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■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ♦❢ ♣♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠s ❛♥❞ ❝♦rr❡❝t✐♦♥ ♦❢ ✈❡❝t♦r ✜❡❧❞s

❏♦r❞② P❛❧❛❢♦① ✭❆ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❏❛❝❦② ❈r❡ss♦♥✮

❏♦✉r♥é❡s ◆❛t✐♦♥❛❧❡s ❞❡ ❈❛❧❝✉❧ ❋♦r♠❡❧ ✷✵✶✼ ❈■❘▼

✶✻✲✷✵ ❏❛♥✉❛r② ✷✵✶✼

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

✶

■♥tr♦❞✉❝t✐♦♥ ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

✷

Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡

• ❡♥❡r❛❧ ♥♦t❛t✐♦♥s

❖✉r r❡s✉❧ts ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ■❧❧✉str❛t✐♦♥s ♦❢ ♦✉r t❤❡♦r❡♠s

✸

Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

■♥tr♦❞✉❝t✐♦♥

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✸ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❲❡ ❝♦♥s✐❞❡r t❤❡ ❝♦♠♣❧❡① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ r❡❛❧ ♣❧❛♥❛r ✈❡❝t♦r ✜❡❧❞ ✇✐t❤ ❛ ❝❡♥t❡r ✐♥ ✵ ❞❡♥♦t❡❞ ❜② Xlin = i(x∂x − y∂y) ✇✐t❤ x, y ∈ C ✇✐t❤ y = ¯ x✳

❋✐❣✉r❡ ✕ ❚❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣♦✐♥t ✵ ✐s ❛ ❝❡♥t❡r✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✹ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❲❤✐❝❤ ♣r♦♣❡rt✐❡s ❛r❡ ♣r❡s❡r✈❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ♣❡rt✉r❜❛t✐♦♥ ♦❢ t❤✐s ✜❡❧❞ ❄ X = Xlin + P(x, y)∂x + Q(x, y)∂y ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ ❛♥❞ ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ♣r♦♣❡rt② t♦ ❜❡ ❛ ❝❡♥t❡r ❄ ❆ ❝❡♥t❡r ✐s ✐s♦❝❤r♦♥♦✉s ✐❢ ❛❧❧ t❤❡ ♦r❜✐ts ❤❛✈❡ t❤❡ s❛♠❡ ♣❡r✐♦❞✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ✐s♦❝❤r♦♥♦✉s ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ ❛♥❞ ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ✐s♦❝❤r♦♥✐❝✐t② ❄

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✺ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❲❤✐❝❤ ♣r♦♣❡rt✐❡s ❛r❡ ♣r❡s❡r✈❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ♣❡rt✉r❜❛t✐♦♥ ♦❢ t❤✐s ✜❡❧❞ ❄ X = Xlin + P(x, y)∂x + Q(x, y)∂y ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ P ❛♥❞ Q ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ♣r♦♣❡rt② t♦ ❜❡ ❛ ❝❡♥t❡r ❄ ❆ ❝❡♥t❡r ✐s ✐s♦❝❤r♦♥♦✉s ✐❢ ❛❧❧ t❤❡ ♦r❜✐ts ❤❛✈❡ t❤❡ s❛♠❡ ♣❡r✐♦❞✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ✐s♦❝❤r♦♥♦✉s ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ ❛♥❞ ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ✐s♦❝❤r♦♥✐❝✐t② ❄

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✺ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❲❤✐❝❤ ♣r♦♣❡rt✐❡s ❛r❡ ♣r❡s❡r✈❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ♣❡rt✉r❜❛t✐♦♥ ♦❢ t❤✐s ✜❡❧❞ ❄ X = Xlin + P(x, y)∂x + Q(x, y)∂y ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ P ❛♥❞ Q ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ♣r♦♣❡rt② t♦ ❜❡ ❛ ❝❡♥t❡r ❄ ❆ ❝❡♥t❡r ✐s ✐s♦❝❤r♦♥♦✉s ✐❢ ❛❧❧ t❤❡ ♦r❜✐ts ❤❛✈❡ t❤❡ s❛♠❡ ♣❡r✐♦❞✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ✐s♦❝❤r♦♥♦✉s ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ ❛♥❞ ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ✐s♦❝❤r♦♥✐❝✐t② ❄

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✺ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❲❤✐❝❤ ♣r♦♣❡rt✐❡s ❛r❡ ♣r❡s❡r✈❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ♣❡rt✉r❜❛t✐♦♥ ♦❢ t❤✐s ✜❡❧❞ ❄ X = Xlin + P(x, y)∂x + Q(x, y)∂y ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ P ❛♥❞ Q ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ♣r♦♣❡rt② t♦ ❜❡ ❛ ❝❡♥t❡r ❄ ❆ ❝❡♥t❡r ✐s ✐s♦❝❤r♦♥♦✉s ✐❢ ❛❧❧ t❤❡ ♦r❜✐ts ❤❛✈❡ t❤❡ s❛♠❡ ♣❡r✐♦❞✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ✐s♦❝❤r♦♥♦✉s ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ P ❛♥❞ Q ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ✐s♦❝❤r♦♥✐❝✐t② ❄

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✺ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

■❢ X ✐s ❛❧s♦ ❍❛♠✐❧t♦♥✐❛♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❥❡❝t✉r❡ ✿ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ✭✷✵✵✷✮ ✶ ❊✈❡r② ❝❡♥t❡r ♦❢ ❛ r❡❛❧ ♣❧❛♥❛r ♣♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ ♦❢ ❡✈❡♥ ❞❡❣r❡❡ ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ▲♦✉❞ ✭✶✾✻✹✮ ✿ tr✉❡ ❢♦r q✉❛❞r❛t✐❝ s②st❡♠s ✱ ❇✳❙❝❤✉♠❛♥ ✭✷✵✵✶✮ ✿ tr✉❡ ✐♥ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝❛s❡✱ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t ✭✷✵✵✷✮ ✿ tr✉❡ ✐♥ t❤❡ q✉❛rt✐❝ ❝❛s❡✱ ❖t❤❡r ❝❛s❡s ✿ t❤❡ ❝♦♥❥❡❝t✉r❡ ✐s ♦♣❡♥ ✦

✶✳ ❳✳❏❛rq✉❡ ❛♥❞ ❏✳❱✐❧❧❛❞❡❧♣r❛t ✱ ✧◆♦♥❡①✐st❡♥❝❡ ♦❢ ■s♦❝❤r♦♥♦✉s ❈❡♥t❡rs ✐♥ P❧❛✲ ♥❛r P♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ ❙②st❡♠s ♦❢ ❉❡❣r❡❡ ❋♦✉r✧✱ ❏♦✉r♥❛❧ ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✶✽✵✱ ✸✸✹✕✸✼✸✱ ✷✵✵✷

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

■❢ X ✐s ❛❧s♦ ❍❛♠✐❧t♦♥✐❛♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❥❡❝t✉r❡ ✿ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ✭✷✵✵✷✮ ✶ ❊✈❡r② ❝❡♥t❡r ♦❢ ❛ r❡❛❧ ♣❧❛♥❛r ♣♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ ♦❢ ❡✈❡♥ ❞❡❣r❡❡ ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ▲♦✉❞ ✭✶✾✻✹✮ ✿ tr✉❡ ❢♦r q✉❛❞r❛t✐❝ s②st❡♠s ✱ ❇✳❙❝❤✉♠❛♥ ✭✷✵✵✶✮ ✿ tr✉❡ ✐♥ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝❛s❡✱ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t ✭✷✵✵✷✮ ✿ tr✉❡ ✐♥ t❤❡ q✉❛rt✐❝ ❝❛s❡✱ ❖t❤❡r ❝❛s❡s ✿ t❤❡ ❝♦♥❥❡❝t✉r❡ ✐s ♦♣❡♥ ✦

✶✳ ❳✳❏❛rq✉❡ ❛♥❞ ❏✳❱✐❧❧❛❞❡❧♣r❛t ✱ ✧◆♦♥❡①✐st❡♥❝❡ ♦❢ ■s♦❝❤r♦♥♦✉s ❈❡♥t❡rs ✐♥ P❧❛✲ ♥❛r P♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ ❙②st❡♠s ♦❢ ❉❡❣r❡❡ ❋♦✉r✧✱ ❏♦✉r♥❛❧ ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✶✽✵✱ ✸✸✹✕✸✼✸✱ ✷✵✵✷

