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◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❙♦♠❡ ♣❛r❛♠❡t❡rs ✐♥ ❧✐♥❡❛r ❧❛♠❜❞❛✲t❡r♠s

❏❡❛♥ P❡②❡♥ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❖❧✐✈✐❡r ❇♦❞✐♥✐ ❛♥❞ ❙❡r❣❡② ❉♦✈❣❛❧ ❈▲❆ ✷✵✶✽✱ ❙♦r❜♦♥♥❡ ❯♥✐✈❡rs✐té

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❚❛❜❧❡ ♦❢ ❝♦♥t❡♥ts

✶

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥ ❈♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ♠❛♣s ❛♥❞ ❧✐♥❡❛r ❧❛♠❜❞❛✲t❡r♠s ❆s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥

✷

❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♣❡❝✐✜❝❛t✐♦♥s ❆s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥

✸

❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss

✹

❘❡❢❡r❡♥❝❡s

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❈♦✉♥t✐♥❣ ♣❧❛✐♥ ❛♥❞ ❝❧♦s❡❞ t❡r♠s

❆ ♣❧❛✐♥ ❧✐♥❡❛r✲❧❛♠❜❞❛ t❡r♠ ❝❛♥ ❡✐t❤❡r ❜❡ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ t❡r♠s✱ ♦r ❛ s♠❛❧❧❡r t❡r♠ ✇❤❡r❡ ❛ ❜✐♥❛r② ♥♦❞❡ ✐s ✐♥s❡rt❡❞ ♦♥ t❤❡ t♦♣ ❧❡❢t ♦r ♦♥ t❤❡ t♦♣ r✐❣❤t ♦❢ ❛♥ ❡①✐st✐♥❣ ♥♦❞❡✱ ❛♥ ❛❜str❛❝t✐♦♥ ✐s ❛❞❞❡❞ ❛❜♦✈❡ t❤❡ r♦♦t ❛♥❞ ❝♦♥♥❡❝t❡❞ t♦ t❤❡ ♥❡✇ ❧❡❛❢✿ Tn+3 =

• T (f)

n+2−p

T (f)

p

• p,l
• InsertBinTn

f(z) = z + z2 + zf 2(z) + 2z4∂zf(z). ❈❧♦s❡❞ t❡r♠s ♦❜❡② t❤❡ s❛♠❡ s♣❡❝✐✜❝❛t✐♦♥ ❛♣❛rt ❢r♦♠ t❤❡ ✐♥✐t✐❛❧ t❡r♠s✿ f(z) = z2 + zf 2(z) + 2z4∂zf(z).

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❈♦♥str✉❝t✐♦♥ ♦❢ ❝❧♦s❡❞ t❡r♠s ♦❢ s✐③❡ ✺ ❢r♦♠ t❤❡ t❡r♠ ♦❢ s✐③❡ ✷✳

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❲❡ ❝❛♥ ❛❧s♦ ❣✐✈❡ ❛ s♣❡❝✐✜❝❛t✐♦♥ ✇❤❡r❡ ✇❡ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ♥✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s✿ T (f)

n+1,m =

• T (f)

n−p,m−l

T (f)

p,l

• p,l
• T (f)

n,m+1

f(z, u) = uz + zf 2(z, u) + 2∂uf(z, u)

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❙♦♠❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❡♥✉♠❡r❛t✐♥❣ ❛rr❛②✳ ❍❍❍❍ ❍ ♥ ♠ ✵ ✶ ✷ ✸ ✹ ✶ ✵ ✶ ✵ ✵ ✵ ✷ ✶ ✵ ✵ ✵ ✵ ✸ ✵ ✵ ✶ ✵ ✵ ✹ ✵ ✹ ✵ ✵ ✵ ✺ ✺ ✵ ✵ ✷ ✵ ✻ ✵ ✵ ✶✻ ✵ ✵ ✼ ✵ ✺✵ ✵ ✵ ✺ ✽ ✻✵ ✵ ✵ ✻✹ ✵ ✾ ✵ ✵ ✸✺✵ ✵ ✵ ✶✵ ✵ ✾✻✵ ✵ ✵ ✷✺✻ ✶✶ ✶✶✵✺ ✵ ✵ ✷✶✵✵ ✵

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❈♦rr❡s♣♦♥❞❡♥❝❡ ✇✐t❤ tr✐✈❛❧❡♥t ♠❛♣s

