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❆♥❛❧②t✐❝ q✉❛♥t✉♠ ✇❡❛❦ ❝♦✐♥ ✢✐♣♣✐♥❣ ♣r♦t♦❝♦❧s ✇✐t❤ ❛r❜✐tr❛r✐❧② s♠❛❧❧ ❜✐❛s

❆t✉❧ ❙✳ ❆r♦r❛✱ ❏éré♠✐❡ ❘♦❧❛♥❞✱ ❈❤r②s♦✉❧❛ ❱❧❛❝❤♦✉ ❛r❳✐✈✿✶✾✶✶✳✶✸✷✽✸

◗❈r②♣t ✷✵✷✵

❙❡❝✉r❡ t✇♦✲♣❛rt② ❝♦♠♣✉t❛t✐♦♥

❚✇♦ ♣❛rt✐❡s ❥♦✐♥t❧② ❝♦♠♣✉t❡ ❛♥ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥ ♦♥ t❤❡✐r ✐♥♣✉ts ✇✐t❤♦✉t s❤❛r✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡✐r ✐♥♣✉ts ✇✐t❤ t❤❡ ♦t❤❡r

❈❧❛ss✐❝❛❧

❖❜❧✐✈✐♦✉s ❚r❛♥s❢❡r⇒ ❇✐t ❈♦♠♠✐t♠❡♥t ⇒ ❈♦✐♥ ❋❧✐♣♣✐♥❣

P❡r❢❡❝t s❡❝✉r✐t② ✐♠♣♦ss✐❜❧❡ ✇✐t❤♦✉t ❡①tr❛ ❛ss✉♠♣t✐♦♥s ✭❡✳❣✳ ❝♦♠♣✉t❛t✐♦♥❛❧ ❤❛r❞♥❡ss✮

◗✉❛♥t✉♠

❖❜❧✐✈✐♦✉s ❚r❛♥s❢❡r⇔ ❇✐t ❈♦♠♠✐t♠❡♥t ⇒ ❈♦✐♥ ❋❧✐♣♣✐♥❣

P❡r❢❡❝t s❡❝✉r✐t② ✐s ✐♠♣♦ss✐❜❧❡ ✭♥♦♥✲r❡❧❛t✐✈✐st✐❝✮

◗✉❛♥t✉♠ ✇❡❛❦ ❝♦✐♥ ✢✐♣♣✐♥❣ ✐s t❤❡ str♦♥❣❡st ❦♥♦✇♥ ♣r✐♠✐t✐✈❡ ✇✐t❤ ❛r❜✐tr❛r✐❧② ♣❡r❢❡❝t s❡❝✉r✐t②

❈♦✐♥ ✢✐♣♣✐♥❣1

♦✈❡r t❤❡ t❡❧❡♣❤♦♥❡

❚✇♦ ❞✐str✉st❢✉❧ ♣❛rt✐❡s✱ ❆❧✐❝❡ ❛♥❞ ❇♦❜✱ ✇✐s❤ t♦ r❡♠♦t❡❧② ❣❡♥❡r❛t❡ ❛♥ ✉♥❜✐❛s❡❞ r❛♥❞♦♠ ❜✐t✳ ◮ ❙tr♦♥❣ ❈♦✐♥ ❋❧✐♣♣✐♥❣ ✭❙❈❋✮ ❚❤❡ ♣❛rt✐❡s ❞♦ ♥♦t ❦♥♦✇ ❛ ♣r✐♦r✐ t❤❡ ♣r❡❢❡rr❡❞ ♦✉t❝♦♠❡ ♦❢ t❤❡ ♦t❤❡r ◮ ❲❡❛❦ ❈♦✐♥ ❋❧✐♣♣✐♥❣ ✭❲❈❋✮ ❚❤❡ ♣❛rt✐❡s ❤❛✈❡ ❛ ♣r✐♦r✐ ❦♥♦✇♥ ♦♣♣♦s✐t❡ ♣r❡❢❡rr❡❞ ♦✉t❝♦♠❡s