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

■❢ X ✐s ❛❧s♦ ❍❛♠✐❧t♦♥✐❛♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❥❡❝t✉r❡ ✿ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ✭✷✵✵✷✮ ✶ ❊✈❡r② ❝❡♥t❡r ♦❢ ❛ r❡❛❧ ♣❧❛♥❛r ♣♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ ♦❢ ❡✈❡♥ ❞❡❣r❡❡ ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ▲♦✉❞ ✭✶✾✻✹✮ ✿ tr✉❡ ❢♦r q✉❛❞r❛t✐❝ s②st❡♠s ✱ ❇✳❙❝❤✉♠❛♥ ✭✷✵✵✶✮ ✿ tr✉❡ ✐♥ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝❛s❡✱ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t ✭✷✵✵✷✮ ✿ tr✉❡ ✐♥ t❤❡ q✉❛rt✐❝ ❝❛s❡✱ ❖t❤❡r ❝❛s❡s ✿ t❤❡ ❝♦♥❥❡❝t✉r❡ ✐s ♦♣❡♥ ✦

✶✳ ❳✳❏❛rq✉❡ ❛♥❞ ❏✳❱✐❧❧❛❞❡❧♣r❛t ✱ ✧◆♦♥❡①✐st❡♥❝❡ ♦❢ ■s♦❝❤r♦♥♦✉s ❈❡♥t❡rs ✐♥ P❧❛✲ ♥❛r P♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ ❙②st❡♠s ♦❢ ❉❡❣r❡❡ ❋♦✉r✧✱ ❏♦✉r♥❛❧ ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✶✽✵✱ ✸✸✹✕✸✼✸✱ ✷✵✵✷

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

■❢ X ✐s ❛❧s♦ ❍❛♠✐❧t♦♥✐❛♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❥❡❝t✉r❡ ✿ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ✭✷✵✵✷✮ ✶ ❊✈❡r② ❝❡♥t❡r ♦❢ ❛ r❡❛❧ ♣❧❛♥❛r ♣♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ ♦❢ ❡✈❡♥ ❞❡❣r❡❡ ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ▲♦✉❞ ✭✶✾✻✹✮ ✿ tr✉❡ ❢♦r q✉❛❞r❛t✐❝ s②st❡♠s ✱ ❇✳❙❝❤✉♠❛♥ ✭✷✵✵✶✮ ✿ tr✉❡ ✐♥ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝❛s❡✱ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t ✭✷✵✵✷✮ ✿ tr✉❡ ✐♥ t❤❡ q✉❛rt✐❝ ❝❛s❡✱ ❖t❤❡r ❝❛s❡s ✿ t❤❡ ❝♦♥❥❡❝t✉r❡ ✐s ♦♣❡♥ ✦

✶✳ ❳✳❏❛rq✉❡ ❛♥❞ ❏✳❱✐❧❧❛❞❡❧♣r❛t ✱ ✧◆♦♥❡①✐st❡♥❝❡ ♦❢ ■s♦❝❤r♦♥♦✉s ❈❡♥t❡rs ✐♥ P❧❛✲ ♥❛r P♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ ❙②st❡♠s ♦❢ ❉❡❣r❡❡ ❋♦✉r✧✱ ❏♦✉r♥❛❧ ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✶✽✵✱ ✸✸✹✕✸✼✸✱ ✷✵✵✷

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

■❢ X ✐s ❛❧s♦ ❍❛♠✐❧t♦♥✐❛♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❥❡❝t✉r❡ ✿ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ✭✷✵✵✷✮ ✶ ❊✈❡r② ❝❡♥t❡r ♦❢ ❛ r❡❛❧ ♣❧❛♥❛r ♣♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ ♦❢ ❡✈❡♥ ❞❡❣r❡❡ ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ▲♦✉❞ ✭✶✾✻✹✮ ✿ tr✉❡ ❢♦r q✉❛❞r❛t✐❝ s②st❡♠s ✱ ❇✳❙❝❤✉♠❛♥ ✭✷✵✵✶✮ ✿ tr✉❡ ✐♥ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝❛s❡✱ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t ✭✷✵✵✷✮ ✿ tr✉❡ ✐♥ t❤❡ q✉❛rt✐❝ ❝❛s❡✱ ❖t❤❡r ❝❛s❡s ✿ t❤❡ ❝♦♥❥❡❝t✉r❡ ✐s ♦♣❡♥ ✦

✶✳ ❳✳❏❛rq✉❡ ❛♥❞ ❏✳❱✐❧❧❛❞❡❧♣r❛t ✱ ✧◆♦♥❡①✐st❡♥❝❡ ♦❢ ■s♦❝❤r♦♥♦✉s ❈❡♥t❡rs ✐♥ P❧❛✲ ♥❛r P♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ ❙②st❡♠s ♦❢ ❉❡❣r❡❡ ❋♦✉r✧✱ ❏♦✉r♥❛❧ ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✶✽✵✱ ✸✸✹✕✸✼✸✱ ✷✵✵✷

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❈♦♥❞✐t✐♦♥ ♦❢ ■s♦❝❤r♦♥✐❝✐t② ✷ ❚❤❡ ✐s♦❝❤r♦♥✐❝✐t② ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❧✐♥❡❛r✐③❛❜✐❧✐t②✳ ❍♦✇ t♦ st✉❞② t❤❡ ❧✐♥❡❛r✐③❛❜✐❧✐t② ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❄

✷✳ ❙❛❜❛t✐♥✐ ❛♥❞ ❈❤❛✈❛rr✐❣❛ ✱ ✧❆ s✉r✈❡② ♦❢ ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs✧✱ ◗✉❛❧✐t❛t✐✈❡ ❚❤❡♦r② ♦❢ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ✶ ✭✶✾✾✾✮

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✼ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❈♦♥❞✐t✐♦♥ ♦❢ ■s♦❝❤r♦♥✐❝✐t② ✷ ❚❤❡ ✐s♦❝❤r♦♥✐❝✐t② ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❧✐♥❡❛r✐③❛❜✐❧✐t②✳ ❍♦✇ t♦ st✉❞② t❤❡ ❧✐♥❡❛r✐③❛❜✐❧✐t② ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❄

✷✳ ❙❛❜❛t✐♥✐ ❛♥❞ ❈❤❛✈❛rr✐❣❛ ✱ ✧❆ s✉r✈❡② ♦❢ ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs✧✱ ◗✉❛❧✐t❛t✐✈❡ ❚❤❡♦r② ♦❢ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ✶ ✭✶✾✾✾✮

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✼ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❈♦rr❡❝t✐♦♥ ❛♥❞ ♠♦✉❧❞ ❝❛❧❝✉❧✉s ❋♦r♠❛❧✐s♠ ✿ ▼♦✉❧❞ ❝❛❧❝✉❧✉s ✐♥tr♦❞✉❝❡❞ ❜② ❏✳❊❝❛❧❧❡ ✐♥ ✼✵✬s✳ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ✿ ❛ ❢♦r♠❛❧ ✈❡❝t♦r ✜❡❧❞ ❞❡✜♥❡❞ ❜② ❏✳❊❝❛❧❧❡ ❛♥❞ ❇✳❱❛❧❧❡t ✸ ✿ ❉❡✜♥✐t✐♦♥ ♦❢ ❈♦rr❡❝t✐♦♥ ❛♥❛❧②t✐❝ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ ❂❧✐♥❡❛r ♣❛rt ♦❢ ❋✐♥❞ ❛ ✈❡❝t♦r ✜❡❧❞ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♠✉t✐♥❣ ♣r♦❜❧❡♠ ✿ ❢♦r♠❛❧❧② ❝♦♥❥✉❣❛t❡ t♦ ✱ ✵✱ ❚❤❡ s♦❧✉t✐♦♥ ✐s t❤❡ ❝♦rr❡❝t✐♦♥ ♦❢ ✳

✸✳ ❏✳❊❝❛❧❧❡ ❛♥❞ ❇✳❱❛❧❧❡t✱ ✧❈♦rr❡❝t✐♦♥ ❛♥❞ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ r❡s♦♥❛♥t ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ❞✐✛❡♦♠♦r♣❤✐s♠s✧✱ ▼❛t❤✳ ❩✳ ✷✷✾✱ ✷✹✾✲✸✶✽ ✭✶✾✾✽✮

