rts rst - - PowerPoint PPT Presentation

■✈❛♥ ❆r③❤❛♥ts❡✈ ✭❍❙❊ ❯♥✐✈❡rs✐t②✱ ▼♦s❝♦✇✮ ■♥✜♥✐t❡ tr❛♥s✐t✐✈✐t②✱ ✜♥✐t❡ ❣❡♥❡r❛t✐♦♥✱ ❛♥❞ ❉❡♠❛③✉r❡ r♦♦ts ✇✐t❤ ❑❛r✐♥❡ ❑✉②✉♠③❤✐②❛♥ ❛♥❞ ▼✐❦❤❛✐❧ ❩❛✐❞❡♥❜❡r❣ ❆❞✈❛♥❝❡s ✐♥ ▼❛t❤❡♠❛t✐❝s ✸✺✶ ✭✷✵✶✾✮ ✶✕✸✷

✶✹✳✵✺✳✷✵✷✵

■♥✜♥✐t❡ ❚r❛♥s✐t✐✈✐t②

❉❡✜♥✐t✐♦♥ ▲❡t G ❜❡ ❛ ❣r♦✉♣✱ X ❛ s❡t✱ ❛♥❞ m ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❆♥ ❛❝t✐♦♥ G × X → X ✐s ❝❛❧❧❡❞ m✲tr❛♥s✐t✐✈❡ ✐❢ ❢♦r ❛♥② t✇♦ t✉♣❧❡s (a✶, . . . , am) ❛♥❞ (b✶, . . . , bm) ♦❢ ♣❛✐r✇✐s❡ ❞✐st✐♥❝t ♣♦✐♥ts ♦♥ X t❤❡r❡ ✐s ❛♥ ❡❧❡♠❡♥t g ∈ G s✉❝❤ t❤❛t (ga✶, . . . , gam) = (b✶, . . . , bm)✳ ❉❡✜♥✐t✐♦♥ ❆♥ ❛❝t✐♦♥ G × X → X ✐s ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡ ✐❢ ✐t ✐s m✲tr❛♥s✐t✐✈❡ ❢♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r m✳ ❊①❛♠♣❧❡ ✶✮ ▲❡t X ❜❡ ❛♥ ✐♥✜♥✐t❡ s❡t ❛♥❞ G t❤❡ ❣r♦✉♣ ♦❢ ❛❧❧ ♣❡r♠✉t❛t✐♦♥s ♦♥ X✳ ✷✮ ▲❡t X ❜❡ ❛♥ ✐♥✜♥✐t❡ s❡t ❛♥❞ G t❤❡ ❣r♦✉♣ ♦❢ ❛❧❧ ♣❡r♠✉t❛t✐♦♥s ✇✐t❤ ✜♥✐t❡ s✉♣♣♦rt ♦♥ X✳

❆✣♥❡ ❙♣❛❝❡s

❚❤❡♦r❡♠ ❚❤❡ ❣r♦✉♣ Aut(An) ✐s ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡ ♦♥ An ❢♦r ❛♥② n ✷✳ ■❞❡❛ ✭n = ✷✮✿ ✉s❡ ♣❛r❛❧❧❡❧ tr❛♥s❧❛t✐♦♥s (x✶ + a, x✷) ✱ (x✶, x✷ + b) ❛♥❞ t❤❡✐r r❡♣❧✐❝❛s (x✶ + af✶(x✷), x✷) ✱ (x✶, x✷ + bf✷(x✶))✱ ✇❤❡r❡ a, b ∈ K✳ ❊①❛♠♣❧❡ ❚❤❡ ❣r♦✉♣ Aut(A✶) ✐s ✐s♦♠♦r♣❤✐❝ t♦ K × ⋌ K✳ ■t ✐s ✷✲tr❛♥s✐t✐✈❡✱ ❜✉t ♥♦t ✸✲tr❛♥s✐t✐✈❡ ♦♥ A✶✳