1▼✳ ❇❧✉♠✱ ❙■●❆❈❚ ◆❡✇s ✶✺✳✶✱ ♣♣✳✷✸✲✷✼ ✭✶✾✽✸✮✳

Pr♦t♦❝♦❧ ❢❡❛t✉r❡s

❍♦♥❡st ✐s ❛ ♣❧❛②❡r ✇❤♦ ❢♦❧❧♦✇s t❤❡ ♣r♦t♦❝♦❧ ❡①❛❝t❧② ❛s ❞❡s❝r✐❜❡❞✳

❆ ❇ ❋❡❛t✉r❡ Pr✭❆ ✇✐♥s✮ Pr✭❇ ✇✐♥s✮ ❍♦♥❡st ❍♦♥❡st ❈♦rr❡❝t♥❡ss PA = 1/2 PB = 1/2 ❈❤❡❛ts ❍♦♥❡st ❆ ❝❛♥ ❜✐❛s P ∗

A

1 − P ∗

A

❍♦♥❡st ❈❤❡❛ts ❇ ❝❛♥ ❜✐❛s 1 − P ∗

B

P ∗

B

❈❤❡❛ts ❈❤❡❛ts ◆♦ ♣r♦t♦❝♦❧ ✕ ✕

❆ ♣r♦t♦❝♦❧ ❤❛s ❜✐❛s ǫ ✐❢ ♥❡✐t❤❡r ♣❧❛②❡r ❝❛♥ ❢♦r❝❡ t❤❡✐r ❞❡s✐r❡❞ ♦✉t❝♦♠❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❤✐❣❤❡r t❤❛♥ 1

2 + ǫ✱ ✐✳❡✳ t❤❡ ❜✐❛s ✐s t❤❡

s♠❛❧❧❡st ǫ s✉❝❤ t❤❛t P ∗

A, P ∗ B ≤ 1 2 + ǫ✳

❇♦✉♥❞s ❛♥❞ ❜❡st ❡①♣❧✐❝✐t ♣r♦t♦❝♦❧s

❈❧❛ss✐❝❛❧

❈♦♠♣❧❡t❡❧② ✐♥s❡❝✉r❡ ǫ = 1

2✱ ✉♥❧❡ss ❡①tr❛ ❛ss✉♠♣t✐♦♥s ❛r❡ ♠❛❞❡

◗✉❛♥t✉♠

❇♦✉♥❞ Pr♦t♦❝♦❧ ❙❈❋ ǫ ≥

1 √ 2 − 1 2 1

ǫ →

1 √ 2 − 1 2 2 ❛♥❞ ǫ = 1 4 3

❲❈❋ ǫ → 04,5 ǫ =

1 10 6✱ ♥✉♠❡r✐❝❛❧❧② ǫ → 06

1❆✳ ❨✳ ❑✐t❛❡✈✱ ◗■P ✇♦r❦s❤♦♣ ✭✷✵✵✸✮✳ 2❆✳ ❈❤❛✐❧❧♦✉① ❛♥❞ ■✳ ❑❡r❡♥✐❞✐s✱ ✺✵t❤ ❋❖❈❙✱ ♣♣✳ ✺✷✼✲✺✸✸ ✭✷✵✵✾✮✳ 3❆✳ ❆♠❜❛✐♥✐s✱ ❏ ❈♦♠♣ ❛♥❞ ❙②s ❙❝✐ ✻✽✳✷✱ ♣♣✳ ✸✾✽✲✹✶✻ ✭✷✵✵✹✮✳ 4❈✳ ▼♦❝❤♦♥✱ ❛r❳✐✈✿✵✼✶✶✳✹✶✶✹ ✭✷✵✵✼✮✳ 5❉✳ ❆❤❛r♦♥♦✈✱ ❆✳ ❈❤❛✐❧❧♦✉①✱ ▼✳ ●❛♥③✱ ■✳ ❑❡r❡♥✐❞✐s ❛♥❞ ▲✳ ▼❛❣♥✐♥✱ ❙■❆▼ ❏ ❈♦♠♣ ✹✺✳✸✱ ♣♣✳