❉❡✜♥✐t✐♦♥ ❯♣ t♦ ❛ ❧❛❜❡❧❧✐♥❣✱ ❛ r♦♦t❡❞ tr✐✈❛❧❡♥t ♠❛♣ ♦❢ s✐③❡ n = 6k + 2 ✐s ❛ r♦♦t ✐♥ ❏1, n❑ ❛♥❞ ❝♦✉♣❧❡ ♦❢ t✇♦ ♣❡r♠✉t❛t✐♦♥s ♦✈❡r ❏1, n❑✱ s✉❝❤ t❤❛t t❤❡ ✜rst ♦♥❡✱ ❞❡♥♦t❡❞ ❜② σ•✱ ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥ ✇✐t❤ ♥♦ ✜①❡❞ ♣♦✐♥t✱ t❤❡ s❡❝♦♥❞ ♦♥❡✱ ❞❡♥♦t❡❞ ❜② σ•✱ ✐s ♦❢ ♦r❞❡r 3 ❛♥❞ ❤❛s t✇♦ ✜①❡❞ ♣♦✐♥ts a ❛♥❞ b ✭✐♥❝❧✉❞✐♥❣ t❤❡ r♦♦t✮✱ ❛♥❞ s✉❝❤ t❤❛t σ•✱ σ• ❛♥❞ (a b) ❣❡♥❡r❛t❡ ❛ tr❛♥s✐t✐✈❡ ❣r♦✉♣✳

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❘❡♠❛r❦ ❲❡ ❝❛♥ r❡♣r❡s❡♥t r♦♦t❡❞ tr✐✈❛❧❡♥t ♠❛♣s ✇✐t❤ ❣r❛♣❤s ✇✐t❤ ❞❡❣r❡❡ 2 ♥♦❞❡s

• ❛♥❞ ❞❡❣r❡❡ 3 ♥♦❞❡s • ✇❤❡r❡ σ•✱ σ• ❛❝t ♦♥ t❤❡ ❡❞❣❡s✳

❊①❛♠♣❧❡ ρ = 7✱ σ• = (12)(38)(56)(74)✱ σ• = (123)(456)✳

• 4
• 7
• ✽
• ✸
• ✶

✷ ✺ ✻

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❚❤❡♦r❡♠ ✭❖✳❇♦❞✐♥✐✱ ❉✳ ●❛r❞②✱ ❆✳ ❏❛❝q✉♦t✱ ✷✵✶✸✮ T (f)

3k+2,0 ✐s ✐♥ ❜✐❥❡❝t✐♦♥ ✇✐t❤ tr✐✈❛❧❡♥t r♦♦t❡❞ ♠❛♣✳ ■♥ ♣❛rt✐❝✉❧❛r t❤❡ ❣❡♥❡r✲

❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❝❧♦s❡❞ t❡r♠s ✐s ❣✐✈❡♥ ❜②✿ [z3k+2, u0] f = [z6k+2]z3∂z log(ez3/3 ⊙ ez2/2) ✇❤❡r❡ ⊙ ✐s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t ✐♥ z✳

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❚❤❡♦r❡♠ ✭❈♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♦❢ ◆✳ ❩❡✐❧❜❡r❣❡r✱ ✷✵✶✼✮ ❲❡ ❝❛♥ ❣❡♥❡r❛❧✐③❡ t❤✐s ❝♦rr❡s♣♦♥❞❡♥❝❡ t♦ ♣❧❛✐♥ t❡r♠s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ T (f) ✐s ❣✐✈❡♥ ❜②✿ [z3k+c(m), um] f = [z6k+

c(m), um]z3∂z log(ez3/3+uz ⊙ ez2/2)

✇❤❡r❡ c(m) = 2 ✐❢ m = 0 mod[3]✱ 1 ✐❢ m = 1 mod[3]✱ 0 ✐❢ m = 2 mod[3]

• c(m) = 2 ✐❢ m = 0 mod[3]✱ 6 ✐❢ m = 1 mod[3]✱ 4 ✐❢ m = 2 mod[3]

❘❡♠❛r❦ ❚❤❛♥❦s t♦ ❛ t❤❡♦r❡♠ ♦❢ ❊✳ ❆✳ ❇❡♥❞❡r✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❡ s✉❜s❡q✉❡♥t ♣r♦♣♦s✐t✐♦♥ ✉s✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿ ez3/3+uz ⊙ ez2/2

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❈♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ♠❛♣s ❛♥❞ ❝❧♦s❡❞ t❡r♠s