• ❡♥❡r❛❧ Pr♦❜❧❡♠s

▲❡t X ❜❡ ❛♥ ❛✣♥❡ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t② ♦✈❡r t❤❡ ✜❡❧❞ C✳ ❲❤❡♥ t❤❡ ❣r♦✉♣ Aut(X) ♦❢ ♣♦❧②♥♦♠✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ X ✐s ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡ ♦♥ X❄ ■❢ X ✐s s✐♥❣✉❧❛r✱ ✇❡ ❛s❦ t❤✐s q✉❡st✐♦♥ ❢♦r t❤❡ s♠♦♦t❤ ❧♦❝✉s X r❡❣ ✳ ■❞❡❛✿ t♦ ✉s❡ Ga✲s✉❜❣r♦✉♣s ✐♥ t❤❡ ❣r♦✉♣ Aut(X) ❛♥❞ t❤❡✐r r❡♣❧✐❝❛s✳ ❍❡r❡ Ga = (C, +)✳ ◆♦t❛t✐♦♥✿ SAut(X) ✐s t❤❡ s✉❜❣r♦✉♣ ♦❢ Aut(X) ❣❡♥❡r❛t❡❞ ❜② ❛❧❧ Ga✲s✉❜❣r♦✉♣s✳

▲♦❝❛❧❧② ◆✐❧♣♦t❡♥t ❉❡r✐✈❛t✐♦♥s

❉❡✜♥✐t✐♦♥ ❆ ❞❡r✐✈❛t✐♦♥ D : A → A ♦❢ ❛♥ ❛❧❣❡❜r❛ A ✐s ❧♦❝❛❧❧② ♥✐❧♣♦t❡♥t ✐❢ ❢♦r ❛♥② a ∈ A t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r k s✉❝❤ t❤❛t Dk(a) = ✵✳ ▲♦❝❛❧❧② ♥✐❧♣♦t❡♥t ❞❡r✐✈❛t✐♦♥s ♦♥ C[X] ⇔ Ga✲s✉❜❣r♦✉♣s ✐♥ Aut(X) D ∈ LND(C[X]) ⇐ ⇒ exp(CD) ⊆ Aut(X) ■❢ D ∈ LND(A) ❛♥❞ f ∈ Ker(D)✱ t❤❡♥ fD ∈ LND(A)✳ Ga✲s✉❜❣r♦✉♣s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ▲◆❉s ♦❢ t❤❡ ❢♦r♠ fD ❛r❡ r❡♣❧✐❝❛s ♦❢ t❤❡ Ga✲s✉❜❣r♦✉♣ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ D✳

❋❧❡①✐❜✐❧✐t② ✈s ■♥✜♥✐t❡ ❚r❛♥s✐t✐✈✐t②

❉❡✜♥✐t✐♦♥ ❆♥ ❛✣♥❡ ✈❛r✐❡t② X ✐s ✢❡①✐❜❧❡ ✐❢ t❤❡ t❛♥❣❡♥t s♣❛❝❡ Tx(X) ❛t ❛♥② s♠♦♦t❤ ♣♦✐♥t x ∈ X r❡❣ ✐s ❣❡♥❡r❛t❡❞ ❜② ✈❡❧♦❝✐t② ✈❡❝t♦rs t♦ ♦r❜✐ts ♦❢ Ga✲s✉❜❣r♦✉♣s ♣❛ss✐♥❣ t❤r♦✉❣❤ x✳ ❚❤❡♦r❡♠ ✭❆✳✲❋❧❡♥♥❡r✲❑❛❧✐♠❛♥✲❑✉t③s❝❤❡❜❛✉❝❤✲❩❛✐❞❡♥❜❡r❣✬✷✵✶✸✮ ▲❡t X ❜❡ ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❛✣♥❡ ✈❛r✐❡t② ♦❢ ❞✐♠❡♥s✐♦♥ ✷✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✭❛✮ t❤❡ ❣r♦✉♣ SAut(X) ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ X r❡❣❀ ✭❜✮ t❤❡ ❣r♦✉♣ SAut(X) ❛❝ts ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ♦♥ X r❡❣❀ ✭❝✮ t❤❡ ✈❛r✐❡t② X ✐s ✢❡①✐❜❧❡✳