✻✸✸✲✻✼✾ ✭✷✵✶✻✮✳

6❆✳ ❙✳ ❆r♦r❛✱ ❏✳ ❘♦❧❛♥❞ ❛♥❞ ❙✳ ❲❡✐s✱ ✺✶st ❆❈▼ ❙■●❆❈❚ ❙❚❖❈✱ ♣♣✳ ✷✵✺✲✷✶✻ ✭✷✵✶✾✮✳

Pr♦t♦❝♦❧ ❞❡s❝r✐♣t✐♦♥

❆ ♥❡✇ ❢r❛♠❡✇♦r❦ ✐s ♥❡❡❞❡❞ ♣❡r♠✐tt✐♥❣ ✉s t♦ ✜♥❞ ❜♦t❤ t❤❡ ♣r♦t♦❝♦❧ ❛♥❞ ✐ts ❜✐❛s✳

❚✐♠❡✲❞❡♣❡♥❞❡♥t ♣♦✐♥t ❣❛♠❡s∗ ✭❚❉P●✮

❙❡q✉❡♥❝❡ ♦❢ ❢r❛♠❡s ✐♥❝❧✉❞✐♥❣ ♣♦✐♥ts ♦♥ x − y ♣❧❛♥❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✇❡✐❣❤ts ❛ss✐❣♥❡❞ ◮ ❙t❛rt✐♥❣ ♣♦✐♥ts✿ (0, 1) ❛♥❞ (1, 0) ✇✐t❤ p = 1/2✳ ◮ ❚r❛♥s✐t✐♦♥s ❜❡t✇❡❡♥ ❢r❛♠❡s✿

• z

pz =

• z′

pz′

• z

λz λ + z pz ≤

• z′

λz′ λ + z′ pz′, ∀λ ≥ 0 ◮ ❋✐♥❛❧ ♣♦✐♥t (β, α) ✇✐t❤ p = 1✳

∗ ▼♦❝❤♦♥ ✐♥ ❛r❳✐✈✿✵✼✶✶✳✹✶✶✹ ❛ttr✐❜✉t❡s t❤❡ ♣♦✐♥t✲❣❛♠❡ ❢♦r♠❛❧✐s♠ t♦ ❆✳ ❨✳ ❑✐t❛❡✈✳

❊①❛♠♣❧❡s ♦❢ ❛❧❧♦✇❡❞ ♠♦✈❡s

❚r❛♥s✐t✐♦♥s ❡①♣r❡ss✐❜❧❡ ❜② ♠❛tr✐❝❡s ✭❊❇▼✮

❈♦♥s✐❞❡r ❛ ❍❡r♠✐t✐❛♥ ♠❛tr✐① Z ≥ 0 ❛♥❞ ❧❡t Π[z] ❜❡ t❤❡ ♣r♦❥❡❝t♦r ♦♥ t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡ z✳ ❚❤❡♥ Z =

z zΠ[z]✳ ▲❡t |ψ ❜❡ ❛ ✈❡❝t♦r

✭♥♦t ♥❡❝❡ss❛r✐❧② ♥♦r♠❛❧✐s❡❞✮✳ ❲❡ ❞❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥ Pr♦❜[Z, |ψ] : [0, ∞) → [0, ∞) ✇✐t❤ ✜♥✐t❡ s✉♣♣♦rt ❛s Pr♦❜[Z, |ψ](z) =

• ψ|Π[z]|ψ ✐❢ z ∈ s♣❡❝tr✉♠(Z)

♦t❤❡r✇✐s❡. ▲❡t g, h : [0, ∞) → [0, ∞) ❜❡ t✇♦ ❢✉♥❝t✐♦♥s ✇✐t❤ ✜♥✐t❡ s✉♣♣♦rts✳ ❚❤❡ ❧✐♥❡ tr❛♥s✐t✐♦♥ g → h ✐s ❝❛❧❧❡❞ ❊❇▼ ✐❢ t❤❡r❡ ❡①✐st t✇♦ ♠❛tr✐❝❡s 0 ≤ G ≤ H ❛♥❞ ❛ ✈❡❝t♦r |ψ s✉❝❤ t❤❛t✿ g = Pr♦❜[G, |ψ] ❛♥❞ h = Pr♦❜[H, |ψ].