• ■♥✐t✐❛❧ t❡r♠✿
• ≃
• ρ
• ❆♣♣❧✐❝❛t✐♦♥ ✭❋❧✐♣ ❝❤❛♥❣❡s r♦♦t ♥♦❞❡ ✐♥t♦ ❛ ❜❧❛❝❦ ♥♦❞❡ ✇✐t❤ ♥♦ r♦♦t

❡❞❣❡✮✿ ❋❧✐♣ ❆

• ❋❧✐♣ ❇

ρ ≃

• ❆

❇

• ❆❞❞ ❛♥ ❛❜str❛❝t✐♦♥ ❛❜♦✈❡ t❤❡ r♦♦t ✭✐❢ ❆ ✐s ♥♦t ❝❧♦s❡❞✮✿
• ❋❧✐♣ ❆

ρ ≃

• ❆

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

• ■♥s❡rt✐♦♥ ♦❢ ❛ ❜✐♥❛r② ♥♦❞❡ ♦♥ t❤❡ t♦♣ r✐❣❤t✿

✭❬ ♠❛r❦s t❤❡ ♥♦❞❡ ❛❜♦✈❡ ✇❤✐❝❤ ✇❡ ♠❛❦❡ t❤❡ ✐♥s❡rt✐♦♥✮ ❆❜♦✈❡ t❤❡ s♦♥ ♦❢ ❛♥ ❛❜str❛❝t✐♦♥

• ❬

− →

• ❬

❆❜♦✈❡ t❤❡ s♦♥ ♦❢ ❛♥ ❛♣♣❧✐❝❛t✐♦♥

• ❬

− →

• ❬

❘❡♠❛r❦ ❲❡ ❤❛✈❡ t♦ ❝❤❛♥❣❡ t❤❡ ❛❞❞❡❞ ❧❡❛❢ ✐♥t♦ ❛ r❡❞ ♦♥❡ ✇❤❡♥ ✇❡ ❝♦♥♥❡❝t ✐t t♦ ❛♥ ❛❜str❛❝t✐♦♥✳

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❆❜♦✈❡ t❤❡ r♦♦t ♥♦❞❡

• −

→

• ❬

❬

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❊①❛♠♣❧❡✿ (λx · x)(λx · x)

• ρ

@

• ρ

≃

• ρ
• ■♥ t❡r♠s ♦❢ ♣❡r♠✉t❛t✐♦♥s✱ ✉♣ t♦ ❛ r❡❧❛❜❡❧❧✐♥❣ ♦❢ t❤❡ ❡❞❣❡s ✇❡ ✇✐❧❧ ❤❛✈❡✿
• ✷
• 1
• ✺
• ✻
• ✼

✽ ✸ ✹ σ• = (234)(678) σ• = (12)(34)(56)(78)

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

• ❡♥❡r❛❧✐s❛t✐♦♥ t♦ ♣❧❛✐♥ t❡r♠s
• ■♥ ♣❧❛✐♥ t❡r♠s ❢r❡❡ ❧❡❛✈❡s ❛r❡ ❜❧❛❝❦ ❝♦♥tr❛r② t♦ ❝❧♦s❡❞ ❧❡❛✈❡s✱ t❤❡② ❛r❡

❛❧s♦ ❢♦❧❧♦✇❡❞ ❜② ❛♥ ❡❞❣❡✳

• ❲❡ ❛❞❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❧❡♠❡♥t✿
• ≃
• ρ
• ❆♣♣❧✐❝❛t✐♦♥✿ t♦ ♠❛❦❡ ❛ ♣r♦❞✉❝t ✇✐t❤ ❛ t❡r♠ ❝♦♥t❛✐♥✐♥❣ t❤❡ ♦❢ t❡r♠ s✐③❡

1✱ ✇❡ ❝♦♥♥❡❝t ✐t ❞✐r❡❝t❧② t♦ t❤❡ r♦♦t ♥♦❞❡ ♦❢ t❤❡ ❛♣♣❧✐❝❛t✐♦♥✳

• ❈♦♥♥❡❝t✐♥❣ ❛♥ ❛❜str❛❝t✐♦♥✿ t♦ ❝♦♥♥❡❝t ❛♥ ❛❜str❛❝t✐♦♥ t♦ ❛ ❜❧❛❝❦ ❧❡❛❢ ♦❢

❛ t❡r♠✱ ✐❢ ✐t ✐s t❤❡ t❡r♠ ♦❢ s✐③❡ 1✱ ✇❡ ❥✉st tr❛♥s❢♦r♠ ✐t ✐♥t♦ t❤❡ t❡r♠ ♦❢ s✐③❡ 2✱ ❡❧s❡ ✇❡ tr❛♥s❢♦r♠ t❤❡ ❧❡❛❢ ✐♥t♦ ❛ r❡❞ ♦♥❡ ❜❡❢♦r❡ t❤❡ ❝♦♥♥❡❝t✐♦♥✳

