❞❡st ♣♦ss✐❜❧❡ ❝❧❛ss ♦❢ s✐♠♣❧❡ ♣♦❧②t♦♣❡s s✳t✳ t❤❡r❡ ❛r❡ ♥♦♥tr✐✈✐❛❧ ❤✐❣❤❡r ▼❛ss❡② ♦♣❡r❛t✐♦♥s ✐♥ ✱ ♦r ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ✐s ♥♦t r❛t✐♦♥❛❧❧② ❢♦r♠❛❧✳ ❋♦r♠❛❧✐t② ♠❡❛♥s✱ t❤❛t ✐ts ❙✉❧❧✐✈❛♥✲❞❡ ❘❤❛♠ ❛❧❣❡❜r❛ ♦❢ P▲✲❢♦r♠s ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ ✐s ❢♦r♠❛❧ ✐♥ ❈❉●❆✱ ✐✳❡✳✱ t❤❡r❡ ❡①✐sts ❛ ③✐❣③❛❣ ♦❢ q✉❛s✐✲✐s♦♠♦r♣❤✐s♠s ✭✇❡❛❦ ❡q✉✐✈❛❧❡♥❝❡✮ ❜❡t✇❡❡♥ ❛♥❞ ✐ts ❝♦❤♦♠♦❧♦❣② ❛❧❣❡❜r❛ ✵ ✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❚❤❡ ♠♦♠❡♥t✲❛♥❣❧❡ ❢✉♥❝t♦r Z r❡♣r❡s❡♥ts t❤❡ ❤♦♠♦t♦♣② t②♣❡ ♦❢ ZP ❛♥❞ t❤❡ r✐♥❣ str✉❝t✉r❡ ♦❢ H∗(ZP; k) ❛s ✐♥✈❛r✐❛♥ts ♦❢ t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ t②♣❡ ✭❢❛❝❡ ❧❛tt✐❝❡ ❡q✉✐✈❛❧❡♥❝❡✮ ♦❢ P✳ ❍❡r❡ ✇❡ ❛r❡ ♠❛✐♥❧② ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ❋♦r♠❛❧✐t② ❛♥❞ ❤✐❣❤❡r ▼❛ss❡② ♣r♦❞✉❝ts ❢♦r ZP ❉❡t❡r♠✐♥❡ t❤❡ ✇✐❞❡st ♣♦ss✐❜❧❡ ❝❧❛ss ♦❢ s✐♠♣❧❡ ♣♦❧②t♦♣❡s P s✳t✳ t❤❡r❡ ❛r❡ ♥♦♥tr✐✈✐❛❧ ❤✐❣❤❡r ▼❛ss❡② ♦♣❡r❛t✐♦♥s ✐♥ H∗(ZP; Q)✱ ♦r ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ZP ✐s ♥♦t r❛t✐♦♥❛❧❧② ❢♦r♠❛❧✳ ❋♦r♠❛❧✐t② ♠❡❛♥s✱ t❤❛t ✐ts ❙✉❧❧✐✈❛♥✲❞❡ ❘❤❛♠ ❛❧❣❡❜r❛ (A, d) ♦❢ P▲✲❢♦r♠s ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ Q ✐s ❢♦r♠❛❧ ✐♥ ❈❉●❆✱ ✐✳❡✳✱ t❤❡r❡ ❡①✐sts ❛ ③✐❣③❛❣ ♦❢ q✉❛s✐✲✐s♦♠♦r♣❤✐s♠s ✭✇❡❛❦ ❡q✉✐✈❛❧❡♥❝❡✮ ❜❡t✇❡❡♥ (A, d) ❛♥❞ ✐ts ❝♦❤♦♠♦❧♦❣② ❛❧❣❡❜r❛ (H∗(ZP; Q), ✵)✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❊①❛♠♣❧❡s s♣❤❡r❡s❀ ❍✲s♣❛❝❡s❀ s②♠♠❡tr✐❝ s♣❛❝❡s❀ ❝♦♠♣❛❝t ❝♦♥♥❡❝t❡❞ ▲✐❡ ❣r♦✉♣s G ❛♥❞ t❤❡✐r ❝❧❛ss✐❢②✐♥❣ s♣❛❝❡s BG❀ ❝♦♠♣❛❝t ❑☎ ❛❤❧❡r ♠❛♥✐❢♦❧❞s ✭P✳❉❡❧✐❣♥❡✱ P❤✳●r✐✣t❤s✱ ❏✳▼♦r❣❛♥✱ ❉✳❙✉❧❧✐✈❛♥✬✼✺✮ ❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ♣r♦❥❡❝t✐✈❡ t♦r✐❝ ✈❛r✐❡t✐❡s❀ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞s ✭❚✳P❛♥♦✈✱ ◆✳❘❛②✬✵✽✮✳ ▼♦r❡♦✈❡r✱ ❢♦r♠❛❧✐t② ✐s ♣r❡s❡r✈❡❞ ❜② ✇❡❞❣❡s✱ ❞✐r❡❝t ♣r♦❞✉❝ts ❛♥❞ ❝♦♥♥❡❝t❡❞ s✉♠s✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❊①❛♠♣❧❡s s♣❤❡r❡s❀ ❍✲s♣❛❝❡s❀ s②♠♠❡tr✐❝ s♣❛❝❡s❀ ❝♦♠♣❛❝t ❝♦♥♥❡❝t❡❞ ▲✐❡ ❣r♦✉♣s G ❛♥❞ t❤❡✐r ❝❧❛ss✐❢②✐♥❣ s♣❛❝❡s BG❀ ❝♦♠♣❛❝t ❑☎ ❛❤❧❡r ♠❛♥✐❢♦❧❞s ✭P✳❉❡❧✐❣♥❡✱ P❤✳●r✐✣t❤s✱ ❏✳▼♦r❣❛♥✱ ❉✳❙✉❧❧✐✈❛♥✬✼✺✮ ❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ♣r♦❥❡❝t✐✈❡ t♦r✐❝ ✈❛r✐❡t✐❡s❀ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞s ✭❚✳P❛♥♦✈✱ ◆✳❘❛②✬✵✽✮✳ ▼♦r❡♦✈❡r✱ ❢♦r♠❛❧✐t② ✐s ♣r❡s❡r✈❡❞ ❜② ✇❡❞❣❡s✱ ❞✐r❡❝t ♣r♦❞✉❝ts ❛♥❞ ❝♦♥♥❡❝t❡❞ s✉♠s✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❊①❛♠♣❧❡s s♣❤❡r❡s❀ ❍✲s♣❛❝❡s❀ s②♠♠❡tr✐❝ s♣❛❝❡s❀ ❝♦♠♣❛❝t ❝♦♥♥❡❝t❡❞ ▲✐❡ ❣r♦✉♣s G ❛♥❞ t❤❡✐r ❝❧❛ss✐❢②✐♥❣ s♣❛❝❡s BG❀ ❝♦♠♣❛❝t ❑☎ ❛❤❧❡r ♠❛♥✐❢♦❧❞s ✭P✳❉❡❧✐❣♥❡✱ P❤✳●r✐✣t❤s✱ ❏✳▼♦r❣❛♥✱ ❉✳❙✉❧❧✐✈❛♥✬✼✺✮ ❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ♣r♦❥❡❝t✐✈❡ t♦r✐❝ ✈❛r✐❡t✐❡s❀ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞s ✭❚✳P❛♥♦✈✱ ◆✳❘❛②✬✵✽✮✳ ▼♦r❡♦✈❡r✱ ❢♦r♠❛❧✐t② ✐s ♣r❡s❡r✈❡❞ ❜② ✇❡❞❣❡s✱ ❞✐r❡❝t ♣r♦❞✉❝ts ❛♥❞ ❝♦♥♥❡❝t❡❞ s✉♠s✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❊①❛♠♣❧❡s s♣❤❡r❡s❀ ❍✲s♣❛❝❡s❀ s②♠♠❡tr✐❝ s♣❛❝❡s❀ ❝♦♠♣❛❝t ❝♦♥♥❡❝t❡❞ ▲✐❡ ❣r♦✉♣s G ❛♥❞ t❤❡✐r ❝❧❛ss✐❢②✐♥❣ s♣❛❝❡s BG❀ ❝♦♠♣❛❝t ❑☎ ❛❤❧❡r ♠❛♥✐❢♦❧❞s ✭P✳❉❡❧✐❣♥❡✱ P❤✳●r✐✣t❤s✱ ❏✳▼♦r❣❛♥✱ ❉✳❙✉❧❧✐✈❛♥✬✼✺✮ ❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ♣r♦❥❡❝t✐✈❡ t♦r✐❝ ✈❛r✐❡t✐❡s❀ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞s ✭❚✳P❛♥♦✈✱ ◆✳❘❛②✬✵✽✮✳ ▼♦r❡♦✈❡r✱ ❢♦r♠❛❧✐t② ✐s ♣r❡s❡r✈❡❞ ❜② ✇❡❞❣❡s✱ ❞✐r❡❝t ♣r♦❞✉❝ts ❛♥❞ ❝♦♥♥❡❝t❡❞ s✉♠s✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