❋♦r ❡❛❝❤ ❊❇▼ ❚❉P● t❤❡r❡ ❡①✐sts ❛ ❲❈❋ ♣r♦t♦❝♦❧ ✇✐t❤ P ∗

A ≤ α, P ∗ B ≤ β✳

❚✐♠❡✲✐♥❞❡♣❡♥❞❡♥t ♣♦✐♥t ❣❛♠❡s ✭❚■P●✮

❋♦r ❛♥ ❊❇▼ tr❛♥s✐t✐♦♥ g → h✱ ✇❡ ❞❡✜♥❡ t❤❡ ❊❇▼ ❢✉♥❝t✐♦♥ g − h✳ ❚❤❡ s❡t ♦❢ ❊❇▼ ❢✉♥❝t✐♦♥s ✐s t❤❡ s❛♠❡ ✭✉♣ t♦ ❝❧♦s✉r❡s✮ ❛s t❤❡ s❡t ♦❢ ✈❛❧✐❞ ❢✉♥❝t✐♦♥s✳ ❆ ❢✉♥❝t✐♦♥ f(x) ✐s ✈❛❧✐❞ ✐❢

x f(x) = 0 ❛♥❞ x f(x) λ+x ≤ 0, ∀λ ≥ 0.

❋♦r ❡❛❝❤ ❚■P● t❤❡r❡ ❡①✐sts ❛♥ ❊❇▼ ❚❉P● ✇✐t❤ t❤❡ s❛♠❡ ✜♥❛❧ ❢r❛♠❡

❊①✐st❡♥❝❡ ♦❢ ❛ ❲❈❋ ♣r♦t♦❝♦❧ ✇✐t❤ ǫ → 01

❋❛♠✐❧② ♦❢ ❚■P●2 ❛♣♣r♦❛❝❤✐♥❣ ❜✐❛s ǫ = 1 4k + 2, ✇❤❡r❡ 2k ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ ♠❛✐♥ ♠♦✈❡ ♦❢ t❤❡ ♣♦✐♥t ❣❛♠❡

1❈✳ ▼♦❝❤♦♥✱ ❛r❳✐✈✿✵✼✶✶✳✹✶✶✹ ✭✷✵✵✼✮✳ 2P✐❝t✉r❡ ❢r♦♠ P✳ ❍ø②❡r ❛♥❞ ❊✳ P❡❧❝❤❛t✱ ▼❆ t❤❡s✐s✱ ❯♥✐✈❡rs✐t② ♦❢ ❈❛❧❣❛r② ✭✷✵✶✸✮✳

❊q✉✐✈❛❧❡♥t ❢r❛♠❡✇♦r❦s ❛♥❞ t❤❡ ♣r♦♦❢ ♦❢ ❡①✐st❡♥❝❡1,2

1❈✳ ▼♦❝❤♦♥✱ ❛r❳✐✈✿✵✼✶✶✳✹✶✶✹ ✭✷✵✵✼✮✳ 2❉✳ ❆❤❛r♦♥♦✈✱ ❆✳ ❈❤❛✐❧❧♦✉①✱ ▼✳ ●❛♥③✱ ■✳ ❑❡r❡♥✐❞✐s ❛♥❞ ▲✳ ▼❛❣♥✐♥✱ ❙■❆▼ ❏ ❈♦♠♣ ✹✺✳✸✱ ♣♣✳

✻✸✸✲✻✼✾ ✭✷✵✶✻✮✳

❚❉P●✲t♦✲❡①♣❧✐❝✐t✲♣r♦t♦❝♦❧ ❢r❛♠❡✇♦r❦ ✭❚❊❋✮1

❈♦♥✈❡rs✐♦♥ ♦❢ ❛ ❚❉P● t♦ ❛♥ ❡①♣❧✐❝✐t ❲❈❋ ♣r♦t♦❝♦❧ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❜✐❛s✱ ❣✐✈❡♥ t❤❛t ❢♦r ❡✈❡r② tr❛♥s✐t✐♦♥ ♦❢ t❤❡ ❚❉P●✱ ❛ ✉♥✐t❛r② s❛t✐s❢②✐♥❣ ❝❡rt❛✐♥ ❝♦♥str❛✐♥ts ❝❛♥ ❜❡ ❢♦✉♥❞

1❆✳ ❙✳ ❆r♦r❛✱ ❏✳ ❘♦❧❛♥❞ ❛♥❞ ❙✳ ❲❡✐s✱ ✺✶st ❆❈▼ ❙■●❆❈❚ ❙❚❖❈✱ ♣♣✳ ✷✵✺✲✷✶✻ ✭✷✵✶✾✮✳

❚❊❋ ❝♦♥str❛✐♥ts

U ✐s ❛ ✉♥✐t❛r②∗ ♠❛tr✐① ❛❝t✐♥❣ ♦♥ s♣❛♥{|g1 , |g2 , . . . , |h1 , |h2 , . . .}✱ s✳ t✳