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❊①❛♠♣❧❡✿ x λy · y

xλy · y ≃

• ≃
• ρ

❯♣ t♦ ❛ r❡❧❛❜❡❧❧✐♥❣ ♦❢ t❤❡ ❡❞❣❡s✱ ✇❡ ❝❛♥ ❣✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥✿

• 1

2 3 4 5 6 σ• = (345) σ• = (23)(45)

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❆✈❡r❛❣❡ ✈❛❧✉❡ ❛♥❞ ✈❛r✐❛♥❝❡

Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❡①♣❡❝t❛t✐♦♥ ❛♥❞ t❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ✐♥ r❛♥❞♦♠ t❡r♠s ♦❢ s✐③❡ n ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛s②♠♣t♦t✐❝✿ EXn ∼ VXn ∼ (2n)1/3 ■t ❢♦❧❧♦✇s ❞✐r❡❝t❧② ❢r♦♠ t❤❡ s♣❡❝✐✜❝❛t✐♦♥s ♦❢ ♣❧❛✐♥ t❡r♠s ❛♥❞ ♦❢ t❡r♠s ✇✐t❤ ❛ ❣✐✈❡♥ ♥✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s✳

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❆s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥

Pr♦♣♦s✐t✐♦♥ Xn − (2n)1/3 (2n)1/6

law

− → N(0, 1) ❲❡ ❝♦♠♣✉t❡ t❤❡ ❝♦❡✣❝✐❡♥ts φn(m) = [zn, um]ez3/3+uz ⊙ ez2/2✳ ❚❤❡♥ ✇❡ s❤♦✇ t❤❛t ❛s②♠♣t♦t✐❝❛❧❧② t❤❡ ❞✐s❝r❡t❡ ❞❡r✐✈❛t✐✈❡ φn(m + 3) − φn(m) ✐s

• (2n)1/3 − m
• /(2n)1/3 t✐♠❡s φn(m)✳

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs

P❛r❛♠❡t❡rs ❊q✉❛t✐♦♥✭❝❧♦s❡❞ t❡r♠s✮ ❊q✉❛t✐♦♥✭♣❧❛✐♥ t❡r♠s✮ ❙✐③❡ ♦♥❧② f = z2 + zf2 + 2z4∂zf f = z + z2 + zf2 + 2z4∂zf ❆❜str❛❝t✐♦♥s f = uz2 + zf2 + 2uz4∂zf f = z + uz2 + zf2 + 2uz4∂zf ❋r❡❡ ✈❛r✐❛❜❧❡s ✴ f = uz + zf2 + z∂uf ❍❡❛❞ ❛❜str❛❝t✐♦♥s f = uz2 + zf2

|u=1 + 2uz4∂zf

f = z + uz2 + zf2

|u=1 + 2uz4∂zf

❘❡♠❛r❦s ✲❈❧♦s❡❞ ❧❛♠❜❞❛✲t❡r♠s ♦❢ s✐③❡ 3k + 2 ❤❛✈❡ ❡①❛❝t❧② k + 1 ❢r❡❡ ✈❛r✐❛❜❧❡s ❜❡❝❛✉s❡ ❡❛❝❤ t✐♠❡ ✇❡ ❛❞❞ ❛ ❜✐♥❛r② ♥♦❞❡ ✇❡ ❤❛✈❡ t♦ ❛❞❞ ❛♥ ❛❜str❛❝t✐♦♥✱ s♦ ✇❡ ❝❛♥ s✐♠♣❧② ✇r✐t❡ ❛ s♣❡❝✐✜❝❛t✐♦♥ ✇✐t❤ t✇♦ ♣❛r❛♠❡t❡rs✳ ✲❆s t❤❡ ♥✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ✐s s♠❛❧❧ ❝♦♠♣❛r❡❞ t♦ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s✱ ♦♥ ❛✈❡r❛❣❡✱ t❤❡ ♥✉♠❜❡r ♦❢ ❛❜str❛❝t✐♦♥s ✐♥ ♣❧❛✐♥ t❡r♠s ♦❢ s✐③❡ n ✐s ❝♦♥❝❡♥tr❛t❡❞ ♥❡❛r n/3✳