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✽ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❈♦rr❡❝t✐♦♥ ❛♥❞ ♠♦✉❧❞ ❝❛❧❝✉❧✉s ❋♦r♠❛❧✐s♠ ✿ ▼♦✉❧❞ ❝❛❧❝✉❧✉s ✐♥tr♦❞✉❝❡❞ ❜② ❏✳❊❝❛❧❧❡ ✐♥ ✼✵✬s✳ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ✿ ❛ ❢♦r♠❛❧ ✈❡❝t♦r ✜❡❧❞ ❞❡✜♥❡❞ ❜② ❏✳❊❝❛❧❧❡ ❛♥❞ ❇✳❱❛❧❧❡t ✸ ✿ ❉❡✜♥✐t✐♦♥ ♦❢ ❈♦rr❡❝t✐♦♥ ❛♥❛❧②t✐❝ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ ❂❧✐♥❡❛r ♣❛rt ♦❢ ❋✐♥❞ ❛ ✈❡❝t♦r ✜❡❧❞ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♠✉t✐♥❣ ♣r♦❜❧❡♠ ✿ ❢♦r♠❛❧❧② ❝♦♥❥✉❣❛t❡ t♦ ✱ ✵✱ ❚❤❡ s♦❧✉t✐♦♥ ✐s t❤❡ ❝♦rr❡❝t✐♦♥ ♦❢ ✳

✸✳ ❏✳❊❝❛❧❧❡ ❛♥❞ ❇✳❱❛❧❧❡t✱ ✧❈♦rr❡❝t✐♦♥ ❛♥❞ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ r❡s♦♥❛♥t ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ❞✐✛❡♦♠♦r♣❤✐s♠s✧✱ ▼❛t❤✳ ❩✳ ✷✷✾✱ ✷✹✾✲✸✶✽ ✭✶✾✾✽✮

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✽ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❈♦rr❡❝t✐♦♥ ❛♥❞ ♠♦✉❧❞ ❝❛❧❝✉❧✉s ❋♦r♠❛❧✐s♠ ✿ ▼♦✉❧❞ ❝❛❧❝✉❧✉s ✐♥tr♦❞✉❝❡❞ ❜② ❏✳❊❝❛❧❧❡ ✐♥ ✼✵✬s✳ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ✿ ❛ ❢♦r♠❛❧ ✈❡❝t♦r ✜❡❧❞ ❞❡✜♥❡❞ ❜② ❏✳❊❝❛❧❧❡ ❛♥❞ ❇✳❱❛❧❧❡t ✸ ✿ ❉❡✜♥✐t✐♦♥ ♦❢ ❈♦rr❡❝t✐♦♥ X ❛♥❛❧②t✐❝ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ Xlin❂❧✐♥❡❛r ♣❛rt ♦❢ X ❋✐♥❞ ❛ ✈❡❝t♦r ✜❡❧❞ Z ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♠✉t✐♥❣ ♣r♦❜❧❡♠ ✿ X − Z ❢♦r♠❛❧❧② ❝♦♥❥✉❣❛t❡ t♦ Xlin✱ [Xlin, Z] = ✵✱ ❚❤❡ s♦❧✉t✐♦♥ Z ✐s t❤❡ ❝♦rr❡❝t✐♦♥ ♦❢ X✳

✸✳ ❏✳❊❝❛❧❧❡ ❛♥❞ ❇✳❱❛❧❧❡t✱ ✧❈♦rr❡❝t✐♦♥ ❛♥❞ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ r❡s♦♥❛♥t ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ❞✐✛❡♦♠♦r♣❤✐s♠s✧✱ ▼❛t❤✳ ❩✳ ✷✷✾✱ ✷✹✾✲✸✶✽ ✭✶✾✾✽✮

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✽ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❈r✐t❡r✐♦♥ ♦❢ ❧✐♥❡❛r✐③❛❜✐❧✐t② ❬❊❝❛❧❧❡✱❱❛❧❧❡t❪ ❆ ✈❡❝t♦r ✜❡❧❞ ✐s ❧✐♥❡❛r✐③❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❝♦rr❡❝t✐♦♥ ✐s ③❡r♦✳ ❚❤❡ ✐♥t❡r❡st ♦❢ t❤✐s ❢♦r♠❛❧✐s♠ ✿ ❆♥ ❛❧❣♦r✐t❤♠✐❝ ❛♥❞ ❡①♣❧✐❝✐t ✇❛② t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ❧✐♥❡❛r✐③❛❜✐❧✐t②✳ ❚♦ ❞✐st✐♥❣✉✐s❤ ✇❤❛t ❞❡♣❡♥❞s ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ❛♥❞ ❛♥❞ ✇❤❛t ✐s ✉♥✐✈❡rs❛❧ ❢♦r t❤❡ ❧✐♥❡❛r✐③❛❜✐❧✐t②✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❈r✐t❡r✐♦♥ ♦❢ ❧✐♥❡❛r✐③❛❜✐❧✐t② ❬❊❝❛❧❧❡✱❱❛❧❧❡t❪ ❆ ✈❡❝t♦r ✜❡❧❞ ✐s ❧✐♥❡❛r✐③❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❝♦rr❡❝t✐♦♥ ✐s ③❡r♦✳ ❚❤❡ ✐♥t❡r❡st ♦❢ t❤✐s ❢♦r♠❛❧✐s♠ ✿ ❆♥ ❛❧❣♦r✐t❤♠✐❝ ❛♥❞ ❡①♣❧✐❝✐t ✇❛② t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ❧✐♥❡❛r✐③❛❜✐❧✐t②✳ ❚♦ ❞✐st✐♥❣✉✐s❤ ✇❤❛t ❞❡♣❡♥❞s ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ❛♥❞ ❛♥❞ ✇❤❛t ✐s ✉♥✐✈❡rs❛❧ ❢♦r t❤❡ ❧✐♥❡❛r✐③❛❜✐❧✐t②✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s

❈r✐t❡r✐♦♥ ♦❢ ❧✐♥❡❛r✐③❛❜✐❧✐t② ❬❊❝❛❧❧❡✱❱❛❧❧❡t❪ ❆ ✈❡❝t♦r ✜❡❧❞ ✐s ❧✐♥❡❛r✐③❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❝♦rr❡❝t✐♦♥ ✐s ③❡r♦✳ ❚❤❡ ✐♥t❡r❡st ♦❢ t❤✐s ❢♦r♠❛❧✐s♠ ✿ ❆♥ ❛❧❣♦r✐t❤♠✐❝ ❛♥❞ ❡①♣❧✐❝✐t ✇❛② t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ❧✐♥❡❛r✐③❛❜✐❧✐t②✳ ❚♦ ❞✐st✐♥❣✉✐s❤ ✇❤❛t ❞❡♣❡♥❞s ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ P ❛♥❞ Q ❛♥❞ ✇❤❛t ✐s ✉♥✐✈❡rs❛❧ ❢♦r t❤❡ ❧✐♥❡❛r✐③❛❜✐❧✐t②✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

• ❡♥❡r❛❧ ♥♦t❛t✐♦♥s

❖✉r r❡s✉❧ts ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ■❧❧✉str❛t✐♦♥s ♦❢ ♦✉r t❤❡♦r❡♠s

❖✉r r❡s✉❧ts

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✵ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

• ❡♥❡r❛❧ ♥♦t❛t✐♦♥s

❖✉r r❡s✉❧ts ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ■❧❧✉str❛t✐♦♥s ♦❢ ♦✉r t❤❡♦r❡♠s

❲❡ ❝♦♥s✐❞❡r ❛ ♣♦❧②♥♦♠✐❛❧ ♣❡rt✉r❜❛t✐♦♥ ❛s ❛❜♦✈❡ ✿ X = Xlin +

l

• r=k

Xr ✇✐t❤ Xr = Pr(x, y)∂x + Qr(x, y)∂y✱ Pr(x, y) =

r

• j=✵

pr−j−✶,jxr−jyj✱ Qr(x, y) =

r

• j=✵

qr−j,j−✶xr−jyj✳ pr−j−✶,j, qr−j,j−✶ ∈ C ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ✿ ❘❡❛❧ s②st❡♠ ❝♦♥❞✐t✐♦♥ ✿ pj,k = ¯ qk,j ✇✐t❤ j + k = r − ✶ ❍❛♠✐❧t♦♥✐❛♥ ❝♦♥❞✐t✐♦♥ ✿ pj−✶,r−j = − r−j+✶

j

qj−✶,r−j ✇✐t❤ j = ✶, ...r✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✶ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

• ❡♥❡r❛❧ ♥♦t❛t✐♦♥s

❖✉r r❡s✉❧ts ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ■❧❧✉str❛t✐♦♥s ♦❢ ♦✉r t❤❡♦r❡♠s

❚❤❡♦r❡♠ ✶ ❬P✳✱❈r❡ss♦♥❪ ▲❡t X ❜❡ ❛ r❡❛❧ ❍❛♠✐❧t♦♥✐❛♥ ✈❡❝t♦r ✜❡❧❞ ♦❢ ❡✈❡♥ ❞❡❣r❡❡ ✷♥ ❣✐✈❡♥ ❜② ✿ X = Xlin +