U |v = |w ❛♥❞

nh

• i=1

xhi |hi hi|−

ng

• i=1

xgiEhU |gi gi| U †Eh ≥ 0,

✇✐t❤ |v :=

• i

√pgi |gi

√

i pgi

❛♥❞ |w :=

• i √phi |hi
• i phi

✱

• {|gi}ng

i=1, {|hinh i=1}

• ♦rt❤♦♥♦r♠❛❧ ❛♥❞ Eh := n

i=1 |hi hi|✳ ❆❧s♦✱ xgi ❛♥❞ xhi ❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢

t❤❡ ng ❛♥❞ nh ♣♦✐♥ts ♦❢ t❤❡ ✐♥✐t✐❛❧ ❛♥❞ ✜♥❛❧ ❢r❛♠❡✱ r❡s♣❡❝t✐✈❡❧②✱ ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦❜❛❜✐❧✐t② ✇❡✐❣❤ts pgi ❛♥❞ phi

❯s✐♥❣ ❚❊❋1 ❛ ♣r♦t♦❝♦❧ ✇✐t❤ ǫ =

1 10 ✇❛s ❝♦♥str✉❝t❡❞ ❛♥❛❧②t✐❝❛❧❧② ❛♥❞ ❛♥

❛❧❣♦r✐t❤♠ ✇❛s ♣r♦♣♦s❡❞ t♦ ♥✉♠❡r✐❝❛❧❧② ❝♦♥str✉❝t U ❢♦r ❧♦✇❡r ❜✐❛s

∗ ✐t ✐s s✉✣❝✐❡♥t t♦ ❝♦♥s✐❞❡r ♦rt❤♦❣♦♥❛❧ ♠❛tr✐❝❡s 1❆✳ ❙✳ ❆r♦r❛✱ ❏✳ ❘♦❧❛♥❞ ❛♥❞ ❙✳ ❲❡✐s✱ ✺✶st ❆❈▼ ❙■●❆❈❚ ❙❚❖❈✱ ♣♣✳ ✷✵✺✲✷✶✻ ✭✷✵✶✾✮✳

f− ❛ss✐❣♥♠❡♥t1

• ✐✈❡♥ ❛ s❡t ♦❢ r❡❛❧ ❝♦♦r❞✐♥❛t❡s 0 ≤ x1 < x2 · · · < xn ❛♥❞ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ ❛t

♠♦st n − 2 s❛t✐s❢②✐♥❣ f(−λ) ≥ 0 ❢♦r ❛❧❧ λ ≥ 0✱ ❛♥ f✲❛ss✐❣♥♠❡♥t ✐s ❣✐✈❡♥ ❜② t❤❡ ❢✉♥❝t✐♦♥

t =

n

• i=1

−f(xi)

• j=i(xj − xi)
• =:pi

[xi] = h − g,

✇❤❡r❡ h ❝♦♥t❛✐♥s t❤❡ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ t ❛♥❞ g t❤❡ ♥❡❣❛t✐✈❡ ♣❛rt ✭✇✐t❤♦✉t ❛♥② ❝♦♠♠♦♥ s✉♣♣♦rt✮✱ ✈✐③✳ h =

i:pi>0 pi [xi] ❛♥❞ g = i:pi<0 (−pi) [xi]✳

◮ ❆♥ ❛ss✐❣♥♠❡♥t ✐s ❜❛❧❛♥❝❡❞ ✐❢ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ✇✐t❤ ♥❡❣❛t✐✈❡ ✇❡✐❣❤ts✱ pi < 0✱ ❡q✉❛❧s t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ✇✐t❤ ♣♦s✐t✐✈❡ ✇❡✐❣❤ts✱ pi > 0✳ ❆♥ ❛ss✐❣♥♠❡♥t ✐s ✉♥❜❛❧❛♥❝❡❞ ✐❢ ✐t ✐s ♥♦t ❜❛❧❛♥❝❡❞✳ ◮ ❲❤❡♥ f ✐s ❛ ♠♦♥♦♠✐❛❧✱ ✈✐③✳ ❤❛s t❤❡ ❢♦r♠ f(x) = cxq✱ ✇❤❡r❡ c > 0 ❛♥❞ q ≥ 0✱ ✇❡ ❝❛❧❧ t❤❡ ❛ss✐❣♥♠❡♥t ❛ ♠♦♥♦♠✐❛❧ ❛ss✐❣♥♠❡♥t✳ ◮ ❆ ♠♦♥♦♠✐❛❧ ❛ss✐❣♥♠❡♥t ✐s ❛❧✐❣♥❡❞ ✐❢ t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ♠♦♥♦♠✐❛❧ ✐s ❛♥ ❡✈❡♥ ♥✉♠❜❡r ✭q = 2(b − 1), b ∈ N✮✳ ❆ ♠♦♥♦♠✐❛❧ ❛ss✐❣♥♠❡♥t ✐s ♠✐s❛❧✐❣♥❡❞ ✐❢ ✐t ✐s ♥♦t ❛❧✐❣♥❡❞✳