❊①❛♠♣❧❡s ♦❢ ❋❧❡①✐❜❧❡ ❱❛r✐❡t✐❡s

• ❙✉s♣❡♥s✐♦♥s ❙✉s♣(X, f ) ❣✐✈❡♥ ❜② uv = f (x) ✐♥ A✷ × X ♦✈❡r

❛ ✢❡①✐❜❧❡ ✈❛r✐❡t② X❀

• ◆♦♥✲❞❡❣❡♥❡r❛t❡ ✭C[X]× = C×✮ ❛✣♥❡ t♦r✐❝ ✈❛r✐❡t✐❡s❀
• ◆♦♥✲❞❡❣❡♥❡r❛t❡ ❤♦r♦s♣❤❡r✐❝❛❧ ✈❛r✐❡t✐❡s ♦❢ r❡❞✉❝t✐✈❡ ❣r♦✉♣s❀
• ❍♦♠♦❣❡♥❡♦✉s s♣❛❝❡s G/F✱ ✇❤❡r❡ G ✐s s❡♠✐s✐♠♣❧❡ ❛♥❞ F ✐s

r❡❞✉❝t✐✈❡❀

• ◆♦r♠❛❧ ❛✣♥❡ ❙▲(✷)✲❡♠❜❡❞❞✐♥❣s❀
• ❆✣♥❡ ❝♦♥❡s ♦✈❡r ✢❛❣ ✈❛r✐❡t✐❡s ❛♥❞ ❞❡❧ P❡③③♦ s✉r❢❛❝❡s✳

❘♦♦t ❙✉❜❣r♦✉♣s ❛♥❞ ❉❡♠❛③✉r❡ ❘♦♦ts

▲❡t X ❜❡ ❛ ✈❛r✐❡t② ✇✐t❤ ❛♥ ❛❝t✐♦♥ ♦❢ ❛ t♦r✉s T✳ ❆ Ga✲s✉❜❣r♦✉♣ H ✐♥ Aut(X) ✐s ❝❛❧❧❡❞ ❛ r♦♦t s✉❜❣r♦✉♣ ✐❢ H ✐s ♥♦r♠❛❧✐③❡❞ ✐♥ Aut(X) ❜② t❤❡ t♦r✉s T✳ ■♥ t❤✐s ❝❛s❡ T ❛❝ts ♦♥ H ❜② s♦♠❡ ❝❤❛r❛❝t❡r e✳ ❙✉❝❤ ❛ ❝❤❛r❛❝t❡r ✐s ❝❛❧❧❡❞ ❛ r♦♦t ♦❢ t❤❡ T✲✈❛r✐❡t② X✳ ❆ss✉♠❡ X ✐s t♦r✐❝ ✇✐t❤ ❛❝t✐♥❣ t♦r✉s T✳ ❲❤❛t ❛r❡ t❤❡ r♦♦ts ♦❢ X❄ ▲❡t p✶, . . . , ps ❜❡ t❤❡ ♣r✐♠✐t✐✈❡ ❧❛tt✐❝❡ ✈❡❝t♦rs ♦♥ r❛②s ♦❢ t❤❡ ❢❛♥ ΣX✳ ❉❡✜♥✐t✐♦♥ ❆ ❉❡♠❛③✉r❡ r♦♦t ♦❢ t❤❡ ❢❛♥ ΣX ✐♥ ❛ ❝❤❛r❛❝t❡r e ∈ M s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ✶ i s ✇✐t❤ e, pi = −✶ ❛♥❞ e, pj ✵ ❢♦r j = i✳ ❚❤❡♦r❡♠ ✭❉❡♠❛③✉r❡✬ ✶✾✼✵✮ ▲❡t X ❜❡ ❛ ❝♦♠♣❧❡t❡ ♦r ❛♥ ❛✣♥❡ t♦r✐❝ ✈❛r✐❡t✐❡s✳ ❚❤❡♥ r♦♦t s✉❜❣r♦✉♣s ♦♥ X ❛r❡ ✐♥ ❜✐❥❡❝t✐♦♥ ✇✐t❤ ❉❡♠❛③✉r❡ r♦♦ts ♦❢ t❤❡ ❢❛♥ ΣX✳