✷n

• r=✷

Xr ■❢ X s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ✿

✶ t❤❡r❡ ❡①✐sts ✶ ≤ k < n − ✶ s✉❝❤ t❤❛t pi,i = ✵ ❢♦r

i = ✶, ..., k − ✶ ❛♥❞ Im(pk,k) > ✵✱

✷ pi,i = ✵ ❢♦r i = ✶, ..., n − ✶✱

❚❤❡♥ t❤❡ ✈❡❝t♦r ✜❡❧❞ X ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✷ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

• ❡♥❡r❛❧ ♥♦t❛t✐♦♥s

❖✉r r❡s✉❧ts ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ■❧❧✉str❛t✐♦♥s ♦❢ ♦✉r t❤❡♦r❡♠s

❚❤❡♦r❡♠ ✷ ❬P✳✱❈r❡ss♦♥❪ ❆ r❡❛❧ ❍❛♠✐❧t♦♥✐❛♥ ✈❡❝t♦r ✜❡❧❞ ♦❢ t❤❡ ❢♦r♠ ✿ X = Xlin + Xk + ... + X✷n, ❢♦r k ≥ ✷ ❛♥❞ n ≤ k − ✶✱ ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✸ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

• ❡♥❡r❛❧ ♥♦t❛t✐♦♥s

❖✉r r❡s✉❧ts ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ■❧❧✉str❛t✐♦♥s ♦❢ ♦✉r t❤❡♦r❡♠s

❇② t❤❡ ❚❤❡♦r❡♠ ✶✱ ✇❡ ❤❛✈❡ ✿ X = Xlin + X✷ ✱

✷ ✸ ✹ ✇✐t❤ ✶ ✶

✵✱

✷ ✸ ✹ ✺ ✻ ✇✐t❤ ✶ ✶

✵ ♦r

✶ ✶

✵ ❛♥❞

✷ ✷

✵ ❡t❝✳✳✳ ❛r❡ ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ❇② t❤❡ ❚❤❡♦r❡♠ ✷✱ ✇❡ ❤❛✈❡ ✿

✷ ✱ ✸ ✹✱ ✹ ✺ ✻✱ ✾✷ ✹✼

❡t❝✳✳✳ ❛r❡ ♥♦♥✐s♦❝❤r♦♥♦✉s ✦

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✹ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

• ❡♥❡r❛❧ ♥♦t❛t✐♦♥s

❖✉r r❡s✉❧ts ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ■❧❧✉str❛t✐♦♥s ♦❢ ♦✉r t❤❡♦r❡♠s

❇② t❤❡ ❚❤❡♦r❡♠ ✶✱ ✇❡ ❤❛✈❡ ✿ X = Xlin + X✷ ✱ X = Xlin + X✷ + X✸ + X✹ ✇✐t❤ Im(p✶,✶) > ✵✱ X = Xlin + X✷ + X✸ + X✹ + X✺ + X✻ ✇✐t❤ Im(p✶,✶) > ✵ ♦r p✶,✶ = ✵ ❛♥❞ Im(p✷,✷) > ✵ ❡t❝✳✳✳ ❛r❡ ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ❇② t❤❡ ❚❤❡♦r❡♠ ✷✱ ✇❡ ❤❛✈❡ ✿

✷ ✱ ✸ ✹✱ ✹ ✺ ✻✱ ✾✷ ✹✼

❡t❝✳✳✳ ❛r❡ ♥♦♥✐s♦❝❤r♦♥♦✉s ✦

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✹ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

• ❡♥❡r❛❧ ♥♦t❛t✐♦♥s

❖✉r r❡s✉❧ts ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ■❧❧✉str❛t✐♦♥s ♦❢ ♦✉r t❤❡♦r❡♠s

❇② t❤❡ ❚❤❡♦r❡♠ ✶✱ ✇❡ ❤❛✈❡ ✿ X = Xlin + X✷ ✱ X = Xlin + X✷ + X✸ + X✹ ✇✐t❤ Im(p✶,✶) > ✵✱ X = Xlin + X✷ + X✸ + X✹ + X✺ + X✻ ✇✐t❤ Im(p✶,✶) > ✵ ♦r p✶,✶ = ✵ ❛♥❞ Im(p✷,✷) > ✵ ❡t❝✳✳✳ ❛r❡ ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ❇② t❤❡ ❚❤❡♦r❡♠ ✷✱ ✇❡ ❤❛✈❡ ✿ X = Xlin + X✷ ✱

✸ ✹✱ ✹ ✺ ✻✱ ✾✷ ✹✼

❡t❝✳✳✳ ❛r❡ ♥♦♥✐s♦❝❤r♦♥♦✉s ✦

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✹ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

• ❡♥❡r❛❧ ♥♦t❛t✐♦♥s

❖✉r r❡s✉❧ts ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ■❧❧✉str❛t✐♦♥s ♦❢ ♦✉r t❤❡♦r❡♠s

❇② t❤❡ ❚❤❡♦r❡♠ ✶✱ ✇❡ ❤❛✈❡ ✿ X = Xlin + X✷ ✱ X = Xlin + X✷ + X✸ + X✹ ✇✐t❤ Im(p✶,✶) > ✵✱ X = Xlin + X✷ + X✸ + X✹ + X✺ + X✻ ✇✐t❤ Im(p✶,✶) > ✵ ♦r p✶,✶ = ✵ ❛♥❞ Im(p✷,✷) > ✵ ❡t❝✳✳✳ ❛r❡ ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ❇② t❤❡ ❚❤❡♦r❡♠ ✷✱ ✇❡ ❤❛✈❡ ✿ X = Xlin + X✷ ✱ X = Xlin + X✸ + X✹✱ X = Xlin + X✹ + X✺ + X✻✱ X = Xlin +

✾✷

• ✹✼

Xr ❡t❝✳✳✳ ❛r❡ ♥♦♥✐s♦❝❤r♦♥♦✉s ✦

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✹ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✺ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

■♥ ♦r❞❡r t♦ ♣r♦✈❡ ♦✉r t✇♦ t❤❡♦r❡♠s ✿ Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ ▼♦✉❧❞ ❡①♣❛♥s✐♦♥ ❙t✉❞② ♦❢ t❤❡ ❈♦rr❡❝t✐♦♥ ❜② ❞❡♣t❤ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

■♥ ♦r❞❡r t♦ ♣r♦✈❡ ♦✉r t✇♦ t❤❡♦r❡♠s ✿ Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ ▼♦✉❧❞ ❡①♣❛♥s✐♦♥ ❙t✉❞② ♦❢ t❤❡ ❈♦rr❡❝t✐♦♥ ❜② ❞❡♣t❤ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

■♥ ♦r❞❡r t♦ ♣r♦✈❡ ♦✉r t✇♦ t❤❡♦r❡♠s ✿ Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ ▼♦✉❧❞ ❡①♣❛♥s✐♦♥ ❙t✉❞② ♦❢ t❤❡ ❈♦rr❡❝t✐♦♥ ❜② ❞❡♣t❤ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

■♥ ♦r❞❡r t♦ ♣r♦✈❡ ♦✉r t✇♦ t❤❡♦r❡♠s ✿ Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ ▼♦✉❧❞ ❡①♣❛♥s✐♦♥ ❙t✉❞② ♦❢ t❤❡ ❈♦rr❡❝t✐♦♥ ❜② ❞❡♣t❤ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ ▼♦✉❧❞ ❡①♣❛♥s✐♦♥ ❲❡ ❝♦♥s✐❞❡r ❛ ✈❡❝t♦r ✜❡❧❞ X = Xlin + Xr✳ ❚❤❡ ♣r❡♣❛r❡❞ ❢♦r♠ ♦❢ X ✐s ✿ X = Xlin +

• n∈A(X)

Bn, ✇❤❡r❡ ▲❡tt❡r ✿

✶ ✷

✱ ❆❧♣❤❛❜❡t ✿

✷ ✱

❍♦♠♦❣❡♥❡♦✉s ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ✿ s❛t✐s❢②✐♥❣

✶ ✷ ✶ ✶ ✷ ✷ ✇✐t❤ ❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✼ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ ▼♦✉❧❞ ❡①♣❛♥s✐♦♥ ❲❡ ❝♦♥s✐❞❡r ❛ ✈❡❝t♦r ✜❡❧❞ X = Xlin + Xr✳ ❚❤❡ ♣r❡♣❛r❡❞ ❢♦r♠ ♦❢ X ✐s ✿ X = Xlin +

• n∈A(X)