1❈✳ ▼♦❝❤♦♥✱ ❛r❳✐✈✿✵✼✶✶✳✹✶✶✹ ✭✷✵✵✼✮✳

❚❤❡ f−❛ss✐❣♥♠❡♥t ❛s ❛ s✉♠ ♦❢ ♠♦♥♦♠✐❛❧ ❛ss✐❣♥♠❡♥ts

❈♦♥s✐❞❡r ❛ s❡t ♦❢ r❡❛❧ ❝♦♦r❞✐♥❛t❡s s❛t✐s❢②✐♥❣ 0 ≤ x1 < x2 · · · < xn ❛♥❞ ❧❡t f(x) = (r1 − x)(r2 − x) . . . (rk − x) ✇❤❡r❡ k ≤ n − 2✳ ▲❡t t = n

i=1 pi [xi] ❜❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ f✲❛ss✐❣♥♠❡♥t✳

❚❤❡♥ t =

k

• l=0

αl n

• i=1

−(−xi)l

• j=i(xj − xi) [xi]
• ,

✇❤❡r❡ αl ≥ 0✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ αl ✐s t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ (−x)l ✐♥ f(x)✳

❙♦❧✈✐♥❣ ❛♥ ❛ss✐❣♥♠❡♥t

• ✐✈❡♥ ❛♥ f− ❛ss✐❣♥♠❡♥t t = nh

i=1 phi [xhi] − ng i=1 pgi [xgi] ❛♥❞

❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s

• |g1 , |g2 . . .
• gng
• , |h1 , |h2 . . . |hnh
• ,

✇❡ s❛② t❤❛t t❤❡ ♦rt❤♦❣♦♥❛❧ ♠❛tr✐① O s♦❧✈❡s t ✐❢ O |v = |w ❛♥❞ Xh ≥ EhOXgOT Eh, ✇❤❡r❡ |v = ng

i=1

√pgi |gi✱ |w = nh

i=1

√phi |hi✱ Xh = nh

i=1 xhi |hi hi|✱ Xg = ng i=1 xgi |gi gi| ❛♥❞

Eh = nh

i=1 |hi hi|✳

▼♦r❡♦✈❡r✱ ✇❡ s❛② t❤❛t t ❤❛s ❛♥ ❡✛❡❝t✐✈❡ s♦❧✉t✐♦♥ ✐❢ t =

i∈I t′ i

❛♥❞ t′

i ❤❛s ❛ s♦❧✉t✐♦♥ ❢♦r ❛❧❧ i ∈ I✱ ✇❤❡r❡ I ✐s ❛ ✜♥✐t❡ s❡t✳ ✹ t②♣❡s ♦❢ ♠♦♥♦♠✐❛❧ ❛ss✐❣♥♠❡♥ts✿ ❜❛❧❛♥❝❡❞✴✉♥❜❛❧❛♥❝❡❞ ✕ ❛❧✐❣♥❡❞✴♠✐s❛❧✐❣♥❡❞