❋✐♥✐t❡ ❣❡♥❡r❛t✐♦♥

❈♦♥❥❡❝t✉r❡ ❆✳ ❆♥② ❣❡♥❡r✐❝❛❧❧② ✢❡①✐❜❧❡ ❛✣♥❡ ✈❛r✐❡t② X ❛❞♠✐ts ❛ ✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥ {H✶, . . . , Hk} ♦❢ Ga✲s✉❜❣r♦✉♣s ♦❢ Aut(X) s✉❝❤ t❤❛t t❤❡ ❣r♦✉♣ G = H✶, . . . , Hk ❛❝ts ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ♦♥ ✐ts ♦♣❡♥ ♦r❜✐t✳ ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢✿ ❙t❡♣ ✶✳ ❋✐♥❞ G = H✶, . . . , Hs t❤❛t ❛❝ts ♦♥ X ✇✐t❤ ❛♥ ♦♣❡♥ ♦r❜✐t❀ ❙t❡♣ ✷✳ Pr♦✈❡ t❤❛t t❤❡ ❝❧♦s✉r❡ G ♦❢ t❤❡ s✉❜❣r♦✉♣ G ✐♥ Aut(X) ✐♥ ✐♥❞✲t♦♣♦❧♦❣② ❝♦♥t❛✐♥s ❵♠❛♥② ♦t❤❡r✬ Ga✲s✉❜❣r♦✉♣s❀ ❙t❡♣ ✸✳ Pr♦✈❡ t❤❛t G ❛❝ts ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ♦♣❡♥ ♦r❜✐t❀ ❙t❡♣ ✹✳ Pr♦✈❡ t❤❛t G ❛❝ts ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ♦♣❡♥ ♦r❜✐t✳ ❙t❡♣ ✸ ⇒ ❙t❡♣ ✹ t✉r♥s ♦✉t t♦ ❜❡ tr✉❡ ✐♥ ❣❡♥❡r❛❧✳

❆ ❈♦♥❥❡❝t✉r❡ ♦♥ ▲♦❝❛❧❧② ◆✐❧♣♦t❡♥t ❉❡r✐✈❛t✐♦♥s

❚♦ ❙t❡♣ ✷✿ ❈♦♥❥❡❝t✉r❡ ❇✳ ▲❡t X ❜❡ ❛♥ ❛✣♥❡ ✈❛r✐❡t②✱ ❛♥❞ A = C[X] ❜❡ ✐ts str✉❝t✉r❡ ❛❧❣❡❜r❛✳ ❈♦♥s✐❞❡r t❤❡ ❣r♦✉♣ G = H✶, . . . , Hk ❣❡♥❡r❛t❡❞ ❜② ❛ ✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ Ga✲s✉❜❣r♦✉♣s Hi = exp(CDi) ⊂ SAut(X)✱ ✇❤❡r❡ Di ∈ LND(A)✱ i = ✶, . . . , k✳ ❚❤❡♥ t❤❡ Ga✲s✉❜❣r♦✉♣ H = exp(CD) ⊂ SAut(X)✱ ✇❤❡r❡ D ∈ LND(A)✱ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ G ✐❢ ❛♥❞ ♦♥❧② ✐❢ D ∈ ▲✐❡ D✶, . . . , Dk✳

❚❤❡ ❚♦r✐❝ ❈❛s❡

❚❤❡♦r❡♠ ✭❆✳✲❑✉②✉♠③❤✐②❛♥✲❩❛✐❞❡♥❜❡r❣✬✷✵✶✾✮ ❋♦r ❛♥② ♥♦♥✲❞❡❣❡r❡r❛t❡ ❛✣♥❡ t♦r✐❝ ✈❛r✐❡t② X ♦❢ ❞✐♠❡♥s✐♦♥ ❛t ❧❡❛st ✷✱ ✇❤✐❝❤ ✐s s♠♦♦t❤ ✐♥ ❝♦❞✐♠❡♥t✐♦♥ ✷✱ ♦♥❡ ❝❛♥ ✜♥❞ ❛ ✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ r♦♦t s✉❜❣r♦✉♣s s✉❝❤ t❤❛t t❤❡ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② t❤❡s❡ s✉❜❣r♦✉♣s ❛❝ts ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ s♠♦♦t❤ ❧♦❝✉s X r❡❣✳ ■♥ t❤❡ ♣r♦♦❢ ✇❡ ✉s❡ ❈♦① r✐♥❣s ❛♥❞ t❤❡ q✉♦t✐❡♥t ♣r❡s❡♥t❛t✐♦♥ π: As → X ❜② ❛♥ ❛❝t✐♦♥ ♦❢ ❛ q✉❛s✐t♦r✉s✳