Bn, ✇❤❡r❡ ▲❡tt❡r ✿ n = (n✶, n✷) ∈ A(X)✱ ❆❧♣❤❛❜❡t ✿

✷ ✱

❍♦♠♦❣❡♥❡♦✉s ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ✿ s❛t✐s❢②✐♥❣

✶ ✷ ✶ ✶ ✷ ✷ ✇✐t❤ ❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✼ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ ▼♦✉❧❞ ❡①♣❛♥s✐♦♥ ❲❡ ❝♦♥s✐❞❡r ❛ ✈❡❝t♦r ✜❡❧❞ X = Xlin + Xr✳ ❚❤❡ ♣r❡♣❛r❡❞ ❢♦r♠ ♦❢ X ✐s ✿ X = Xlin +

• n∈A(X)

Bn, ✇❤❡r❡ ▲❡tt❡r ✿ n = (n✶, n✷) ∈ A(X)✱ ❆❧♣❤❛❜❡t ✿ A(X) ⊂ Z✷ ✱ ❍♦♠♦❣❡♥❡♦✉s ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ✿ s❛t✐s❢②✐♥❣

✶ ✷ ✶ ✶ ✷ ✷ ✇✐t❤ ❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✼ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❛♥❞ ▼♦✉❧❞ ❡①♣❛♥s✐♦♥ ❲❡ ❝♦♥s✐❞❡r ❛ ✈❡❝t♦r ✜❡❧❞ X = Xlin + Xr✳ ❚❤❡ ♣r❡♣❛r❡❞ ❢♦r♠ ♦❢ X ✐s ✿ X = Xlin +

• n∈A(X)

Bn, ✇❤❡r❡ ▲❡tt❡r ✿ n = (n✶, n✷) ∈ A(X)✱ ❆❧♣❤❛❜❡t ✿ A(X) ⊂ Z✷ ✱ ❍♦♠♦❣❡♥❡♦✉s ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ✿ Bn s❛t✐s❢②✐♥❣ Bn(xm✶ym✷) = βnxm✶+n✶ym✷+n✷ ✇✐t❤ βn ∈ C

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✼ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❊①❛♠♣❧❡ ♦❢ ❞❡❝♦♠♣♦s✐t✐♦♥ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❡❝t♦r ✜❡❧❞ ✿ X = Xlin + X✷ ✇❤❡r❡ X✷ =

• p✶,✵x✷ + p✵,✶xy + p−✶,✷y✷

∂x +

• q−✶,✷x✷ + q✶,✵xy + q✵,✶y✷

∂y, ❚❤❡ ❛❧♣❤❛❜❡t ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❛r❡ ❣✐✈❡♥ ❜② ✿

✶ ✵ ✶ ✵ ✵ ✶

✱

✵ ✶ ✵ ✶ ✵ ✶

✱

✷ ✶ ✷ ✶ ✷

✱

✶ ✷ ✶ ✷ ✷

✳ ✷ ✶ ✶ ✵ ✵ ✶ ✶ ✷ ✱

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✽ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❊①❛♠♣❧❡ ♦❢ ❞❡❝♦♠♣♦s✐t✐♦♥ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❡❝t♦r ✜❡❧❞ ✿ X = Xlin + X✷ ✇❤❡r❡ X✷ =

• p✶,✵x✷ + p✵,✶xy + p−✶,✷y✷

∂x +

• q−✶,✷x✷ + q✶,✵xy + q✵,✶y✷

∂y, ❚❤❡ ❛❧♣❤❛❜❡t ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❛r❡ ❣✐✈❡♥ ❜② ✿ B(✶,✵) = x(p✶,✵x∂x + p✵,✶y∂y)✱ B(✵,✶) = y(p✵,✶x∂x + p✵,✶y∂y)✱ B(✷,−✶) = p✷,−✶x✷∂y✱ B(−✶,✷) = p−✶,✷y✷∂x✳ A(X) = {(✷, −✶), (✶, ✵), (✵, ✶), (−✶, ✷)}✱

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✽ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❊①❛♠♣❧❡ ♦❢ ❞❡❝♦♠♣♦s✐t✐♦♥ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❡❝t♦r ✜❡❧❞ ✿ X = Xlin + X✷ ✇❤❡r❡ X✷ =

• p✶,✵x✷ + p✵,✶xy + p−✶,✷y✷

∂x +

• q−✶,✷x✷ + q✶,✵xy + q✵,✶y✷

∂y, ❚❤❡ ❛❧♣❤❛❜❡t ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❛r❡ ❣✐✈❡♥ ❜② ✿ B(✶,✵) = x(p✶,✵x∂x + p✵,✶y∂y)✱ B(✵,✶) = y(p✵,✶x∂x + p✵,✶y∂y)✱ B(✷,−✶) = p✷,−✶x✷∂y✱ B(−✶,✷) = p−✶,✷y✷∂x✳ A(X) = {(✷, −✶), (✶, ✵), (✵, ✶), (−✶, ✷)}✱

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✽ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❲❡ ✇r✐t❡ X ❛s ❛ s❡r✐❡s ✿ X = Xlin +

• ♥∈A∗(X)

I ♥B♥ ✇❤❡r❡ ✿ ✿ s❡t ♦❢ ✇♦r❞s ♦♥ ✱ ♥

✶

✇♦r❞ ❜② ❝♦♥❝❛t❡♥❛t✐♦♥✱ ✇✐t❤ ✱ ♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✇♦r❞ ♥✱

♥

✶

✱ t❤❡ ✉s✉❛❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✱ t❤❡ ❝♦❡✣❝✐❡♥t ✐s ❛ ♠♦✉❧❞ ✿ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ❢r♦♠ t♦ ✳ ❚❤✐s ♦♣❡r❛t✐♦♥ ✐s ❝❛❧❧❡❞ ♠♦✉❧❞ ❡①♣❛♥s✐♦♥✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❲❡ ✇r✐t❡ X ❛s ❛ s❡r✐❡s ✿ X = Xlin +

• ♥∈A∗(X)

I ♥B♥ ✇❤❡r❡ ✿ A∗(X) ✿ s❡t ♦❢ ✇♦r❞s ♦♥ A(X)✱ ♥

✶

✇♦r❞ ❜② ❝♦♥❝❛t❡♥❛t✐♦♥✱ ✇✐t❤ ✱ ♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✇♦r❞ ♥✱

♥

✶

✱ t❤❡ ✉s✉❛❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✱ t❤❡ ❝♦❡✣❝✐❡♥t ✐s ❛ ♠♦✉❧❞ ✿ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ❢r♦♠ t♦ ✳ ❚❤✐s ♦♣❡r❛t✐♦♥ ✐s ❝❛❧❧❡❞ ♠♦✉❧❞ ❡①♣❛♥s✐♦♥✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❲❡ ✇r✐t❡ X ❛s ❛ s❡r✐❡s ✿ X = Xlin +

• ♥∈A∗(X)

I ♥B♥ ✇❤❡r❡ ✿ A∗(X) ✿ s❡t ♦❢ ✇♦r❞s ♦♥ A(X)✱ ♥ = n✶ · ... · nr ✇♦r❞ ❜② ❝♦♥❝❛t❡♥❛t✐♦♥✱ ✇✐t❤ nj ∈ A(X)✱ ♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✇♦r❞ ♥✱

♥

✶

✱ t❤❡ ✉s✉❛❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✱ t❤❡ ❝♦❡✣❝✐❡♥t ✐s ❛ ♠♦✉❧❞ ✿ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ❢r♦♠ t♦ ✳ ❚❤✐s ♦♣❡r❛t✐♦♥ ✐s ❝❛❧❧❡❞ ♠♦✉❧❞ ❡①♣❛♥s✐♦♥✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❲❡ ✇r✐t❡ X ❛s ❛ s❡r✐❡s ✿ X = Xlin +

• ♥∈A∗(X)

I ♥B♥ ✇❤❡r❡ ✿ A∗(X) ✿ s❡t ♦❢ ✇♦r❞s ♦♥ A(X)✱ ♥ = n✶ · ... · nr ✇♦r❞ ❜② ❝♦♥❝❛t❡♥❛t✐♦♥✱ ✇✐t❤ nj ∈ A(X)✱ ℓ(♥) = r t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✇♦r❞ ♥✱

♥

✶

✱ t❤❡ ✉s✉❛❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✱ t❤❡ ❝♦❡✣❝✐❡♥t ✐s ❛ ♠♦✉❧❞ ✿ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ❢r♦♠ t♦ ✳ ❚❤✐s ♦♣❡r❛t✐♦♥ ✐s ❝❛❧❧❡❞ ♠♦✉❧❞ ❡①♣❛♥s✐♦♥✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❲❡ ✇r✐t❡ X ❛s ❛ s❡r✐❡s ✿ X = Xlin +