▲❡t k ❜❡ ❛ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣ ✇✐t❤ ❛ ✉♥✐t ❛♥❞ ❝♦♥s✐❞❡r ❛ (n − ✶)✲❞✐♠❡♥s✐♦♥❛❧ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① K ♦♥ t❤❡ ♦r❞❡r❡❞ s❡t [m] = {✶, . . . , m}✳ ▲❡t k[m] = k[v✶, . . . , vm] ❜❡ t❤❡ ❣r❛❞❡❞ ♣♦❧②♥♦♠✐❛❧ ❛❧❣❡❜r❛ ♦♥ m ✈❛r✐❛❜❧❡s✱ ❞❡❣(vi) = ✷✳ ❋❛❝❡ r✐♥❣s ❆ ❢❛❝❡ r✐♥❣ ✭♦r ❛ ❙t❛♥❧❡②✲❘❡✐s♥❡r r✐♥❣✮ ♦❢ ✐s t❤❡ q✉♦t✐❡♥t r✐♥❣
✶
✇❤❡r❡ ✐s t❤❡ ✐❞❡❛❧ ❣❡♥❡r❛t❡❞ ❜② t❤♦s❡ sq✉❛r❡ ❢r❡❡ ♠♦♥♦♠✐❛❧s
✶
❢♦r ✇❤✐❝❤
✶
✐s ♥♦t ❛ s✐♠♣❧❡① ♦❢ ✳ ❲❡ ❞❡♥♦t❡ ✳ ◆♦t❡ t❤❛t ✐s ❛ ♠♦❞✉❧❡ ♦✈❡r
✶
✈✐❛ t❤❡ q✉♦t✐❡♥t ♣r♦❥❡❝t✐♦♥✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
▲❡t k ❜❡ ❛ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣ ✇✐t❤ ❛ ✉♥✐t ❛♥❞ ❝♦♥s✐❞❡r ❛ (n − ✶)✲❞✐♠❡♥s✐♦♥❛❧ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① K ♦♥ t❤❡ ♦r❞❡r❡❞ s❡t [m] = {✶, . . . , m}✳ ▲❡t k[m] = k[v✶, . . . , vm] ❜❡ t❤❡ ❣r❛❞❡❞ ♣♦❧②♥♦♠✐❛❧ ❛❧❣❡❜r❛ ♦♥ m ✈❛r✐❛❜❧❡s✱ ❞❡❣(vi) = ✷✳ ❋❛❝❡ r✐♥❣s ❆ ❢❛❝❡ r✐♥❣ ✭♦r ❛ ❙t❛♥❧❡②✲❘❡✐s♥❡r r✐♥❣✮ ♦❢ K ✐s t❤❡ q✉♦t✐❡♥t r✐♥❣ k[K] := k[v✶, . . . , vm]/IK ✇❤❡r❡ IK ✐s t❤❡ ✐❞❡❛❧ ❣❡♥❡r❛t❡❞ ❜② t❤♦s❡ sq✉❛r❡ ❢r❡❡ ♠♦♥♦♠✐❛❧s vi✶ · · · vis ❢♦r ✇❤✐❝❤ {i✶, . . . , is} ✐s ♥♦t ❛ s✐♠♣❧❡① ♦❢ K✳ ❲❡ ❞❡♥♦t❡ k[P] = k[∂P∗]✳ ◆♦t❡ t❤❛t k[K] ✐s ❛ ♠♦❞✉❧❡ ♦✈❡r k[v✶, . . . , vm] ✈✐❛ t❤❡ q✉♦t✐❡♥t ♣r♦❥❡❝t✐♦♥✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t r❡❧❛t❡s ❝♦❤♦♠♦❧♦❣② ♦❢ ZP t♦ ❝♦♠❜✐♥❛t♦r✐❝s ♦❢ t❤❡ ♣♦❧②t♦♣❡ P✿ ❚❤❡♦r❡♠ ✭❱✳❇✉❝❤st❛❜❡r✱ ❚✳P❛♥♦✈✬✾✽✮ ■❢ ✇❡ ❞❡✜♥❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❣r❛❞❡❞ ❛❧❣❡❜r❛ R(P) = Λ[u✶, . . . , um] ⊗ k[P]/(v✷
i = uivi = ✵, ✶ ≤ i ≤ m) ✇✐t❤
❜✐❞❡❣ ui = (−✶, ✷), ❜✐❞❡❣ vi = (✵, ✷); dui = vi, dvi = ✵✱ t❤❡♥✿ H∗,∗(ZP; k) ∼ = H∗,∗[R(P), d] ∼ = ❚♦r∗,∗
k[v✶,...,vm](k[P], k).
❚❤❡s❡ ❛❧❣❡❜r❛s ❛❞♠✐t ✲♠✉❧t✐❣r❛❞✐♥❣ ❛♥❞ ✇❡ ❤❛✈❡ ❚♦r
✷❛
✶
✷❛
✇❤❡r❡ ❚♦r
✷
✶
✶
❢♦r ✳ ❍❡r❡ ✇❡ ❞❡♥♦t❡ ✳ ❚❤❡ ♠✉❧t✐❣r❛❞❡❞ ❝♦♠♣♦♥❡♥t ❚♦r
✷❛
✶
✵✱ ✐❢ ❛ ✐s ♥♦t ❛ ✵ ✶ ✲✈❡❝t♦r ♦❢ ❧❡♥❣t❤ ✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t r❡❧❛t❡s ❝♦❤♦♠♦❧♦❣② ♦❢ ZP t♦ ❝♦♠❜✐♥❛t♦r✐❝s ♦❢ t❤❡ ♣♦❧②t♦♣❡ P✿ ❚❤❡♦r❡♠ ✭❱✳❇✉❝❤st❛❜❡r✱ ❚✳P❛♥♦✈✬✾✽✮ ■❢ ✇❡ ❞❡✜♥❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❣r❛❞❡❞ ❛❧❣❡❜r❛ R(P) = Λ[u✶, . . . , um] ⊗ k[P]/(v✷
i = uivi = ✵, ✶ ≤ i ≤ m) ✇✐t❤
❜✐❞❡❣ ui = (−✶, ✷), ❜✐❞❡❣ vi = (✵, ✷); dui = vi, dvi = ✵✱ t❤❡♥✿ H∗,∗(ZP; k) ∼ = H∗,∗[R(P), d] ∼ = ❚♦r∗,∗
k[v✶,...,vm](k[P], k).
❚❤❡s❡ ❛❧❣❡❜r❛s ❛❞♠✐t N ⊕ Zm✲♠✉❧t✐❣r❛❞✐♥❣ ❛♥❞ ✇❡ ❤❛✈❡ ❚♦r−i,✷❛
k[v✶,...,vm](k[P], k) ∼
= H−i,✷❛(R(P), d), ✇❤❡r❡ ❚♦r−i,✷J
k[v✶,...,vm](k[P], k) ∼
= H|J|−i−✶(PJ; k) ❢♦r J ⊂ [m]✳ ❍❡r❡ ✇❡ ❞❡♥♦t❡ PJ = ∪j∈J Fj✳ ❚❤❡ ♠✉❧t✐❣r❛❞❡❞ ❝♦♠♣♦♥❡♥t ❚♦r−i,✷❛
k[v✶,...,vm](k[P], k) = ✵✱ ✐❢ ❛ ✐s ♥♦t ❛ (✵, ✶)✲✈❡❝t♦r ♦❢ ❧❡♥❣t❤ m✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❲❡ ♥♦✇ t✉r♥ t♦ ❛ ❞✐s❝✉ss✐♦♥ ♦❢ ✢❛❣ ♥❡st♦❤❡❞r❛✳ ❇✉✐❧❞✐♥❣ s❡t ▲❡t S = {✶, ✷, . . . , n + ✶}✱ n ≥ ✷✳ ❆ ❜✉✐❧❞✐♥❣ s❡t ♦♥ ❙ ✐s ❛ ❢❛♠✐❧② ♦❢ s✉❜s❡ts B = {Bk ⊆ S}✱ s✉❝❤ t❤❛t✿ ✶✮ {i} ∈ B ❢♦r ❛❧❧ ✶ ≤ i ≤ n + ✶❀ ✷✮ ✐❢ Bi ∩ Bj = ∅✱ t❤❡♥ Bi ∪ Bj ∈ B✳ ◆❡st♦❤❡❞r❛ ◆❡st♦❤❡❞r♦♥ ✐s ❛ s✐♠♣❧❡ ❝♦♥✈❡① ✲❞✐♠❡♥s✐♦♥❛❧ ♣♦❧②t♦♣❡ ✱ ✇❤❡r❡ ❝♦♥✈
✶✳
❊①❛♠♣❧❡✿ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r❛ ❆ ❣r❛♣❤✐❝❛❧ ❜✉✐❧❞✐♥❣ s❡t ❢♦r ❛ ❣r❛♣❤ ♦♥ t❤❡ ✈❡rt❡① s❡t ❝♦♥s✐sts ♦❢ s✉❝❤ t❤❛t ✐s ❛ ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤ ♦❢ ✳ ❚❤❡♥ ✐s ❝❛❧❧❡❞ ❛ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r♦♥✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❲❡ ♥♦✇ t✉r♥ t♦ ❛ ❞✐s❝✉ss✐♦♥ ♦❢ ✢❛❣ ♥❡st♦❤❡❞r❛✳ ❇✉✐❧❞✐♥❣ s❡t ▲❡t S = {✶, ✷, . . . , n + ✶}✱ n ≥ ✷✳ ❆ ❜✉✐❧❞✐♥❣ s❡t ♦♥ ❙ ✐s ❛ ❢❛♠✐❧② ♦❢ s✉❜s❡ts B = {Bk ⊆ S}✱ s✉❝❤ t❤❛t✿ ✶✮ {i} ∈ B ❢♦r ❛❧❧ ✶ ≤ i ≤ n + ✶❀ ✷✮ ✐❢ Bi ∩ Bj = ∅✱ t❤❡♥ Bi ∪ Bj ∈ B✳ ◆❡st♦❤❡❞r❛ ◆❡st♦❤❡❞r♦♥ ✐s ❛ s✐♠♣❧❡ ❝♦♥✈❡① n✲❞✐♠❡♥s✐♦♥❛❧ ♣♦❧②t♦♣❡ PB =