❆♥❛❧②t✐❝ s♦❧✉t✐♦♥

❇❛❧❛♥❝❡❞ ❛♥❞ ❛❧✐❣♥❡❞ ♠♦♥♦♠✐❛❧ ❛ss✐❣♥♠❡♥ts

▲❡t m = 2b ∈ Z, t = n

i=1 xm hiphi

• xhi
• − n

i=1 xm gipgi [xgi] ❛ ♠♦♥♦♠✐❛❧

❛ss✐❣♥♠❡♥t ♦✈❡r 0 < x1 < x2 · · · < x2n✱ {|h1 , |h2 . . . |hn , |g1 , |g2 . . . |gn} ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s✱ ❛♥❞ Xg :=

n

• i=1

xgi |gi gi| . = ❞✐❛❣(0, 0, . . . 0

• n ③❡r♦s

, xg1, xg2 . . . xgn), Xh :=

n

• i=1

xhi |hi hi| . = ❞✐❛❣(xh1, xh2 . . . xhn, 0, 0 . . . 0

• n ③❡r♦s

), |v :=

n

• i=1

√pgi |gi . = (0, 0, . . . 0

• n ③❡r♦s

, √pg1, √pg2 . . . √pgn)T ❛♥❞

• v′

:= (Xg)b |v . |w :=

n

• i=1

√phi |hi . = (√ph1, √ph2 . . . √phn, 0, 0, . . . 0

• n ③❡r♦s

)T ❛♥❞

• w′

:= (Xh)b |w ,

❆♥❛❧②t✐❝ s♦❧✉t✐♦♥

❇❛❧❛♥❝❡❞ ❛♥❞ ❛❧✐❣♥❡❞ ♠♦♥♦♠✐❛❧ ❛ss✐❣♥♠❡♥ts

❚❤❡♥✱ O :=

n−b−1

• i=−b

Π⊥

hi(Xh)i |w′ v′| (Xg)iΠ⊥ gi

√chicgi + ❤✳❝✳

• s❛t✐s✜❡s

Xh ≥ EhOXgOT Eh and EhO

• v′

=

• w′

, ✇❤❡r❡ Eh := n

i=1 |hi hi|✱ chi := w′| (Xh)iΠ⊥ hi(Xh)i |w′✱ ❛♥❞ Π⊥

hi :=

     ♣r♦❥❡❝t♦r ♦rt❤♦❣♦♥❛❧ t♦ s♣❛♥{(Xh)−|i|+1 w′ , (Xh)−|i|+2 w′ . . . ,

• w′

} i < 0 ♣r♦❥❡❝t♦r ♦rt❤♦❣♦♥❛❧ t♦ s♣❛♥{(Xh)−b w′ , (Xh)−b+1 w′ , . . . (Xh)i−1 w′ } i > 0 I i = 0.

❆♥❛❧♦❣♦✉s ❛r❡ t❤❡ ❢♦r♠s ♦❢ Π⊥

gi ❛♥❞ cgi✳

❚❤❡ ❡①♣r❡ss✐♦♥s ❢♦r t❤❡ s♦❧✉t✐♦♥ O ❢♦r t❤❡ ♦t❤❡r ♣♦ss✐❜❧❡ t②♣❡s ♦❢ ♠♦♥♦♠✐❛❧ ❛ss✐❣♥♠❡♥ts ❛r❡ s✐♠✐❧❛r

❆♥❛❧②t✐❝ s♦❧✉t✐♦♥

❇❛❧❛♥❝❡❞ ❛♥❞ ❛❧✐❣♥❡❞ ♠♦♥♦♠✐❛❧ ❛ss✐❣♥♠❡♥ts

❙✉♠♠❛r② ❛♥❞ ❝♦♥❝❧✉s✐♦♥s

◮ ❆♥❛❧②t✐❝❛❧ ❝♦♥str✉❝t✐♦♥ ♦❢ ❲❈❋ ♣r♦t♦❝♦❧s ✇✐t❤ ❛r❜✐tr❛r✐❧② ❝❧♦s❡ t♦ ③❡r♦ ❜✐❛s ◮ ❖✉r ❛♣♣r♦❛❝❤ ✐s s✐♠♣❧❡r ❛s ✐t ❛✈♦✐❞s t❤❡ ✕ q✉✐t❡ t❡❝❤♥✐❝❛❧ ✕ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ❢r♦♠ ❊❇▼ t♦ ✈❛❧✐❞ ❢✉♥❝t✐♦♥s ◮ ❆♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥s ✐♥ ❢❡✇❡r ❞✐♠❡♥s✐♦♥s❄