• ♥∈A∗(X)

I ♥B♥ ✇❤❡r❡ ✿ A∗(X) ✿ s❡t ♦❢ ✇♦r❞s ♦♥ A(X)✱ ♥ = n✶ · ... · nr ✇♦r❞ ❜② ❝♦♥❝❛t❡♥❛t✐♦♥✱ ✇✐t❤ nj ∈ A(X)✱ ℓ(♥) = r t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✇♦r❞ ♥✱ B♥ = Bn✶ ◦ · · · ◦ Bnr ✱ t❤❡ ✉s✉❛❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✱ t❤❡ ❝♦❡✣❝✐❡♥t ✐s ❛ ♠♦✉❧❞ ✿ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ❢r♦♠ t♦ ✳ ❚❤✐s ♦♣❡r❛t✐♦♥ ✐s ❝❛❧❧❡❞ ♠♦✉❧❞ ❡①♣❛♥s✐♦♥✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❲❡ ✇r✐t❡ X ❛s ❛ s❡r✐❡s ✿ X = Xlin +

• ♥∈A∗(X)

I ♥B♥ ✇❤❡r❡ ✿ A∗(X) ✿ s❡t ♦❢ ✇♦r❞s ♦♥ A(X)✱ ♥ = n✶ · ... · nr ✇♦r❞ ❜② ❝♦♥❝❛t❡♥❛t✐♦♥✱ ✇✐t❤ nj ∈ A(X)✱ ℓ(♥) = r t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✇♦r❞ ♥✱ B♥ = Bn✶ ◦ · · · ◦ Bnr ✱ t❤❡ ✉s✉❛❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✱ t❤❡ ❝♦❡✣❝✐❡♥t I • ✐s ❛ ♠♦✉❧❞ ✿ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ❢r♦♠ A∗(X) t♦ C✳ ❚❤✐s ♦♣❡r❛t✐♦♥ ✐s ❝❛❧❧❡❞ ♠♦✉❧❞ ❡①♣❛♥s✐♦♥✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❲❡ ✇r✐t❡ X ❛s ❛ s❡r✐❡s ✿ X = Xlin +

• ♥∈A∗(X)

I ♥B♥ ✇❤❡r❡ ✿ A∗(X) ✿ s❡t ♦❢ ✇♦r❞s ♦♥ A(X)✱ ♥ = n✶ · ... · nr ✇♦r❞ ❜② ❝♦♥❝❛t❡♥❛t✐♦♥✱ ✇✐t❤ nj ∈ A(X)✱ ℓ(♥) = r t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✇♦r❞ ♥✱ B♥ = Bn✶ ◦ · · · ◦ Bnr ✱ t❤❡ ✉s✉❛❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✱ t❤❡ ❝♦❡✣❝✐❡♥t I • ✐s ❛ ♠♦✉❧❞ ✿ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ❢r♦♠ A∗(X) t♦ C✳ ❚❤✐s ♦♣❡r❛t✐♦♥ ✐s ❝❛❧❧❡❞ ♠♦✉❧❞ ❡①♣❛♥s✐♦♥✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❘❡s♦♥❛♥t ❧❡tt❡rs ❛♥❞ ✇♦r❞s ❲❡ ❞❡♥♦t❡❞ ❜② λ = (i, −i) t❤❡ ❡✐❣❡♥s②st❡♠ ♦❢ Xlin✱ ❚❤❡ ✇❡✐❣❤t ♦❢ ❛ ❧❡tt❡r

✶ ✷ ✐s ❞❡✜♥❡❞ ❜② ✿

✱ ❋♦r ❛ ✇♦r❞ ♥

✶

✱ ♥

✶

❘❡s♦♥❛♥t ✇♦r❞s ❆ ✇♦r❞ ♥ ✐s r❡s♦♥❛♥t ✐❢ ♥ ✵✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✵ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❘❡s♦♥❛♥t ❧❡tt❡rs ❛♥❞ ✇♦r❞s ❲❡ ❞❡♥♦t❡❞ ❜② λ = (i, −i) t❤❡ ❡✐❣❡♥s②st❡♠ ♦❢ Xlin✱ ❚❤❡ ✇❡✐❣❤t ♦❢ ❛ ❧❡tt❡r n = (n✶, n✷) ✐s ❞❡✜♥❡❞ ❜② ✿ ω(n) = n, λ ✱ ❋♦r ❛ ✇♦r❞ ♥

✶

✱ ♥

✶

❘❡s♦♥❛♥t ✇♦r❞s ❆ ✇♦r❞ ♥ ✐s r❡s♦♥❛♥t ✐❢ ♥ ✵✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✵ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❘❡s♦♥❛♥t ❧❡tt❡rs ❛♥❞ ✇♦r❞s ❲❡ ❞❡♥♦t❡❞ ❜② λ = (i, −i) t❤❡ ❡✐❣❡♥s②st❡♠ ♦❢ Xlin✱ ❚❤❡ ✇❡✐❣❤t ♦❢ ❛ ❧❡tt❡r n = (n✶, n✷) ✐s ❞❡✜♥❡❞ ❜② ✿ ω(n) = n, λ ✱ ❋♦r ❛ ✇♦r❞ ♥ = n✶ · ... · nr✱ ω(♥) = ω(n✶) + ... + ω(nr) ❘❡s♦♥❛♥t ✇♦r❞s ❆ ✇♦r❞ ♥ ✐s r❡s♦♥❛♥t ✐❢ ♥ ✵✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✵ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❘❡s♦♥❛♥t ❧❡tt❡rs ❛♥❞ ✇♦r❞s ❲❡ ❞❡♥♦t❡❞ ❜② λ = (i, −i) t❤❡ ❡✐❣❡♥s②st❡♠ ♦❢ Xlin✱ ❚❤❡ ✇❡✐❣❤t ♦❢ ❛ ❧❡tt❡r n = (n✶, n✷) ✐s ❞❡✜♥❡❞ ❜② ✿ ω(n) = n, λ ✱ ❋♦r ❛ ✇♦r❞ ♥ = n✶ · ... · nr✱ ω(♥) = ω(n✶) + ... + ω(nr) ❘❡s♦♥❛♥t ✇♦r❞s ❆ ✇♦r❞ ♥ ✐s r❡s♦♥❛♥t ✐❢ ω(♥) = ✵✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✵ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❚❤❡ ❝♦rr❡❝t✐♦♥ ❛♥❞ ✐ts ♠♦✉❧❞ ❚❤❡♦r❡♠ ❬❊❝❛❧❧❡✱❱❛❧❧❡t❪ ❚❤❡ ❝♦rr❡❝t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✿ Carr(X) =

• ♥∈A∗(X)

Carr♥B♥ =

• k≥✶

✶ k

• ♥∈A∗(X)

ℓ(♥)=k

Carr♥[B♥] ✇❤❡r❡ ✿

♥

✶ ✶ ✷ ✸

✱ ✐s t❤❡ ♠♦✉❧❞ ♦❢ t❤❡ ❝♦rr❡❝t✐♦♥✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✶ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❚❤❡ ❝♦rr❡❝t✐♦♥ ❛♥❞ ✐ts ♠♦✉❧❞ ❚❤❡♦r❡♠ ❬❊❝❛❧❧❡✱❱❛❧❧❡t❪ ❚❤❡ ❝♦rr❡❝t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✿ Carr(X) =

• ♥∈A∗(X)

Carr♥B♥ =

• k≥✶

✶ k

• ♥∈A∗(X)

ℓ(♥)=k

Carr♥[B♥] ✇❤❡r❡ ✿ [B♥] = [Bn✶·...·nr ] = [....[[Bn✶, Bn✷], Bn✸], ...], Bnr ]✱ ✐s t❤❡ ♠♦✉❧❞ ♦❢ t❤❡ ❝♦rr❡❝t✐♦♥✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✶ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❚❤❡ ❝♦rr❡❝t✐♦♥ ❛♥❞ ✐ts ♠♦✉❧❞ ❚❤❡♦r❡♠ ❬❊❝❛❧❧❡✱❱❛❧❧❡t❪ ❚❤❡ ❝♦rr❡❝t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✿ Carr(X) =

• ♥∈A∗(X)

Carr♥B♥ =

• k≥✶

✶ k

• ♥∈A∗(X)