Bk∈B
∆Bk✱ ✇❤❡r❡ ∆Bk = ❝♦♥✈{ej| j ∈ Bk} ⊂ Rn+✶✳ ❊①❛♠♣❧❡✿ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r❛ ❆ ❣r❛♣❤✐❝❛❧ ❜✉✐❧❞✐♥❣ s❡t ❢♦r ❛ ❣r❛♣❤ ♦♥ t❤❡ ✈❡rt❡① s❡t ❝♦♥s✐sts ♦❢ s✉❝❤ t❤❛t ✐s ❛ ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤ ♦❢ ✳ ❚❤❡♥ ✐s ❝❛❧❧❡❞ ❛ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r♦♥✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❲❡ ♥♦✇ t✉r♥ t♦ ❛ ❞✐s❝✉ss✐♦♥ ♦❢ ✢❛❣ ♥❡st♦❤❡❞r❛✳ ❇✉✐❧❞✐♥❣ s❡t ▲❡t S = {✶, ✷, . . . , n + ✶}✱ n ≥ ✷✳ ❆ ❜✉✐❧❞✐♥❣ s❡t ♦♥ ❙ ✐s ❛ ❢❛♠✐❧② ♦❢ s✉❜s❡ts B = {Bk ⊆ S}✱ s✉❝❤ t❤❛t✿ ✶✮ {i} ∈ B ❢♦r ❛❧❧ ✶ ≤ i ≤ n + ✶❀ ✷✮ ✐❢ Bi ∩ Bj = ∅✱ t❤❡♥ Bi ∪ Bj ∈ B✳ ◆❡st♦❤❡❞r❛ ◆❡st♦❤❡❞r♦♥ ✐s ❛ s✐♠♣❧❡ ❝♦♥✈❡① n✲❞✐♠❡♥s✐♦♥❛❧ ♣♦❧②t♦♣❡ PB =
Bk∈B
∆Bk✱ ✇❤❡r❡ ∆Bk = ❝♦♥✈{ej| j ∈ Bk} ⊂ Rn+✶✳ ❊①❛♠♣❧❡✿ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r❛ ❆ ❣r❛♣❤✐❝❛❧ ❜✉✐❧❞✐♥❣ s❡t B(Γ) ❢♦r ❛ ❣r❛♣❤ Γ ♦♥ t❤❡ ✈❡rt❡① s❡t S ❝♦♥s✐sts ♦❢ s✉❝❤ Bk t❤❛t ΓBk ✐s ❛ ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤ ♦❢ Γ✳ ❚❤❡♥ PΓ = PB(Γ) ✐s ❝❛❧❧❡❞ ❛ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r♦♥✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❙✳❉❡✈❛❞♦ss ✭✷✵✵✻✮ ✐♥ t❤❡✐r st✉❞② ♦❢ ❈♦①❡t❡r ❝♦♠♣❧❡①❡s✳ ❊①❛♠♣❧❡s Γ ✐s ❛ ❝♦♠♣❧❡t❡ ❣r❛♣❤ ♦♥ [n + ✶]✳ ❚❤❡♥ PΓ = Pen ✐s ❛ ♣❡r♠✉t♦❤❡❞r♦♥✳ Γ ✐s ❛ st❡❧❧❛r ❣r❛♣❤ ♦♥ [n + ✶]✳ ❚❤❡♥ PΓ = Stn ✐s ❛ st❡❧❧❛❤❡❞r♦♥✳ Γ ✐s ❛ ❝②❝❧❡ ❣r❛♣❤ ♦♥ [n + ✶]✳ ❚❤❡♥ PΓ = Cyn ✐s ❛ ❝②❝❧♦❤❡❞r♦♥ ✭♦r ❇♦tt✲❚❛✉❜❡s ♣♦❧②t♦♣❡✮✳ Γ ✐s ❛ ❝❤❛✐♥ ❣r❛♣❤ ♦♥ [n + ✶]✳ ❚❤❡♥ PΓ = Asn ✐s ❛♥ ❛ss♦❝✐❛❤❡❞r♦♥ ✭♦r ❙t❛s❤❡✛ ♣♦❧②t♦♣❡✮✳
❤❛s ❛♥ ❡♠♣t② ✐♥t❡rs❡❝t✐♦♥ t❤❡♥ s♦♠❡ ♣❛✐r ♦❢ t❤❡s❡ ❢❛❝❡ts ❤❛s ❛♥ ❡♠♣t② ✐♥t❡rs❡❝t✐♦♥✳ ▼♦r❡♦✈❡r✱ t❤❡② ❛r❡ ❉❡❧③❛♥t ♣♦❧②t♦♣❡s ✭❛❧❧ ♥❡st♦❤❡❞r❛ ❛r❡✱ ❞✉❡ t♦ t❤❡ r❡s✉❧t ♦❢ ❆✳❩❡❧❡✈✐♥s❦②✮✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❙✳❉❡✈❛❞♦ss ✭✷✵✵✻✮ ✐♥ t❤❡✐r st✉❞② ♦❢ ❈♦①❡t❡r ❝♦♠♣❧❡①❡s✳ ❊①❛♠♣❧❡s Γ ✐s ❛ ❝♦♠♣❧❡t❡ ❣r❛♣❤ ♦♥ [n + ✶]✳ ❚❤❡♥ PΓ = Pen ✐s ❛ ♣❡r♠✉t♦❤❡❞r♦♥✳ Γ ✐s ❛ st❡❧❧❛r ❣r❛♣❤ ♦♥ [n + ✶]✳ ❚❤❡♥ PΓ = Stn ✐s ❛ st❡❧❧❛❤❡❞r♦♥✳ Γ ✐s ❛ ❝②❝❧❡ ❣r❛♣❤ ♦♥ [n + ✶]✳ ❚❤❡♥ PΓ = Cyn ✐s ❛ ❝②❝❧♦❤❡❞r♦♥ ✭♦r ❇♦tt✲❚❛✉❜❡s ♣♦❧②t♦♣❡✮✳ Γ ✐s ❛ ❝❤❛✐♥ ❣r❛♣❤ ♦♥ [n + ✶]✳ ❚❤❡♥ PΓ = Asn ✐s ❛♥ ❛ss♦❝✐❛❤❡❞r♦♥ ✭♦r ❙t❛s❤❡✛ ♣♦❧②t♦♣❡✮✳
❤❛s ❛♥ ❡♠♣t② ✐♥t❡rs❡❝t✐♦♥ t❤❡♥ s♦♠❡ ♣❛✐r ♦❢ t❤❡s❡ ❢❛❝❡ts ❤❛s ❛♥ ❡♠♣t② ✐♥t❡rs❡❝t✐♦♥✳ ▼♦r❡♦✈❡r✱ t❤❡② ❛r❡ ❉❡❧③❛♥t ♣♦❧②t♦♣❡s ✭❛❧❧ ♥❡st♦❤❡❞r❛ ❛r❡✱ ❞✉❡ t♦ t❤❡ r❡s✉❧t ♦❢ ❆✳❩❡❧❡✈✐♥s❦②✮✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
■♥ ♦r❞❡r t♦ ✇♦r❦ ✇✐t❤ ❝♦♠❜✐♥❛t♦r✐❛❧ t②♣❡s ♦❢ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r❛ ✇❡ s❤♦✉❧❞ ❞❡s❝r✐❜❡ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡✐r ❢❛❝❡ ❧❛tt✐❝❡s✳ ❋❛❝❡ ♣♦s❡t ❋❛❝❡ts ♦❢ PΓ ❛r❡ ✐♥ ✶✲✶ ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇✐t❤ ♥♦♥ ♠❛①✐♠❛❧ ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤s ♦❢ Γ✳ ▼♦r❡♦✈❡r✱ ❛ s❡t ♦❢ ❢❛❝❡ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ s✉❝❤ s✉❜❣r❛♣❤s Γi✶, . . . , Γis ❤❛s ❛ ♥♦♥❡♠♣t② ✐♥t❡rs❡❝t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢✿ ✭✶✮ ❋♦r ❛♥② t✇♦ s✉❜❣r❛♣❤s Γik, Γil✱ ❡✐t❤❡r t❤❡② ❞♦ ♥♦t ❤❛✈❡ ❛ ❝♦♠♠♦♥ ✈❡rt❡① ♦r ♦♥❡ ✐s ❛ s✉❜❣r❛♣❤ ♦❢ ❛♥♦t❤❡r❀ ✭✷✮ ■❢ ❛♥② t✇♦ ♦❢ t❤❡ s✉❜❣r❛♣❤s Γik✶, . . . , Γikl , l ✷ ❞♦ ♥♦t ❤❛✈❡ ❝♦♠♠♦♥ ✈❡rt✐❝❡s✱ t❤❡♥ t❤❡✐r ✉♥✐♦♥ ❣r❛♣❤ ✐s ❞✐s❝♦♥♥❡❝t❡❞✳ ■❢ t❤❡♥ ✐ts ❢❛❝❡ts