❖♣❡♥ q✉❡st✐♦♥s

◮ Pr♦t♦❝♦❧s ❢♦r t❤❡ P❡❧❝❤❛t✲❍ø②❡r ❢❛♠✐❧②1 ♦❢ ♣♦✐♥t ❣❛♠❡s❄ ◮ ●✐✈❡♥ t❤❡ r❡❝❡♥t ❜♦✉♥❞ ♦♥ t❤❡ r♦✉♥❞s ♦❢ ❝♦♠♠✉♥✐❝❛t✐♦♥2✱ ❝❛♥ ✇❡ ✜♥❞ ♣r♦t♦❝♦❧s ♠❛t❝❤✐♥❣ t❤❡ ❜♦✉♥❞s ♦♥ r❡s♦✉r❝❡s❄ ◮ ◆♦✐s❡ r♦❜✉st♥❡ss ♦❢ t❤❡ ♣r♦t♦❝♦❧s✳ ◮ ❉❡✈✐❝❡ ✐♥❞❡♣❡♥❞❡♥t ♣r♦t♦❝♦❧s3

1P✳ ❍ø②❡r ❛♥❞ ❊✳ P❡❧❝❤❛t✱ ▼❆ t❤❡s✐s✱ ❯♥✐✈❡rs✐t② ♦❢ ❈❛❧❣❛r② ✭✷✵✶✸✮✳ 2❈✳ ❆✳ ▼✐❧❧❡r✱ ✺✷♥❞ ❆❈▼ ❙■●❆❈❚ ❙❚❖❈✱ ♣♣✳ ✾✶✻✲✾✷✾ ✭✷✵✷✵✮✳ 3◆✳ ❆❤❛r♦♥✱ ❆✳ ❈❤❛✐❧❧♦✉①✱ ■✳ ❑❡r❡♥✐❞✐s✱ ❙✳ ▼❛ss❛r✱ ❙✳ P✐r♦♥✐♦ ❛♥❞ ❏✳ ❙✐❧♠❛♥✱ ✻t❤ ❚◗❈ ✭✷✵✶✶✮✳

❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts

❲❡ ❛r❡ t❤❛♥❦❢✉❧ t♦ ❚♦♠ ❱❛♥ ❍✐♠❜❡❡❝❦✱ ❑✐s❤♦r ❇❤❛rt✐✱ ❙t❡❢❛♥♦ P✐r♦♥✐♦ ❛♥❞ ❖❣♥②❛♥ ❖r❡s❤❦♦✈ ❢♦r ✈❛r✐♦✉s ✐♥s✐❣❤t❢✉❧ ❞✐s❝✉ss✐♦♥s✳ ❲❡ ❛❝❦♥♦✇❧❡❞❣❡ s✉♣♣♦rt ❢r♦♠ t❤❡ ❇❡❧❣✐❛♥ ❋♦♥❞s ❞❡ ❧❛ ❘❡❝❤❡r❝❤❡ ❙❝✐❡♥t✐✜q✉❡ ✕ ❋◆❘❙ ✉♥❞❡r ❣r❛♥t ♥♦ ❘✳✺✵✳✵✺✳✶✽✳❋ ✭◗✉❛♥t❆❧❣♦✮✳ ❚❤❡ ◗✉❛♥t❆❧❣♦ ♣r♦❥❡❝t ❤❛s r❡❝❡✐✈❡❞ ❢✉♥❞✐♥❣ ❢r♦♠ t❤❡ ◗✉❛♥t❊❘❆ ❊❘❆✲◆❊❚ ❈♦❢✉♥❞ ✐♥ ◗✉❛♥t✉♠ ❚❡❝❤♥♦❧♦❣✐❡s ✐♠♣❧❡♠❡♥t❡❞ ✇✐t❤✐♥ t❤❡ ❊✉r♦♣❡❛♥ ❯♥✐♦♥✬s ❍♦r✐③♦♥ ✷✵✷✵ Pr♦❣r❛♠♠❡✳ ❆❙❆ ❢✉rt❤❡r ❛❝❦♥♦✇❧❡❞❣❡s t❤❡ ❋◆❘❙ ❢♦r s✉♣♣♦rt t❤r♦✉❣❤ t❤❡ ❋❘■❆ ❣r❛♥ts✱ ✸✴✺✴✺ ✕ ▼❈❋✴❳❍✴❋❈ ✕ ✶✻✼✺✹ ❛♥❞ ❋ ✸✴✺✴✺ ✕ ❋❘■❆✴❋❈ ✕ ✻✼✵✵ ❋❈ ✷✵✼✺✾✳