❋✐♥✐t❡ ●❡♥❡r❛t✐♦♥ ❢♦r ❆✣♥❡ ❙♣❛❝❡s ✲ ■

❚❤❡♦r❡♠ ✭❇♦❞♥❛r❝❤✉❦✬✷✵✵✶✮ ❋♦r ❛♥② n ✸ ❛♥❞ ❛♥② tr✐❛♥❣✉❧❛r h ∈ Aut(An) \ ❆✛n ✇❡ ❤❛✈❡ ❆✛n, h = ❚❛♠❡n✳ ❈♦r♦❧❧❛r② ❋♦r ❛♥② n ✸ ❛♥❞ ❛♥② ♥♦♥✲❛✣♥❡ r♦♦t s✉❜❣r♦✉♣ H ✐♥ Aut(An) t❤❡ ❣r♦✉♣ ❆✛n, H ❛❝ts ♦♥ An ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧②✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ♦♥❡ ❝❛♥ ✜♥❞ n + ✷ r♦♦t s✉❜❣r♦✉♣s ✇❤✐❝❤ ❣❡♥❡r❛t❡ ❛ s✉❜❣r♦✉♣ ❛❝t✐♥❣ ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ♦♥ An✳ ❚❤❡♦r❡♠ ✭❆✳✲❑✉②✉♠③❤✐②❛♥✲❩❛✐❞❡♥❜❡r❣✬✷✵✶✾✮ ❋♦r ❛♥② n ✸ ♦♥❡ ❝❛♥ ✜♥❞ t❤r❡❡ Ga✲s✉❜❣r♦✉♣s ♦❢ Aut(An) ✇❤✐❝❤ ❣❡♥❡r❛t❡ ❛ s✉❜❣r♦✉♣ ❛❝t✐♥❣ ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ♦♥ An✳

❋✐♥✐t❡ ●❡♥❡r❛t✐♦♥ ❢♦r ❆✣♥❡ ❙♣❛❝❡s ✲ ■■

▲❡t H ❜❡ t❤❡ Ga✲s✉❜❣r♦✉♣ ♦❢ Aut(An) ❣✐✈❡♥ ❜② (x✶ + ax✷

✷, x✷, . . . , xn).

❚❤❡♦r❡♠ ✭❆✳✲❑✉②✉♠③❤✐②❛♥✲❩❛✐❞❡♥❜❡r❣✬✷✵✶✾✮ ❈♦♥s✐❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ S(n) ♦♥ An ❜② ♣❡r♠✉t❛t✐♦♥s✳ ❚❤❡♥ ❢♦r ❛♥② n ✸ t❤❡ s✉❜❣r♦✉♣ G = H, S(n) ⊂ Aut(An) ❛❝ts ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ✐♥ An \ {✵}✳

❋✐♥✐t❡ ●❡♥❡r❛t✐♦♥ ❢♦r ❆✣♥❡ P❧❛♥❡ ✲ ■

▲❡t Hk ❛♥❞ Rs ❜❡ t❤❡ Ga✲s✉❜❣r♦✉♣s ♦❢ Aut(A✷) ❣✐✈❡♥ ❜② (x✶ + axk

✷ , x✷) ❛♥❞ (x✶, x✷ + bxs ✶), r❡s♣❡❝t✐✈❡❧②✳

▲❡t Gk,s = Hk, Rs✳ ❲❡ ❝❧❛✐♠ t❤❛t ✐❢ ks = ✷ t❤❡♥ Gk,s ❝❛♥ ♥♦t ❜❡ ✷✲tr❛♥s✐t✐✈❡✳ ■♥❞❡❡❞✱ ✐❢ k = ✵ ♦r s = ✵✱ t❤❡♥ t❤❡r❡ ❛r❡ ♦♥❧② ♣❛r❛❧❧❡❧ tr❛♥s❧❛t✐♦♥s ❛❧♦♥❣ ♦♥❡ ❝♦♦r❞✐♥❛t❡✳ ■❢ k = s = ✶✱ t❤❡♥ G✶,✶ ✐s t❤❡ ❣r♦✉♣ ❙▲(✷) ❛♥❞ ✐t ♣r❡s❡r✈❡s ñ♦❧❧✐♥❡❛r✐t②✳ ■❢ ks > ✷✱ ✇❡ t❛❦❡ ❛ r♦♦t ♦❢ ✉♥✐t② ω ♦❢ ❞❡❣r❡❡ ks − ✶ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ s❡t S = {(P, Q) ∈ A✷ × A✷ | P = (x✶, x✷), Q = (ωx✶, ωsx✷)} P′ = (x✶ + axk