✶ ✷
✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣
✶ ❛♥❞ ✷ ❛r❡ s✉❜❣r❛♣❤s ♦❢ ♦♥❡ ❛♥♦t❤❡r❀
■❢ ✶ ❛r❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡♥
✶
✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
■♥ ♦r❞❡r t♦ ✇♦r❦ ✇✐t❤ ❝♦♠❜✐♥❛t♦r✐❛❧ t②♣❡s ♦❢ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r❛ ✇❡ s❤♦✉❧❞ ❞❡s❝r✐❜❡ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡✐r ❢❛❝❡ ❧❛tt✐❝❡s✳ ❋❛❝❡ ♣♦s❡t ❋❛❝❡ts ♦❢ PΓ ❛r❡ ✐♥ ✶✲✶ ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇✐t❤ ♥♦♥ ♠❛①✐♠❛❧ ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤s ♦❢ Γ✳ ▼♦r❡♦✈❡r✱ ❛ s❡t ♦❢ ❢❛❝❡ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ s✉❝❤ s✉❜❣r❛♣❤s Γi✶, . . . , Γis ❤❛s ❛ ♥♦♥❡♠♣t② ✐♥t❡rs❡❝t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢✿ ✭✶✮ ❋♦r ❛♥② t✇♦ s✉❜❣r❛♣❤s Γik, Γil✱ ❡✐t❤❡r t❤❡② ❞♦ ♥♦t ❤❛✈❡ ❛ ❝♦♠♠♦♥ ✈❡rt❡① ♦r ♦♥❡ ✐s ❛ s✉❜❣r❛♣❤ ♦❢ ❛♥♦t❤❡r❀ ✭✷✮ ■❢ ❛♥② t✇♦ ♦❢ t❤❡ s✉❜❣r❛♣❤s Γik✶, . . . , Γikl , l ✷ ❞♦ ♥♦t ❤❛✈❡ ❝♦♠♠♦♥ ✈❡rt✐❝❡s✱ t❤❡♥ t❤❡✐r ✉♥✐♦♥ ❣r❛♣❤ ✐s ❞✐s❝♦♥♥❡❝t❡❞✳ ■❢ P = Pen t❤❡♥ ✐ts ❢❛❝❡ts F✶ ∩ F✷ = ∅ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ Γ✶ ❛♥❞ Γ✷ ❛r❡ s✉❜❣r❛♣❤s ♦❢ ♦♥❡ ❛♥♦t❤❡r❀ ■❢ Γi, ✶ ≤ i ≤ r ❛r❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ♦❢ Γ t❤❡♥ PΓ = PΓ✶ × . . . × PΓr ✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❉❡✜♥✐t✐♦♥✿ s♣❡❝✐❛❧ s✉❜❣r❛♣❤s ✐♥ ❛ ❣r❛♣❤ ✶✮ ❙✉♣♣♦s❡ Γ ✐s ❛ ❣r❛♣❤✳ ❋♦r ❛♥② ♦❢ ✐ts ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤s γ ♦♥❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ ♥✉♠❜❡r i(γ) ♦❢ s✉❝❤ ❝♦♥♥❡❝t❡❞ s✉❜r❛♣❤s ˜ γ ✐♥ Γ t❤❛t ❡✐t❤❡r γ ∩ ˜ γ = ∅, γ, ˜ γ ♦r γ ∩ ˜ γ = ∅, ❛♥❞ γ ⊔ ˜ γ ✐s ❛ ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤ ✐♥ Γ✳ ✷✮ ❲❡ ❞❡♥♦t❡ ❜② t❤❡ ♠❛①✐♠❛❧ ✈❛❧✉❡ ♦❢ ♦✈❡r ❛❧❧ ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤s ✐♥ ✳ ❆ ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤ ✱ ♦♥ ✇❤✐❝❤ ✐s ❛❝❤✐❡✈❡❞✱ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ❛ s♣❡❝✐❛❧ s✉❜❣r❛♣❤✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❉❡✜♥✐t✐♦♥✿ s♣❡❝✐❛❧ s✉❜❣r❛♣❤s ✐♥ ❛ ❣r❛♣❤ ✶✮ ❙✉♣♣♦s❡ Γ ✐s ❛ ❣r❛♣❤✳ ❋♦r ❛♥② ♦❢ ✐ts ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤s γ ♦♥❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ ♥✉♠❜❡r i(γ) ♦❢ s✉❝❤ ❝♦♥♥❡❝t❡❞ s✉❜r❛♣❤s ˜ γ ✐♥ Γ t❤❛t ❡✐t❤❡r γ ∩ ˜ γ = ∅, γ, ˜ γ ♦r γ ∩ ˜ γ = ∅, ❛♥❞ γ ⊔ ˜ γ ✐s ❛ ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤ ✐♥ Γ✳ ✷✮ ❲❡ ❞❡♥♦t❡ ❜② imax = imax(Γ) t❤❡ ♠❛①✐♠❛❧ ✈❛❧✉❡ ♦❢ i(γ) ♦✈❡r ❛❧❧ ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤s γ ✐♥ Γ✳ ❆ ❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤ γ✱ ♦♥ ✇❤✐❝❤ imax ✐s ❛❝❤✐❡✈❡❞✱ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ❛ s♣❡❝✐❛❧ s✉❜❣r❛♣❤✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❯s✐♥❣ t❤❡ ❢❛❝❡ ♣♦s❡t str✉❝t✉r❡ ♦❢ PΓ ✇❡ ❣❡t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✿ ❚❤❡♦r❡♠ ▲❡t P = PΓ ❜❡ ❛ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r♦♥ ♦❢ ❞✐♠❡♥s✐♦♥ n ≥ ✸ ❢♦r ❛ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ Γ✳ ❚❤❡♥ ❢♦r i > imax✿ β−i,✷(i+✶)(P) = ✵. ❉❡♥♦t❡ t❤❡ ♥✉♠❜❡r ♦❢ s♣❡❝✐❛❧ s✉❜❣r❛♣❤s ✐♥ Γ ❜② s✳ ❚❤❡♥ β−imax,✷(imax+✶)(P) = s.