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❊①♣❡❝t❛t✐♦♥ ❛♥❞ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❛❜str❛❝t✐♦♥s

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❙♣❡❝✐✜❝❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ r❡❞❡①❡s

• s✐③❡✱ ♯❛❜str❛❝t✐♦♥s✱ ♯r❡❞❡①❡s ✐♥ ❝❧♦s❡❞ t❡r♠s

f = uz2 + zf 2 + 2(v − 1)uz4f∂zf + 2uz4∂zf + (v − 1)u2z3∂uf

• s✐③❡✱ ♯❛❜str❛❝t✐♦♥s✱ ♯r❡❞❡①❡s ✐♥ ♣❧❛✐♥ t❡r♠s

f = z + uz2 + zf 2 + 2(v − 1)uz4f∂zf + 2uz4∂zf + (v − 1)u2z3∂uf

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❉✐str✐❜✉t✐♦♥ ♦❢ r❡❞❡①❡s ✐♥ ❝❧♦s❡❞ t❡r♠s

→ ✐❢ ✇❡ ♦♥❧② ❝♦♥s✐❞❡r t❤❡ t❡r♠s ♦❢ t❤❡ ❢♦r♠ λx · (✐♥s❡rt❜✐♥Tn−3) ♦r Tn−3(λx · x) ♦r (λx · x)Tn−3✱ ✇❡ ❝❛♥ ❛♣♣r♦①✐♠❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ r❡❞❡①❡s ✐♥ ❝❧♦s❡❞ t❡r♠s ♦❢ s✐③❡ n = 3k + 2 ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❝❡ss✿ Rn = Rn−3 + Bn ✇❤❡r❡ R2 = 0 ❛♥❞ {Bn} ❛r❡ ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s ✇✐t❤ ♣❛r❛♠❡t❡r P(Bn = 1) = 1 − Tn/Tn−3 6 + Tn/Tn−3 2 ∼ 1 − e/2n 6 + e 4n✳

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss

• ❚r❛♥s❧❛t❡ ♣❛r❛♠❡t❡rs ✐♥ ❧❛♠❜❞❛✲t❡r♠s ✐♥t♦ ♣❛r❛♠❡t❡rs ✐♥ ♠❛♣s✳
• ●✐✈❡ ❛ s②st❡♠❛t✐❝ ♠❡t❤♦❞ t♦ s♦❧✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥

❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ♣❛r❛♠❡t❡r ♦❢ ✐♥t❡r❡st✳

• ❊①t❡♥❞ t❤✐s ✇♦r❦ t♦ ❛✣♥❡ t❡r♠s✳
• ❲♦r❦ ✇✐t❤ ♥♦♥ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥s ♦❢ t❡r♠s✳

◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s

❖✳ ❇♦❞✐♥✐✱ ❉✳ ●❛r❞②✱ ❆✳ ❏❛❝q✉♦t✳ ✧❆s②♠♣t♦t✐❝s ❛♥❞ r❛♥❞♦♠ s❛♠♣❧✐♥❣ ❢♦r ❇❈■ ❛♥❞ ❇❈❑ ❧❛♠❜❞❛ t❡r♠s✧ ◆✳ ❩❡✐❧❜❡r❣❡r✳ ✧▲✐♥❡❛r ❧❛♠❜❞❛ t❡r♠s ❛s ✐♥✈❛r✐❛♥ts ♦❢ r♦♦t❡❞ tr✐✈❛❧❡♥t ♠❛♣s✧ ❙✳ ❆✳ ❱✐❞❛❧✳ ✧❆♥ ♦♣t✐♠❛❧ ❛❧❣♦r✐t❤♠ t♦ ❣❡♥❡r❛t❡ r♦♦t❡❞ tr✐✈❛❧❡♥t ❞✐❛❣r❛♠s ❛♥❞ r♦♦t❡❞ tr✐❛♥❣✉❧❛r ♠❛♣✧ ❖✳ ❇♦❞✐♥✐✱ ❙✳ ❉♦✈❣❛❧✱ ❏✳ ❈♦✉rt✐❡❧✱ ❍✲❑✳ ❍✇❛♥❣✳ ✧❆s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ✐♥ r❛♥❞♦♠ ♠❛♣s✧