ℓ(♥)=k

Carr♥[B♥] ✇❤❡r❡ ✿ [B♥] = [Bn✶·...·nr ] = [....[[Bn✶, Bn✷], Bn✸], ...], Bnr ]✱ Carr• ✐s t❤❡ ♠♦✉❧❞ ♦❢ t❤❡ ❝♦rr❡❝t✐♦♥✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✶ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❚❤❡ ♠♦✉❧❞ Carr• ✐s ❣✐✈❡♥ ❢♦r ❛♥② ✇♦r❞ ♥ = n✶ · ... · nr ❜② ✹ ✿

✶

✶ ✷ ✶ ✷ ✸ ✶ ❜ ❝

♥

✶ ❝

❜✱

■❢ ♥ ✐s ♥♦t ❛ r❡s♦♥❛♥t ✇♦r❞✱

♥

✵ ❋♦r ♥ ✵ ✱ ■❢ ♥ ✶✱

♥

✶✱ ■❢ ♥ ✷✱ ♥

✶ ✷✱ ♥ ✶

✶

✹✳ ■t ✐s ♥♦t ❛ tr✐✈✐❛❧ ❢♦r♠✉❧❛ ✿ r❡❧❛t❡❞ t♦ t❤❡ ♥♦t✐♦♥ ♦❢ ✈❛r✐❛♥❝❡ ♦❢ ✈❡❝t♦r ✜❡❧❞s✱ s❡❡ ❏✳❊❝❛❧❧❡ ❛♥❞ ❇✳❱❛❧❧❡t✱ ✧❈♦rr❡❝t✐♦♥ ❛♥❞ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ r❡s♦♥❛♥t ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ❞✐✛❡♦♠♦r♣❤✐s♠s✧✱ ▼❛t❤✳ ❩✳ ✷✷✾✱ ✷✹✾✲✸✶✽ ✭✶✾✾✽✮

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✷ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❚❤❡ ♠♦✉❧❞ Carr• ✐s ❣✐✈❡♥ ❢♦r ❛♥② ✇♦r❞ ♥ = n✶ · ... · nr ❜② ✹ ✿ ω(n✶)Carrn✶·n✷·...·nr + Carrn✶+n✷·n✸·...·nr =

• n✶·❜·❝=♥

Carrn✶·❝Carr❜✱ ■❢ ♥ ✐s ♥♦t ❛ r❡s♦♥❛♥t ✇♦r❞✱

♥

✵ ❋♦r ♥ ✵ ✱ ■❢ ♥ ✶✱

♥

✶✱ ■❢ ♥ ✷✱ ♥

✶ ✷✱ ♥ ✶

✶

✹✳ ■t ✐s ♥♦t ❛ tr✐✈✐❛❧ ❢♦r♠✉❧❛ ✿ r❡❧❛t❡❞ t♦ t❤❡ ♥♦t✐♦♥ ♦❢ ✈❛r✐❛♥❝❡ ♦❢ ✈❡❝t♦r ✜❡❧❞s✱ s❡❡ ❏✳❊❝❛❧❧❡ ❛♥❞ ❇✳❱❛❧❧❡t✱ ✧❈♦rr❡❝t✐♦♥ ❛♥❞ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ r❡s♦♥❛♥t ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ❞✐✛❡♦♠♦r♣❤✐s♠s✧✱ ▼❛t❤✳ ❩✳ ✷✷✾✱ ✷✹✾✲✸✶✽ ✭✶✾✾✽✮

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✷ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❚❤❡ ♠♦✉❧❞ Carr• ✐s ❣✐✈❡♥ ❢♦r ❛♥② ✇♦r❞ ♥ = n✶ · ... · nr ❜② ✹ ✿ ω(n✶)Carrn✶·n✷·...·nr + Carrn✶+n✷·n✸·...·nr =

• n✶·❜·❝=♥

Carrn✶·❝Carr❜✱ ■❢ ♥ ✐s ♥♦t ❛ r❡s♦♥❛♥t ✇♦r❞✱ Carr♥ = ✵ ❋♦r ♥ ✵ ✱ ■❢ ♥ ✶✱

♥

✶✱ ■❢ ♥ ✷✱ ♥

✶ ✷✱ ♥ ✶

✶

✹✳ ■t ✐s ♥♦t ❛ tr✐✈✐❛❧ ❢♦r♠✉❧❛ ✿ r❡❧❛t❡❞ t♦ t❤❡ ♥♦t✐♦♥ ♦❢ ✈❛r✐❛♥❝❡ ♦❢ ✈❡❝t♦r ✜❡❧❞s✱ s❡❡ ❏✳❊❝❛❧❧❡ ❛♥❞ ❇✳❱❛❧❧❡t✱ ✧❈♦rr❡❝t✐♦♥ ❛♥❞ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ r❡s♦♥❛♥t ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ❞✐✛❡♦♠♦r♣❤✐s♠s✧✱ ▼❛t❤✳ ❩✳ ✷✷✾✱ ✷✹✾✲✸✶✽ ✭✶✾✾✽✮

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✷ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❚❤❡ ♠♦✉❧❞ Carr• ✐s ❣✐✈❡♥ ❢♦r ❛♥② ✇♦r❞ ♥ = n✶ · ... · nr ❜② ✹ ✿ ω(n✶)Carrn✶·n✷·...·nr + Carrn✶+n✷·n✸·...·nr =

• n✶·❜·❝=♥

Carrn✶·❝Carr❜✱ ■❢ ♥ ✐s ♥♦t ❛ r❡s♦♥❛♥t ✇♦r❞✱ Carr♥ = ✵ ❋♦r ω(♥) = ✵ ✱ ■❢ ℓ(♥) = ✶✱ Carr♥ = ✶✱ ■❢ ℓ(♥) = ✷✱ ♥ = n✶ · n✷✱ Carr♥ =

−✶ ω(n✶)

✹✳ ■t ✐s ♥♦t ❛ tr✐✈✐❛❧ ❢♦r♠✉❧❛ ✿ r❡❧❛t❡❞ t♦ t❤❡ ♥♦t✐♦♥ ♦❢ ✈❛r✐❛♥❝❡ ♦❢ ✈❡❝t♦r ✜❡❧❞s✱ s❡❡ ❏✳❊❝❛❧❧❡ ❛♥❞ ❇✳❱❛❧❧❡t✱ ✧❈♦rr❡❝t✐♦♥ ❛♥❞ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ r❡s♦♥❛♥t ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ❞✐✛❡♦♠♦r♣❤✐s♠s✧✱ ▼❛t❤✳ ❩✳ ✷✷✾✱ ✷✹✾✲✸✶✽ ✭✶✾✾✽✮

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✷ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

◆❡✇ ✇r✐t✐♥❣ ♦❢ t❤❡ ❈♦rr❡❝t✐♦♥ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❞❡♣t❤ ✿ ❚❤❡ ❞❡♣t❤ ♦❢ ❛ ❧❡tt❡r n = (n✶, n✷) ✐s p(n) = n✶ + n✷✱ ❚❤❡ ❞❡♣t❤ ♦❢ ❛ ✇♦r❞ ♥

✶

✐s ❣✐✈❡♥ ❜② ♥

✶

✱ ❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❝♦rr❡❝t✐♦♥ ✉s✐♥❣ t❤❡ ❞❡♣t❤ ❈♦rr❡❝t✐♦♥ ✈✐❛ t❤❡ ❞❡♣t❤

✶

✇✐t❤

♥ ♥ ✶ ♥ ♥ ♥

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✸ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

◆❡✇ ✇r✐t✐♥❣ ♦❢ t❤❡ ❈♦rr❡❝t✐♦♥ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❞❡♣t❤ ✿ ❚❤❡ ❞❡♣t❤ ♦❢ ❛ ❧❡tt❡r n = (n✶, n✷) ✐s p(n) = n✶ + n✷✱ ❚❤❡ ❞❡♣t❤ ♦❢ ❛ ✇♦r❞ ♥ = n✶ · ... · nr ✐s ❣✐✈❡♥ ❜② p(♥) = p(n✶) + ... + p(nr)✱ ❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❝♦rr❡❝t✐♦♥ ✉s✐♥❣ t❤❡ ❞❡♣t❤ ❈♦rr❡❝t✐♦♥ ✈✐❛ t❤❡ ❞❡♣t❤

✶

✇✐t❤

♥ ♥ ✶ ♥ ♥ ♥

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✸ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