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❋♦r t❤❡ ✹ ❝❧❛ss✐❝❛❧ s❡r✐❡s ♦❢ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r❛ t❤❡ t❤❡♦r❡♠ ❣✐✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛❧✉❡s ♦❢ imax ❛♥❞ s✳ ❆ss♦❝✐❛❤❡❞r♦♥ β−q,✷(q+✶)(P) =
✐❢ n ✐s ❡✈❡♥❀
n+✸ ✷ ,
✐❢ n ✐s ♦❞❞❀ β−i,✷(i+✶)(P) = ✵ ❢♦r i ≥ q + ✶, ✇❤❡r❡ q = q(n) ✐s✿ q = q(n) = n(n+✷)
✹
, ✐❢ n ✐s ❡✈❡♥❀
(n+✶)✷ ✹
, ✐❢ n ✐s ♦❞❞✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❈②❝❧♦❤❡❞r♦♥ β−q,✷(q+✶)(P) =
✐❢ n ✐s ❡✈❡♥❀ n + ✶, ✐❢ n ✐s ♦❞❞❀ β−i,✷(i+✶)(P) = ✵ ❢♦r i ≥ q + ✶, ✇❤❡r❡ q = q(n) ✐s✿ q = q(n) = n(n+✷)−✷
✷
, ✐❢ n ✐s ❡✈❡♥❀
(n+✶)✷−✷ ✷
, ✐❢ n ✐s ♦❞❞✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
P❡r♠✉t♦❤❡❞r♦♥ β−q,✷(q+✶)(P) = n + ✶ [ n+✶
✷ ]
❢♦r i ≥ q + ✶, ✇❤❡r❡ q = q(n) = ✷n+✶ − ✷[ n+✶
✷ ] − ✷[ n+✷ ✷ ] + ✶
❙t❡❧❧❛❤❡❞r♦♥ β−q,✷(q+✶)(P) = n [ n
✷]
❢♦r i ≥ q + ✶, ✇❤❡r❡ q = q(n) = ✷n − ✷[ n
✷] − ✷[ n+✶ ✷ ] + [ n+✸
✷ ]✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❇✐❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs ♦❢ t❤❡ t②♣❡ β−i,✷(i+✶)(P) ❤❛✈❡ ❛♥♦t❤❡r t♦♣♦❧♦❣✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥ ❜② ♠❡❛♥s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✳ ▲♦♦♣ ❤♦♠♦❧♦❣② ♦❢ ♠♦♠❡♥t✲❛♥❣❧❡✲♠❛♥✐❢♦❧❞s ❏✳●r❜✐✁ ❝✱ ❚✳P❛♥♦✈✱ ❙✳❚❤❡r✐❛✉❧t✱ ❏✳❲✉ ✭✷✵✶✷✮ ♣r♦✈❡❞ t❤❛t
m−n
β−i,✷(i+✶)(P) ❡q✉❛❧s t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ♦❢ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❣❡♥❡r❛t♦rs ♦❢ t❤❡ P♦♥tr②❛❣✐♥ ❛❧❣❡❜r❛ H∗(ΩZP; k) ❢♦r ❛♥② ✢❛❣ s✐♠♣❧❡ ♣♦❧②t♦♣❡ P✳ ❘❡♠❛r❦✿ t♦rs✐♦♥ ✐♥ ❝♦❤♦♠♦❧♦❣② ❈♦♥s✐❞❡r t❤❡ ♣r✐♥❝✐♣❛❧ ✲❜✉♥❞❧❡ ❢♦r ✳ ❋♦r ❛♥❞ ✺ ♠❛② ❤❛✈❡ ❛♥ ❛r❜✐tr❛r② ✜♥✐t❡ ❣r♦✉♣ ❛s ❛ ❞✐r❡❝t s✉♠♠❛♥❞✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❞✉❡ t♦ ❉❛♥✐❧♦✈✲❏✉r❦✐❡✇✐❝③ t❤❡♦r❡♠✱ ✐s ❛❧✇❛②s ❢r❡❡✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❇✐❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs ♦❢ t❤❡ t②♣❡ β−i,✷(i+✶)(P) ❤❛✈❡ ❛♥♦t❤❡r t♦♣♦❧♦❣✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥ ❜② ♠❡❛♥s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✳ ▲♦♦♣ ❤♦♠♦❧♦❣② ♦❢ ♠♦♠❡♥t✲❛♥❣❧❡✲♠❛♥✐❢♦❧❞s ❏✳●r❜✐✁ ❝✱ ❚✳P❛♥♦✈✱ ❙✳❚❤❡r✐❛✉❧t✱ ❏✳❲✉ ✭✷✵✶✷✮ ♣r♦✈❡❞ t❤❛t
m−n
β−i,✷(i+✶)(P) ❡q✉❛❧s t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ♦❢ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❣❡♥❡r❛t♦rs ♦❢ t❤❡ P♦♥tr②❛❣✐♥ ❛❧❣❡❜r❛ H∗(ΩZP; k) ❢♦r ❛♥② ✢❛❣ s✐♠♣❧❡ ♣♦❧②t♦♣❡ P✳ ❘❡♠❛r❦✿ t♦rs✐♦♥ ✐♥ ❝♦❤♦♠♦❧♦❣② ❈♦♥s✐❞❡r t❤❡ ♣r✐♥❝✐♣❛❧ Tm−n✲❜✉♥❞❧❡ ZP MP ❢♦r Pn = PΓ✳ ❋♦r P = Pen ❛♥❞ n ≥ ✺ H∗(ZP) ♠❛② ❤❛✈❡ ❛♥ ❛r❜✐tr❛r② ✜♥✐t❡ ❣r♦✉♣ ❛s ❛ ❞✐r❡❝t s✉♠♠❛♥❞✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❞✉❡ t♦ ❉❛♥✐❧♦✈✲❏✉r❦✐❡✇✐❝③ t❤❡♦r❡♠✱ H∗(MP) ✐s ❛❧✇❛②s ❢r❡❡✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❉❡✜♥✐♥❣ s②st❡♠ ❙✉♣♣♦s❡ (A, d) ✐s ❛ ❞❣❛✱ αi = [ai] ∈ H∗[A, d] ❛♥❞ ai ∈ Ani ❢♦r ✶ ≤ i ≤ k✳ ❚❤❡♥ ❛ ❞❡✜♥✐♥❣ s②st❡♠ ❢♦r (α✶, . . . , αk) ✐s ❛ (k + ✶) × (k + ✶)✲♠❛tr✐① C✱ s✳t✳ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❤♦❧❞✿ ✭✶✮ ci,j = ✵✱ ✐❢ i ≥ j✱ ✭✷✮ ci,i+✶ = ai✱ ✭✸✮ a · E✶,k+✶ = dC − ¯ C · C ❢♦r s♦♠❡ a = a(C) ∈ A✱ ✇❤❡r❡ ¯ ci,j = (−✶)degci,j · ci,j✳ ❚❤✐s ✐♠♣❧✐❡s✿ d(a) = ✵ ❛♥❞ a ∈ Am✱ m = n✶ + . . . + nk − k + ✷✳ ❉❡✜♥✐t✐♦♥ ❆ ▼❛ss❡② ✲♣r♦❞✉❝t
✶
✐s s❛✐❞ t♦ ❜❡ ❞❡✜♥❡❞✱ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❞❡✜♥✐♥❣ s②st❡♠ ❢♦r ✐t✳ ■❢ s♦✱ t❤✐s ▼❛ss❡② ♣r♦❞✉❝t ❝♦♥s✐sts ♦❢ ❛❧❧ ❢♦r ❡❛❝❤ ❞❡✜♥✐♥❣ s②st❡♠ ✳ ■t ✐s ❝❛❧❧❡❞ tr✐✈✐❛❧✱ ✐❢ ✵ ❢♦r s♦♠❡ ✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❉❡✜♥✐♥❣ s②st❡♠ ❙✉♣♣♦s❡ (A, d) ✐s ❛ ❞❣❛✱ αi = [ai] ∈ H∗[A, d] ❛♥❞ ai ∈ Ani ❢♦r ✶ ≤ i ≤ k✳ ❚❤❡♥ ❛ ❞❡✜♥✐♥❣ s②st❡♠ ❢♦r (α✶, . . . , αk) ✐s ❛ (k + ✶) × (k + ✶)✲♠❛tr✐① C✱ s✳t✳ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❤♦❧❞✿ ✭✶✮ ci,j = ✵✱ ✐❢ i ≥ j✱ ✭✷✮ ci,i+✶ = ai✱ ✭✸✮ a · E✶,k+✶ = dC − ¯ C · C ❢♦r s♦♠❡ a = a(C) ∈ A✱ ✇❤❡r❡ ¯ ci,j = (−✶)degci,j · ci,j✳ ❚❤✐s ✐♠♣❧✐❡s✿ d(a) = ✵ ❛♥❞ a ∈ Am✱ m = n✶ + . . . + nk − k + ✷✳ ❉❡✜♥✐t✐♦♥ ❆ ▼❛ss❡② k✲♣r♦❞✉❝t α✶, . . . , αk ✐s s❛✐❞ t♦ ❜❡ ❞❡✜♥❡❞✱ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❞❡✜♥✐♥❣ s②st❡♠ C ❢♦r ✐t✳ ■❢ s♦✱ t❤✐s ▼❛ss❡② ♣r♦❞✉❝t ❝♦♥s✐sts ♦❢ ❛❧❧ α = [a(C)] ❢♦r ❡❛❝❤ ❞❡✜♥✐♥❣ s②st❡♠ C✳ ■t ✐s ❝❛❧❧❡❞ tr✐✈✐❛❧✱ ✐❢ [a(C)] = ✵ ❢♦r s♦♠❡ C✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❦❂✷ ■❢ α✶, α✷ ✐s ❞❡✜♥❡❞✱ t❤❡♥ ✇❡ ❤❛✈❡✿ a = d(c✶,✸) − ¯ a✶ · a✷. ❦❂✸ ■❢ α✶, α✷, α✸ ✐s ❞❡✜♥❡❞✱ t❤❡♥ ✇❡ ❤❛✈❡✿ a = d(c✶,✹) − ¯ a✶ · c✷,✹ − ¯ c✶,✸ · a✸, d(c✶,✸) = ¯ a✶ · a✷, d(c✷,✹) = ¯ a✷ · a✸.
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❦❂✹ ■❢ α✶, α✷, α✸, α✹ ✐s ❞❡✜♥❡❞✱ t❤❡♥ ✇❡ ❤❛✈❡✿ a = d(c✶,✺) − ¯ a✶ · c✷,✺ − ¯ c✶,✸ · c✸,✺ − ¯ c✶,✹ · a✹, d(c✶,✸) = ¯ a✶ · a✷, d(c✶,✹) = ¯ a✶ · c✷,✹ + ¯ c✶,✸ · a✸, d(c✷,✹) = ¯ a✷ · a✸, d(c✷,✺) = ¯ a✷ · c✸,✺ + ¯ c✷,✹ · a✹, d(c✸,✺) = ¯ a✸ · a✹.