✷ , x✷), Q′ = (ωx✶ + a(ωsx✷)k, ωsx✷) = (ω(x✶ + axk ✷ ), ωsx✷)

P′′ = (x✶, x✷ + bxs

✶), Q′′ = (ωx✶, ωsx✷ + b(ωx✶)s) = (ωx✶, ωs(x✷ + bxs ✶))

❋✐♥✐t❡ ●❡♥❡r❛t✐♦♥ ❢♦r ❆✣♥❡ P❧❛♥❡ ✲ ■■

❚❤❡♦r❡♠ ✭▲❡✇✐s✲P❡rr②✲❙tr❛✉❜✬✷✵✶✾✮ ❚❤❡ ❣r♦✉♣ G✶,✷ ❣❡♥❡r❛t❡❞ ❜② t✇♦ s✉❜❣r♦✉♣s (x✶ + ax✷, x✷) ❛♥❞ (x✶, x✷ + bx✷

✶)

❛❝ts ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ♦♥ A✷ \ {✵}✳ ❚❤❡ ♣r♦♦❢ ✐s ❜❛s❡❞ ♦♥ ❛ ❞❡t❛✐❧❡❞ st✉❞② ♦❢ t❤❡ P♦❧②❞❡❣r❡❡ ❈♦♥❥❡❝t✉r❡ ❢♦r ♣❧❛♥❡ ♣♦❧②♥♦♠✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s✳

❚✐ts ❆❧t❡r♥❛t✐✈❡

❚❤❡♦r❡♠ ✭❆✳✲❩❛✐❞❡♥❜❡r❣✬✷✵✷✵✮ ▲❡t X ❜❡ ❛♥ ❛✣♥❡ t♦r✐❝ ✈❛r✐❡t②✳ ❈♦♥s✐❞❡r ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♦♥❡✲♣❛r❛♠❡t❡r ✉♥✐♣♦t❡♥t s✉❜❣r♦✉♣s U✶, . . . , Us ♦❢ ❆✉t(X) ✇❤✐❝❤ ❛r❡ ♥♦r♠❛❧✐③❡❞ ❜② t❤❡ t♦r✉s ❛❝t✐♥❣ ♦♥ X✳ ❚❤❡♥ t❤❡ ❣r♦✉♣ G ❣❡♥❡r❛t❡❞ ❜② U✶, . . . , Us ✈❡r✐✜❡s t❤❡ ❚✐ts ❛❧t❡r♥❛t✐✈❡✱ ❛♥❞✱ ♠♦r❡♦✈❡r✱ ❡✐t❤❡r ✐s ❛ ✉♥✐♣♦t❡♥t ❛❧❣❡❜r❛✐❝ ❣r♦✉♣✱ ♦r ❝♦♥t❛✐♥s ❛ ♥♦♥❛❜❡❧✐❛♥ ❢r❡❡ s✉❜❣r♦✉♣✳ ❈♦r♦❧❧❛r② ■❢ G ✐s ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡✱ t❤❡♥ G ❝♦♥t❛✐♥s ❛ ♥♦♥❛❜❡❧✐❛♥ ❢r❡❡ s✉❜❣r♦✉♣✱ ❛♥❞ s♦✱ ✐s ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤✳

❙♦♠❡ r❡❢❡r❡♥❝❡s ✲ ■

❬✶❪ ❆r③❤❛♥ts❡✈✲❋❧❡♥♥❡r✲❑❛❧✐♠❛♥✲❑✉t③s❝❤❡❜❛✉❝❤✲❩❛✐❞❡♥❜❡r❣✳ ❋❧❡①✐❜❧❡ ✈❛r✐❡t✐❡s ❛♥❞ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s✳ ❉✉❦❡ ▼❛t❤✳ ❏✳ ✶✻✷ ✭✷✵✶✸✮✱ ✼✻✼✕✽✷✸ ❬✷❪ ❆r③❤❛♥ts❡✈✲❋❧❡♥♥❡r✲❑❛❧✐♠❛♥✲❑✉t③s❝❤❡❜❛✉❝❤✲❩❛✐❞❡♥❜❡r❣✳ ■♥✜♥✐t❡ tr❛♥s✐t✐✈✐t② ♦♥ ❛✣♥❡ ✈❛r✐❡t✐❡s✳ ■♥✿ ❇✐r❛t✐♦♥❛❧ ●❡♦♠❡tr②✱ ❘❛t✐♦♥❛❧ ❈✉r✈❡s✱ ❛♥❞ ❆r✐t❤♠❡t✐❝✱ ✶✕✶✹✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✷✵✶✸ ❬✸❪ ❆r③❤❛♥ts❡✈✲❑✉②✉♠③❤✐②❛♥✲❩❛✐❞❡♥❜❡r❣✳ ❋❧❛❣ ✈❛r✐❡t✐❡s✱ t♦r✐❝ ✈❛r✐❡t✐❡s✱ ❛♥❞ s✉s♣❡♥s✐♦♥s✿ t❤r❡❡ ✐♥st❛♥❝❡s ♦❢ ✐♥✜♥✐t❡ tr❛♥s✐t✐✈✐t②✳ ❙❜✳ ▼❛t❤✳ ✷✵✸ ✭✷✵✶✷✮✱ ✸✕✸✵ ❬✹❪ ❆r③❤❛♥ts❡✈✲P❡r❡♣❡❝❤❦♦✲❙☎ ✉ss✳ ■♥✜♥✐t❡ tr❛♥s✐t✐✈✐t② ♦♥ ✉♥✐✈❡rs❛❧ t♦rs♦rs✳ ❏✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ✽✾ ✭✷✵✶✹✮✱ ✼✻✷✕✼✼✽ ❬✺❪ ❇❡r❡st✲❊s❤♠❛t♦✈✲❊s❤♠❛t♦✈✳ ▼✉❧t✐tr❛♥s✐t✐✈✐t② ♦❢ ❈❛❧♦❣❡r♦✕▼♦s❡r s♣❛❝❡s✳ ❚r❛♥s❢♦r♠✳ ●r♦✉♣s ✷✶ ✭✷✵✶✻✮✱ ✸✺✕✺✵

❙♦♠❡ r❡❢❡r❡♥❝❡s ✲ ■■

❬✻❪ ❇♦❣♦♠♦❧♦✈✲❑❛r③❤❡♠❛♥♦✈✲❑✉②✉♠③❤✐②❛♥✳ ❯♥✐r❛t✐♦♥❛❧✐t② ❛♥❞ ❡①✐st❡♥❝❡ ♦❢ ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡ ♠♦❞❡❧s✳ ■♥✿ ❇✐r❛t✐♦♥❛❧ ●❡♦♠❡tr②✱ ❘❛t✐♦♥❛❧ ❈✉r✈❡s✱ ❛♥❞ ❆r✐t❤♠❡t✐❝✱ ✼✼✕✾✷✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✷✵✶✸ ❬✼❪ ❉❡♠❛③✉r❡✳ ❙♦✉s✲❣r♦✉♣❡s ❛❧❣✁ ❡❜r✐q✉❡s ❞❡ r❛♥❣ ♠❛①✐♠✉♠ ❞✉ ❣r♦✉♣❡ ❞❡ ❈r❡♠♦♥❛✳ ❆♥♥✳ ❙❝✐✳ ✁ ❊❝♦❧❡ ◆♦r♠✳ ❙✉♣✳ ✭✹✮ ✭✶✾✼✵✮✱ ✺✵✼✕✺✽✽ ❬✽❪ ❋❧❡♥♥❡r✲❑❛❧✐♠❛♥✲❩❛✐❞❡♥❜❡r❣✳ ❚❤❡ ●r♦♠♦✈✲❲✐♥❦❡❧♠❛♥♥ t❤❡♦r❡♠ ❢♦r ✢❡①✐❜❧❡ ✈❛r✐❡t✐❡s✳ ❏✳ ❊✉r✳ ▼❛t❤✳ ❙♦❝✳ ✶✽ ✭✷✵✶✻✮✱ ✷✹✽✸✕✷✺✶✵ ❬✾❪ ❋r❡✉❞❡♥❜✉r❣✳ ❆❧❣❡❜r❛✐❝ t❤❡♦r② ♦❢ ❧♦❝❛❧❧② ♥✐❧♣♦t❡♥t ❞❡r✐✈❛t✐♦♥s✳ ❊♥❝②❝❧♦♣❛❡❞✐❛ ▼❛t❤✳ ❙❝✐✳ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✷✵✵✻ ❬✶✵❪ ❋✉rt❡r✲❑r❛❢t✳ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ ❛✣♥❡ ♥✲s♣❛❝❡✳ ❛r❳✐✈✿✶✽✵✾✳✵✹✶✼✺