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❘❡♠❛r❦s ✭✶✮ ❱✳❇✉❝❤st❛❜❡r ❛♥❞ ❱✳❱♦❧♦❞✐♥ ✭✷✵✶✶✮ ❝♦♥str✉❝t❡❞ r❡❛❧✐③❛t✐♦♥s ♦❢ ❛❧❧ ✢❛❣ ♥❡st♦❤❡❞r❛ ❛s ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s✱ ✐✳❡✳ ❛ r❡s✉❧t ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ tr✉♥❝❛t✐♦♥s ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ✷ ❢❛❝❡s ♦♥❧②✱ st❛rt✐♥❣ ✇✐t❤ ❛ ❝✉❜❡✱ ❛♥❞ ♣r♦✈❡❞ t❤❡ ●❛❧ ❝♦♥❥❡❝t✉r❡ ♦♥ γ✲✈❡❝t♦rs ❢♦r t❤❡♠❀ ✭✷✮ ●✳❉❡♥❤❛♠ ❛♥❞ ❆✳❙✉❝✐✉ ✭✷✵✵✺✮ ❞❡s❝r✐❜❡❞ ✺ ❣r❛♣❤s✱ s✳t✳ t❤❡r❡ ✐s ❛ ♥♦♥tr✐✈✐❛❧ tr✐♣❧❡ ▼❛ss❡② ♣r♦❞✉❝t ♦❢ ✸✲❞✐♠❡♥s✐♦♥❛❧ ❝❧❛ss❡s ✐♥ ✐✛ ♦♥❡ ♦❢ t❤❡s❡ ❣r❛♣❤s ✐s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ✐♥
✶
✳ ❆❧❧ s✉❝❤ ♣r♦❞✉❝ts ❛r❡ ❞❡❝♦♠♣♦s❛❜❧❡✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❘❡♠❛r❦s ✭✶✮ ❱✳❇✉❝❤st❛❜❡r ❛♥❞ ❱✳❱♦❧♦❞✐♥ ✭✷✵✶✶✮ ❝♦♥str✉❝t❡❞ r❡❛❧✐③❛t✐♦♥s ♦❢ ❛❧❧ ✢❛❣ ♥❡st♦❤❡❞r❛ ❛s ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s✱ ✐✳❡✳ ❛ r❡s✉❧t ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ tr✉♥❝❛t✐♦♥s ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ✷ ❢❛❝❡s ♦♥❧②✱ st❛rt✐♥❣ ✇✐t❤ ❛ ❝✉❜❡✱ ❛♥❞ ♣r♦✈❡❞ t❤❡ ●❛❧ ❝♦♥❥❡❝t✉r❡ ♦♥ γ✲✈❡❝t♦rs ❢♦r t❤❡♠❀ ✭✷✮ ●✳❉❡♥❤❛♠ ❛♥❞ ❆✳❙✉❝✐✉ ✭✷✵✵✺✮ ❞❡s❝r✐❜❡❞ ✺ ❣r❛♣❤s✱ s✳t✳ t❤❡r❡ ✐s ❛ ♥♦♥tr✐✈✐❛❧ tr✐♣❧❡ ▼❛ss❡② ♣r♦❞✉❝t ♦❢ ✸✲❞✐♠❡♥s✐♦♥❛❧ ❝❧❛ss❡s ✐♥ H∗(ZP) ✐✛ ♦♥❡ ♦❢ t❤❡s❡ ❣r❛♣❤s ✐s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ✐♥ sk✶(∂P∗)✳ ❆❧❧ s✉❝❤ ♣r♦❞✉❝ts ❛r❡ ❞❡❝♦♠♣♦s❛❜❧❡✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❤♦❧❞s ❢♦r tr✐♣❧❡ ▼❛ss❡② ♣r♦❞✉❝ts ✐♥ t❤❡ ❝♦❤♦♠♦❧♦❣② r✐♥❣ ♦❢ ZP✳ ❚❤❡♦r❡♠ ▲❡t P ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❛ss♦❝✐❛❤❡❞r♦♥ ♦❢ t②♣❡ A✱ B(C)✱ D✱ ♦r P = PΓ✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ ❞❡✜♥❡❞ ❛♥❞ ♥♦♥tr✐✈✐❛❧ tr✐♣❧❡ ▼❛ss❡② ♣r♦❞✉❝t α✶, α✷, α✸ ♦❢ s♦♠❡ ❝❧❛ss❡s αi ∈ H✸(ZP)✱ i = ✶, ✷, ✸ ✐❢ ❛♥❞ ♦♥❧② ✐❢ P ✐s ❛ ❣❡♥❡r❛❧✐③❡❞ ❛ss♦❝✐❛❤❡❞r♦♥ ♦r ❛ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r♦♥ PΓ ❛♥❞ ✐♥ t❤❡ ❣r❛♣❤ Γ t❤❡r❡ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦♥ n + ✶ = ✹ ✈❡rt✐❝❡s✱ ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❝♦♠♣❧❡t❡ ❣r❛♣❤ K✹✳ ❆❧❧ s✉❝❤ ▼❛ss❡② ♣r♦❞✉❝ts ❛r❡ ❞❡❝♦♠♣♦s❛❜❧❡✳ ❚❤❡ ❢❛❝❡ ❧❛tt✐❝❡ ❛❧❧♦✇s ✉s t♦ r❡❞✉❝❡ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ t♦ t❤❡ ❝❛s❡ ♦❢ ✸ ❛♥❞ ❛♣♣❧② t❤❡ r❡s✉❧t ♦❢ ❉❡♥❤❛♠ ❛♥❞ ❙✉❝✐✉✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❤♦❧❞s ❢♦r tr✐♣❧❡ ▼❛ss❡② ♣r♦❞✉❝ts ✐♥ t❤❡ ❝♦❤♦♠♦❧♦❣② r✐♥❣ ♦❢ ZP✳ ❚❤❡♦r❡♠ ▲❡t P ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❛ss♦❝✐❛❤❡❞r♦♥ ♦❢ t②♣❡ A✱ B(C)✱ D✱ ♦r P = PΓ✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ ❞❡✜♥❡❞ ❛♥❞ ♥♦♥tr✐✈✐❛❧ tr✐♣❧❡ ▼❛ss❡② ♣r♦❞✉❝t α✶, α✷, α✸ ♦❢ s♦♠❡ ❝❧❛ss❡s αi ∈ H✸(ZP)✱ i = ✶, ✷, ✸ ✐❢ ❛♥❞ ♦♥❧② ✐❢ P ✐s ❛ ❣❡♥❡r❛❧✐③❡❞ ❛ss♦❝✐❛❤❡❞r♦♥ ♦r ❛ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r♦♥ PΓ ❛♥❞ ✐♥ t❤❡ ❣r❛♣❤ Γ t❤❡r❡ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦♥ n + ✶ = ✹ ✈❡rt✐❝❡s✱ ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❝♦♠♣❧❡t❡ ❣r❛♣❤ K✹✳ ❆❧❧ s✉❝❤ ▼❛ss❡② ♣r♦❞✉❝ts ❛r❡ ❞❡❝♦♠♣♦s❛❜❧❡✳ ❚❤❡ ❢❛❝❡ ❧❛tt✐❝❡ ❛❧❧♦✇s ✉s t♦ r❡❞✉❝❡ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ t♦ t❤❡ ❝❛s❡ ♦❢ n = ✸ ❛♥❞ ❛♣♣❧② t❤❡ r❡s✉❧t ♦❢ ❉❡♥❤❛♠ ❛♥❞ ❙✉❝✐✉✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❆♥② ♥❡st♦❤❡❞r♦♥ ♦♥ ❛ ❝♦♥♥❡❝t❡❞ ❜✉✐❧❞✐♥❣ s❡t ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❛ s✐♠♣❧❡① ❛s ❛ r❡s✉❧t ♦❢ ❛ tr✉♥❝❛t✐♦♥ s❡q✉❡♥❝❡ ♦❢ t❤❡ s✐♠♣❧❡①✬s ❢❛❝❡s ♦♥❧②✳ ❚❤❡♦r❡♠ ■❢ P = Pen, n ≥ ✷ ❛♥❞ t❤❡ ❝❧❛ss❡s αi ∈ H✸(ZP), ✶ ≤ i ≤ n + ✶ ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② (n + ✶) ♣❛✐rs ♦❢ t❤❡ ♦♣♣♦s✐t❡ ♣❡r♠✉t♦❤❡❞r❛ ❢❛❝❡ts✱ t❤❡♥ α✶, . . . , αn+✶ ✐s ❞❡✜♥❡❞ ❛♥❞ tr✐✈✐❛❧❀ ■❢ P = Stn, n ≥ ✷ ❛♥❞ t❤❡ ❝❧❛ss❡s αi ∈ H✸(ZP), ✶ ≤ i ≤ n ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② n ♣❛✐rs ♦❢ t❤❡ ♦♣♣♦s✐t❡ st❡❧❧❛❤❡❞r❛ ❢❛❝❡ts✱ t❤❡♥ α✶, . . . , αn ✐s ❞❡✜♥❡❞ ❛♥❞ tr✐✈✐❛❧✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❊①❛♠♣❧❡✿ n = ✷ P = As✷ ✐s ❛ ✺✲❣♦♥✱ ZP = (S✸ × S✹)#✺ ❛♥❞ t❤❡ ✈❛♥✐s❤✐♥❣ ❝✉♣ ♣r♦❞✉❝t ❝♦rr❡s♣♦♥❞s t♦ ✷ ♣❛✐rs ♦❢ ♥♦♥✲❛❞❥❛❝❡♥t ❡❞❣❡s ✐♥ ❛ ✺✲❣♦♥✳ P = Pe✷ ✐s ❛ ✻✲❣♦♥✱ ZP = (S✸ × S✺)#✻#(S✹ × S✹)#✽#(S✺ × S✸)#✸ ❛♥❞ t❤❡ ✈❛♥✐s❤✐♥❣ tr✐♣❧❡ ▼❛ss❡② ♣r♦❞✉❝t ❝♦rr❡s♣♦♥❞s t♦ ✸ ♣❛✐rs ♦❢ ♣❛r❛❧❧❡❧ ❡❞❣❡s ✐♥ ❛ r❡❣✉❧❛r ✻✲❣♦♥✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❲❡ ♥❡①t ❝♦♥s✐❞❡r ❛ ♣❛rt✐❝✉❧❛r ❢❛♠✐❧② ♦❢ ✷✲tr✉♥❝❛t❡❞ n✲❝✉❜❡s P✱ ♦♥❡ ❢♦r ❡❛❝❤ ❞✐♠❡♥s✐♦♥ n✱ ❢♦r ✇❤✐❝❤ ZP ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ ▼❛ss❡② ♣r♦❞✉❝t ♦❢ ♦r❞❡r n✳ ❉❡✜♥✐t✐♦♥ ❙✉♣♣♦s❡ I n ✐s ❛♥ n✲❞✐♠❡♥s✐♦♥❛❧ ❝✉❜❡ ✇✐t❤ ❢❛❝❡ts F✶, . . . , F✷n✱ s✉❝❤ t❤❛t Fi ❛♥❞ Fn+i✱ ✶ ≤ i ≤ n ❛r❡ ♣❛r❛❧❧❡❧ ✭❞♦ ♥♦t ✐♥t❡rs❡❝t✮✳ ❚❤❡♥ ✇❡ ❞❡✜♥❡ P ❛s ❛ r❡s✉❧t ♦❢ ❛ ❝♦♥s❡❝✉t✐✈❡ ❝✉t ♦❢ ❢❛❝❡s ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ✷ ❢r♦♠ I n✱ ❤❛✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❙t❛♥❧❡②✲❘❡✐s♥❡r ✐❞❡❛❧✿ I = (v✶vn+✶, . . . , vnv✷n, v✶vn+✷, . . . , vn−✶v✷n, . . . , v✶v✷n−✶, v✷v✷n, . . .), ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ I = (vkvn+k+i, ✵ ≤ i ≤ n − ✷, ✶ ≤ k ≤ n − i, . . .), ✇❤❡r❡ vi ❝♦rr❡s♣♦♥❞ t♦ Fi✱ ✶ ≤ i ≤ ✷n ❛♥❞ ✐♥ t❤❡ ❧❛st ❞♦ts ❛r❡ t❤❡ ♠♦♥♦♠✐❛❧s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ♥❡✇ ❢❛❝❡ts✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❘❡♠❛r❦s ❋♦r n = ✷ ✇❡ ❣❡t ❛ ✷✲❞✐♠❡♥s✐♦♥❛❧ ❝✉❜❡ ✭t❤❡ sq✉❛r❡✮ ❛♥❞ ❢♦r n = ✸ ✇❡ ❣❡t ❛ s✐♠♣❧❡ ✸✲♣♦❧②t♦♣❡ P ✇✐t❤ ✽ ❢❛❝❡ts ❣✐✈✐♥❣ ❛ ♥♦♥tr✐✈✐❛❧ tr✐♣❧❡ ▼❛ss❡② ♣r♦❞✉❝t ❞✉❡ t♦ ■✳❇❛s❦❛❦♦✈ r❡s✉❧t ✭✷✵✵✸✮❀ ✐s ❛ ✢❛❣ ♥❡st♦❤❡❞r♦♥✿ ✇❡ ❝❛♥ ❡❛s✐❧② ❝♦♥str✉❝t t❤❡ ❜✉✐❧❞✐♥❣ s❡t ❢♦r ♦♥ t❤❡ ✈❡rt❡① s❡t ✶ ❜② ✐❞❡♥t✐❢②✐♥❣ ✇✐t❤ ✶ ❢♦r ✶ ❛♥❞ ✐❞❡♥t✐❢②✐♥❣ ✇✐t❤ ✶ ❢♦r ✶ ✷ ✳ ❚❤❡♥ ✇❡ ❝♦♥s❡❝✉t✐✈❡❧② ❝✉t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝❡s✿ ✶ ✸ ✶ ✷ ✹ ✶ ✶ ✶ ✶ ✶ ✷ ✶
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❘❡♠❛r❦s ❋♦r n = ✷ ✇❡ ❣❡t ❛ ✷✲❞✐♠❡♥s✐♦♥❛❧ ❝✉❜❡ ✭t❤❡ sq✉❛r❡✮ ❛♥❞ ❢♦r n = ✸ ✇❡ ❣❡t ❛ s✐♠♣❧❡ ✸✲♣♦❧②t♦♣❡ P ✇✐t❤ ✽ ❢❛❝❡ts ❣✐✈✐♥❣ ❛ ♥♦♥tr✐✈✐❛❧ tr✐♣❧❡ ▼❛ss❡② ♣r♦❞✉❝t ❞✉❡ t♦ ■✳❇❛s❦❛❦♦✈ r❡s✉❧t ✭✷✵✵✸✮❀ P ✐s ❛ ✢❛❣ ♥❡st♦❤❡❞r♦♥✿ ✇❡ ❝❛♥ ❡❛s✐❧② ❝♦♥str✉❝t t❤❡ ❜✉✐❧❞✐♥❣ s❡t B ❢♦r P ♦♥ t❤❡ ✈❡rt❡① s❡t S = [n + ✶] ❜② ✐❞❡♥t✐❢②✐♥❣ Fi ✇✐t❤ {✶, . . . , i} ❢♦r ✶ ≤ i ≤ n ❛♥❞ ✐❞❡♥t✐❢②✐♥❣ Fi ✇✐t❤ {i − n + ✶} ❢♦r n + ✶ ≤ i ≤ ✷n✳ ❚❤❡♥ ✇❡ ❝♦♥s❡❝✉t✐✈❡❧② ❝✉t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝❡s✿ {✶} ⊔ {✸}, {✶, ✷} ⊔ {✹}, . . . , {✶, . . . , n − ✶} ⊔ {n + ✶} · · · {✶} ⊔ {n}, {✶, ✷} ⊔ {n + ✶}.
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❘❡♠❛r❦s ❚❤✉s✱ P = PB ❢♦r t❤❡ ❜✉✐❧❞✐♥❣ s❡t B ❝♦♥s✐st✐♥❣ ♦❢ t❤❡ ❜✉✐❧❞✐♥❣ s❡t B✵ = {{i}n+✶
✶
, {✶, ✷}, {✶, ✷, ✸}, . . . , [n + ✶]} ♦❢ ❛♥ n✲❝✉❜❡✱ t❤❡ ❛❜♦✈❡ s✉❜s❡ts ✐♥ [n + ✶] ❛♥❞ ❛❧❧ t❤❡ s✉❜s❡ts ✐♥ [n + ✶] ✇❤✐❝❤ ❛r❡ t❤❡ ✉♥✐♦♥s ♦❢ ♥♦♥tr✐✈✐❛❧❧② ✐♥t❡rs❡❝t✐♥❣ ❡❧❡♠❡♥ts ✐♥ B❀ ✐s ♥♦t ❛ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r♦♥✿ ✐ts ♥✉♠❜❡r ♦❢ ❢❛❝❡ts
✵ ✸ ✷
✶
✵ ✸ ✷
✱ t❤✉s ✇❡ ❝❛♥ ❛♣♣❧② t❤❡ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ❢♦r ❢✲✈❡❝t♦rs ♦❢ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r❛ ♣r♦✈❡❞ ❜② ❇✉❝❤st❛❜❡r ❛♥❞ ❱♦❧♦❞✐♥✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❘❡♠❛r❦s ❚❤✉s✱ P = PB ❢♦r t❤❡ ❜✉✐❧❞✐♥❣ s❡t B ❝♦♥s✐st✐♥❣ ♦❢ t❤❡ ❜✉✐❧❞✐♥❣ s❡t B✵ = {{i}n+✶
✶
, {✶, ✷}, {✶, ✷, ✸}, . . . , [n + ✶]} ♦❢ ❛♥ n✲❝✉❜❡✱ t❤❡ ❛❜♦✈❡ s✉❜s❡ts ✐♥ [n + ✶] ❛♥❞ ❛❧❧ t❤❡ s✉❜s❡ts ✐♥ [n + ✶] ✇❤✐❝❤ ❛r❡ t❤❡ ✉♥✐♦♥s ♦❢ ♥♦♥tr✐✈✐❛❧❧② ✐♥t❡rs❡❝t✐♥❣ ❡❧❡♠❡♥ts ✐♥ B❀ P ✐s ♥♦t ❛ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r♦♥✿ ✐ts ♥✉♠❜❡r ♦❢ ❢❛❝❡ts f✵(P) = n(n+✸)
✷