❙♦♠❡ r❡❢❡r❡♥❝❡s ✲■■■

❬✶✶❪ ●❛✐❢✉❧❧✐♥✲❙❤❛❢❛r❡✈✐❝❤✳ ❋❧❡①✐❜✐❧✐t② ♦❢ ♥♦r♠❛❧ ❛✣♥❡ ❤♦r♦s♣❤❡r✐❝❛❧ ✈❛r✐❡t✐❡s✳ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✶✹✼ ✭✷✵✶✾✮✱ ✸✸✶✼✕✸✸✸✵ ❬✶✷❪ ❑♦✈❛❧❡♥❦♦✳ ❚r❛♥s✐t✐✈✐t② ♦❢ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ ●✐③❛t✉❧❧✐♥ s✉r❢❛❝❡s✳ ■♥t✳ ▼❛t❤✳ ❘❡s✳ ◆♦t✐❝❡s ✷✶ ✭✷✵✶✺✮✱ ✶✶✹✸✸✕✶✶✹✽✹ ❬✶✸❪ ▲❡✇✐s✲P❡rr②✲❙tr❛✉❜✳ ❆♥ ❛❧❣♦r✐t❤♠✐❝ ❛♣♣r♦❛❝❤ t♦ t❤❡ ♣♦❧②❞❡❣r❡❡ ❝♦♥❥❡❝t✉r❡ ❢♦r ♣❧❛♥❡ ♣♦❧②♥♦♠✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s✳ ❏✳ P✉r❡ ❆♣♣❧✳ ❆❧❣❡❜r❛ ✷✷✸ ✭✷✵✶✾✮✱ ✺✸✹✻✕✺✸✺✾ ❬✶✹❪ P❛r❦✲❲♦♥✳ ❋❧❡①✐❜❧❡ ❛✣♥❡ ❝♦♥❡s ♦✈❡r ❞❡❧ P❡③③♦ s✉r❢❛❝❡s ♦❢ ❞❡❣r❡❡ ✹✳ ❊✉r✳ ❏✳ ▼❛t❤✳ ✷ ✭✷✵✶✻✮✱ ✸✵✹✕✸✶✽ ❬✶✺❪ P❡r❡♣❡❝❤❦♦✳ ❋❧❡①✐❜✐❧✐t② ♦❢ ❛✣♥❡ ❝♦♥❡s ♦✈❡r ❞❡❧ P❡③③♦ s✉r❢❛❝❡s ♦❢ ❞❡❣r❡❡ ✹ ❛♥❞ ✺✳ ❋✉♥❝✳ ❆♥❛❧✳ ❆♣♣❧✳ ✹✼ ✭✷✵✶✸✮✱ ✹✺✕✺✷ ❬✶✻❪ ❙❤❛❢❛r❡✈✐❝❤✳ ❋❧❡①✐❜✐❧✐t② ♦❢ ❛✣♥❡ ❤♦r♦s♣❤❡r✐❝❛❧ ✈❛r✐❡t✐❡s ♦❢ s❡♠✐s✐♠♣❧❡ ❣r♦✉♣s✳ ❙❜♦r♥✐❦✿ ▼❛t❤✳ ✷✵✽ ✭✷✵✶✼✮✱ ✷✽✺✕✸✶✵