− ✶ < f✵(Asn) = n(n+✸)
✷
✱ t❤✉s ✇❡ ❝❛♥ ❛♣♣❧② t❤❡ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ❢♦r ❢✲✈❡❝t♦rs ♦❢ ❣r❛♣❤✲❛ss♦❝✐❛❤❡❞r❛ ♣r♦✈❡❞ ❜② ❇✉❝❤st❛❜❡r ❛♥❞ ❱♦❧♦❞✐♥✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❖✉r ♠❛✐♥ r❡s✉❧t ♦♥ ♥♦♥tr✐✈✐❛❧ ❤✐❣❤❡r ▼❛ss❡② ♣r♦❞✉❝ts ❢♦r ♠♦♠❡♥t✲❛♥❣❧❡ ♠❛♥✐❢♦❧❞s ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ αi ∈ H✸(ZP) ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ✸✲❞✐♠❡♥s✐♦♥❛❧ ❝♦❝②❝❧❡ viun+i ∈ R−✶,✹(P) ❢♦r ✶ ≤ i ≤ n ❛♥❞ n ≥ ✷✳ ❚❤❡♥ t❤❡ n✲❢♦❧❞ ▼❛ss❡② ♣r♦❞✉❝t α✶, . . . , αn ✐s ❞❡✜♥❡❞ ❛♥❞ ♥♦♥tr✐✈✐❛❧✳ ❆♥② ❡❧❡♠❡♥t
✶
✐s s✳t✳ ✐s ❛ s✉♠ ♦❢ ✐ts ♠✉❧t✐❣r❛❞❡❞ ❝♦♠♣♦♥❡♥ts ❛♥❞
✷ ✶ ✷
✳ ❚❤✉s✱ ✐s ❛ ❝♦❜♦✉♥❞❛r② ✐✛ ❡❛❝❤ ♦❢ ✐ts ♠✉❧t✐❣r❛❞❡❞ ❝♦♠♣♦♥❡♥ts ✐s ❛ ❝♦❜♦✉♥❞❛r②❀ ❋♦r ❛♥② s✉❝❤ ✐ts ❝♦♠♣♦♥❡♥t ✐♥
✷ ✷ ✷ ✶ ✶ ✵ ✵
✇✐t❤ ✷ ✧✶✧✬s ✐s ❛❧✇❛②s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❝♦❝②❝❧❡
✶ ✷ ✷ ✸ ✷ ✶ ✭✉♣ t♦ s✐❣♥✮✱ ✇❤✐❝❤ ✐s ♥♦t ❛
❝♦❜♦✉♥❞❛r②✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❖✉r ♠❛✐♥ r❡s✉❧t ♦♥ ♥♦♥tr✐✈✐❛❧ ❤✐❣❤❡r ▼❛ss❡② ♣r♦❞✉❝ts ❢♦r ♠♦♠❡♥t✲❛♥❣❧❡ ♠❛♥✐❢♦❧❞s ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ αi ∈ H✸(ZP) ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ✸✲❞✐♠❡♥s✐♦♥❛❧ ❝♦❝②❝❧❡ viun+i ∈ R−✶,✹(P) ❢♦r ✶ ≤ i ≤ n ❛♥❞ n ≥ ✷✳ ❚❤❡♥ t❤❡ n✲❢♦❧❞ ▼❛ss❡② ♣r♦❞✉❝t α✶, . . . , αn ✐s ❞❡✜♥❡❞ ❛♥❞ ♥♦♥tr✐✈✐❛❧✳ ❆♥② ❡❧❡♠❡♥t α = [a] ∈ α✶, . . . , αn ✐s s✳t✳ a ∈ R∗(P) ✐s ❛ s✉♠ ♦❢ ✐ts ♠✉❧t✐❣r❛❞❡❞ ❝♦♠♣♦♥❡♥ts ❛♥❞ d : R−i,✷J(P) R−(i−✶),✷J(P)✳ ❚❤✉s✱ a ✐s ❛ ❝♦❜♦✉♥❞❛r② ✐✛ ❡❛❝❤ ♦❢ ✐ts ♠✉❧t✐❣r❛❞❡❞ ❝♦♠♣♦♥❡♥ts ✐s ❛ ❝♦❜♦✉♥❞❛r②❀ ❋♦r ❛♥② s✉❝❤ a ∈ R∗(P) ✐ts ❝♦♠♣♦♥❡♥t ✐♥ R−(✷n−✷),✷(✶,...,✶,✵,...,✵)(P) ✇✐t❤ ✷n ✧✶✧✬s ✐s ❛❧✇❛②s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❝♦❝②❝❧❡ v✶v✷nu✷u✸ . . . u✷n−✶ ✭✉♣ t♦ s✐❣♥✮✱ ✇❤✐❝❤ ✐s ♥♦t ❛ ❝♦❜♦✉♥❞❛r②✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
IP = (v✶v✹, v✷v✺, v✸v✻, v✶v✺, v✷v✻, . . .). ❚❤❡♥ ❢♦r ai = viun+i ✇❡ ❤❛✈❡ ✭✉♣ t♦ s✐❣♥✮✿ c✶,✸ = v✶u✷u✹u✺, c✷,✹ = v✷u✸u✺u✻, a = v✶v✻u✷u✸u✹u✺. ❚❤✉s✱ α = [a] = −[v✶u✹u✺] · [v✻u✷u✸]✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❯s✐♥❣ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞✳ ❚❤❡♦r❡♠ ❚❤❡r❡ ❡①✐sts ❛ ✢❛❣ ♥❡st♦❤❡❞r♦♥ P = PB✱ s✉❝❤ t❤❛t t❤❡r❡ ❛r❡ ♥♦♥tr✐✈✐❛❧ ❤✐❣❤❡r ▼❛ss❡② ♣r♦❞✉❝ts ♦❢ ❛♥② ♣r❡s❝r✐❜❡❞ ♦r❞❡rs n✶, . . . , nr, r ≥ ✷ ✐♥ H∗(ZP)✳ ❈♦♥str✉❝t✐♦♥✿ s✉❜st✐t✉t✐♦♥ ♦❢ ❜✉✐❧❞✐♥❣ s❡ts ▲❡t
✶ ✶ ❜❡ ❝♦♥♥❡❝t❡❞ ❜✉✐❧❞✐♥❣ s❡ts ♦♥ ✶ ✶ ✳
❚❤❡♥✱ ❢♦r ❡✈❡r② ♦♥ ✶ ✱ ❞❡✜♥❡
✶ ✶ ♦♥ ✶ ✶ ✱ ❝♦♥s✐st✐♥❣ ♦❢ ❡❧❡♠❡♥ts
❛♥❞ ✱ ✇❤❡r❡ ✳ ❚❤❡♥
✶ ✶✳
❲❡ t❛❦❡ ✱
✶
✱ ✇❤❡r❡ ✶ ✐s ❛ ❜✉✐❧❞✐♥❣ s❡t ❢♦r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❛♥❞ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❜✉✐❧❞✐♥❣ s❡t ♦❢ ❛ ✶ ✲❞✐♠❡♥s✐♦♥❛❧ ❝✉❜❡✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❯s✐♥❣ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞✳ ❚❤❡♦r❡♠ ❚❤❡r❡ ❡①✐sts ❛ ✢❛❣ ♥❡st♦❤❡❞r♦♥ P = PB✱ s✉❝❤ t❤❛t t❤❡r❡ ❛r❡ ♥♦♥tr✐✈✐❛❧ ❤✐❣❤❡r ▼❛ss❡② ♣r♦❞✉❝ts ♦❢ ❛♥② ♣r❡s❝r✐❜❡❞ ♦r❞❡rs n✶, . . . , nr, r ≥ ✷ ✐♥ H∗(ZP)✳ ❈♦♥str✉❝t✐♦♥✿ s✉❜st✐t✉t✐♦♥ ♦❢ ❜✉✐❧❞✐♥❣ s❡ts ▲❡t B✶, . . . , Bn+✶ ❜❡ ❝♦♥♥❡❝t❡❞ ❜✉✐❧❞✐♥❣ s❡ts ♦♥ [k✶], . . . , [kn+✶]✳ ❚❤❡♥✱ ❢♦r ❡✈❡r② B ♦♥ [n + ✶]✱ ❞❡✜♥❡ B′ = B(B✶, . . . , Bn+✶) ♦♥ [k✶] ⊔ · · · ⊔ [kn+✶]✱ ❝♦♥s✐st✐♥❣ ♦❢ ❡❧❡♠❡♥ts Si ∈ Bi ❛♥❞
i∈S
[ki]✱ ✇❤❡r❡ S ∈ B✳ ❚❤❡♥ PB′ = PB × PB✶ × · · · × PBn+✶✳ ❲❡ t❛❦❡ P = PB✱ B = B′(B✶, . . . , Br)✱ ✇❤❡r❡ Bs, ✶ ≤ s ≤ r ✐s ❛ ❜✉✐❧❞✐♥❣ s❡t ❢♦r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ P ✐♥ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❛♥❞ B′ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❜✉✐❧❞✐♥❣ s❡t ♦❢ ❛ (r − ✶)✲❞✐♠❡♥s✐♦♥❛❧ ❝✉❜❡✳
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s
❚❍❆◆❑ ❨❖❯ ❋❖❘ ❨❖❯❘ ❆❚❚❊◆❚■❖◆✦
■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s