◆❡✇ ✇r✐t✐♥❣ ♦❢ t❤❡ ❈♦rr❡❝t✐♦♥ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❞❡♣t❤ ✿ ❚❤❡ ❞❡♣t❤ ♦❢ ❛ ❧❡tt❡r n = (n✶, n✷) ✐s p(n) = n✶ + n✷✱ ❚❤❡ ❞❡♣t❤ ♦❢ ❛ ✇♦r❞ ♥ = n✶ · ... · nr ✐s ❣✐✈❡♥ ❜② p(♥) = p(n✶) + ... + p(nr)✱ ⇒ ❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❝♦rr❡❝t✐♦♥ ✉s✐♥❣ t❤❡ ❞❡♣t❤ ❈♦rr❡❝t✐♦♥ ✈✐❛ t❤❡ ❞❡♣t❤

✶

✇✐t❤

♥ ♥ ✶ ♥ ♥ ♥

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✸ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

◆❡✇ ✇r✐t✐♥❣ ♦❢ t❤❡ ❈♦rr❡❝t✐♦♥ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❞❡♣t❤ ✿ ❚❤❡ ❞❡♣t❤ ♦❢ ❛ ❧❡tt❡r n = (n✶, n✷) ✐s p(n) = n✶ + n✷✱ ❚❤❡ ❞❡♣t❤ ♦❢ ❛ ✇♦r❞ ♥ = n✶ · ... · nr ✐s ❣✐✈❡♥ ❜② p(♥) = p(n✶) + ... + p(nr)✱ ⇒ ❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❝♦rr❡❝t✐♦♥ ✉s✐♥❣ t❤❡ ❞❡♣t❤ ❈♦rr❡❝t✐♦♥ ✈✐❛ t❤❡ ❞❡♣t❤ Carr(X) =

p≥✶

Carrp(X) ✇✐t❤ Carrp(X) =

• ♥∈A∗(X)

p(♥)=p ✶ l(♥)Carr♥[B♥]

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✸ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

▲✐♥❡❛r✐③❛❜✐❧✐t② ❛♥❞ ♠❛✐♥ ♣r♦♣❡rt② ❈r✐t❡r✐♦♥ ♦❢ ❧✐♥❡❛r✐③❛❜✐❧✐t② ❆ ✈❡❝t♦r ✜❡❧❞ X ❛s ❛❜♦✈❡ ✐s ❧✐♥❡❛r✐③❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ Carrp(X) = ✵ ❢♦r ❛❧❧ p ≥ ✶✳ Pr♦♣❡rt② ♦❢ ❋♦r ❛♥② ♦❞❞ ✐♥t❡❣❡r ✱ ✵✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✹ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

▲✐♥❡❛r✐③❛❜✐❧✐t② ❛♥❞ ♠❛✐♥ ♣r♦♣❡rt② ❈r✐t❡r✐♦♥ ♦❢ ❧✐♥❡❛r✐③❛❜✐❧✐t② ❆ ✈❡❝t♦r ✜❡❧❞ X ❛s ❛❜♦✈❡ ✐s ❧✐♥❡❛r✐③❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ Carrp(X) = ✵ ❢♦r ❛❧❧ p ≥ ✶✳ Pr♦♣❡rt② ♦❢ Carrp(X) ❋♦r ❛♥② ♦❞❞ ✐♥t❡❣❡r p✱ Carrp(X) = ✵✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✹ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

▼❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢s ❲❡ ❝♦♥s✐❞❡r X = Xlin +

✷n

• r=k

Xr✱ ❚✇♦ ❝❛s❡s ✿ ✷ ♦r ✷ ✶✳ ❍♦✇ t♦ ❝❛❧❝✉❧❛t❡ ❄ ❱❡❝t♦r ✜❡❧❞

✷ ✷ ✶

✳✳✳

✷

❉❡♣t❤ ✷ ✶ ✷ ✳✳✳ ✷ ✶ ❋♦r ❛ ❣✐✈❡♥ ❞❡♣t❤ ✱ ✇❤✐❝❤ ❝♦♥tr✐❜✉t❡s t♦ ❄ ◆♦t❛t✐♦♥ ✿ t❤❡ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ ✐♥ ❞❡♣t❤ ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✇♦r❞s✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✺ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

▼❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢s ❲❡ ❝♦♥s✐❞❡r X = Xlin +

✷n

• r=k

Xr✱ ❚✇♦ ❝❛s❡s ✿ k = ✷l ♦r k = ✷l + ✶✳ ❍♦✇ t♦ ❝❛❧❝✉❧❛t❡ ❄ ❱❡❝t♦r ✜❡❧❞

✷ ✷ ✶

✳✳✳

✷

❉❡♣t❤ ✷ ✶ ✷ ✳✳✳ ✷ ✶ ❋♦r ❛ ❣✐✈❡♥ ❞❡♣t❤ ✱ ✇❤✐❝❤ ❝♦♥tr✐❜✉t❡s t♦ ❄ ◆♦t❛t✐♦♥ ✿ t❤❡ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ ✐♥ ❞❡♣t❤ ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✇♦r❞s✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✺ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

▼❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢s ❲❡ ❝♦♥s✐❞❡r X = Xlin +

✷n

• r=k

Xr✱ ❚✇♦ ❝❛s❡s ✿ k = ✷l ♦r k = ✷l + ✶✳ ❍♦✇ t♦ ❝❛❧❝✉❧❛t❡ Carrp(X) ❄ ❱❡❝t♦r ✜❡❧❞ X✷l X✷l+✶ ✳✳✳ X✷n ❉❡♣t❤ ✷l − ✶ ✷l ✳✳✳ ✷n − ✶ ❋♦r ❛ ❣✐✈❡♥ ❞❡♣t❤ ✱ ✇❤✐❝❤ ❝♦♥tr✐❜✉t❡s t♦ ❄ ◆♦t❛t✐♦♥ ✿ t❤❡ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ ✐♥ ❞❡♣t❤ ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✇♦r❞s✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✺ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

▼❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢s ❲❡ ❝♦♥s✐❞❡r X = Xlin +

✷n

• r=k

Xr✱ ❚✇♦ ❝❛s❡s ✿ k = ✷l ♦r k = ✷l + ✶✳ ❍♦✇ t♦ ❝❛❧❝✉❧❛t❡ Carrp(X) ❄ ❱❡❝t♦r ✜❡❧❞ X✷l X✷l+✶ ✳✳✳ X✷n ❉❡♣t❤ ✷l − ✶ ✷l ✳✳✳ ✷n − ✶ ❋♦r ❛ ❣✐✈❡♥ ❞❡♣t❤ p✱ ✇❤✐❝❤ Xr ❝♦♥tr✐❜✉t❡s t♦ Carrp(X) ❄ ◆♦t❛t✐♦♥ ✿ t❤❡ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ ✐♥ ❞❡♣t❤ ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✇♦r❞s✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✺ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

▼❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢s ❲❡ ❝♦♥s✐❞❡r X = Xlin +

✷n

• r=k

Xr✱ ❚✇♦ ❝❛s❡s ✿ k = ✷l ♦r k = ✷l + ✶✳ ❍♦✇ t♦ ❝❛❧❝✉❧❛t❡ Carrp(X) ❄ ❱❡❝t♦r ✜❡❧❞ X✷l X✷l+✶ ✳✳✳ X✷n ❉❡♣t❤ ✷l − ✶ ✷l ✳✳✳ ✷n − ✶ ❋♦r ❛ ❣✐✈❡♥ ❞❡♣t❤ p✱ ✇❤✐❝❤ Xr ❝♦♥tr✐❜✉t❡s t♦ Carrp(X) ❄ ◆♦t❛t✐♦♥ ✿ Carrp,ℓ(Xi) t❤❡ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ Xi ✐♥ ❞❡♣t❤ p ❛♥❞ ℓ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✇♦r❞s✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✺ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

■❢ k = ✷l + ✶ ✿ Carr✷l+✷q(X) = Carr✷l+✷q,✶(X✷l+✷q+✶)✱ ❢♦r ✵ ≤ q ≤ l − ✶✱ ❛♥❞

✹ ✹ ✶ ✹ ✶ ✹ ✷ ✷

• ❡♥❡r❛❧ ❢♦r♠✉❧❛s

✷ ✶ ✷ ✶

✱

✷ ✷ ✶ ✶ ✷

✶

✱ ✇❤❡r❡

✶ ✷ ✷ ✶ ✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

■❢ k = ✷l + ✶ ✿ Carr✷l+✷q(X) = Carr✷l+✷q,✶(X✷l+✷q+✶)✱ ❢♦r ✵ ≤ q ≤ l − ✶✱ ❛♥❞ Carr✹l(X) = Carr✹l,✶(X✹l+✶) + Carr✹l,✷(X✷l)

• ❡♥❡r❛❧ ❢♦r♠✉❧❛s

✷ ✶ ✷ ✶

✱

✷ ✷ ✶ ✶ ✷

✶

✱ ✇❤❡r❡

✶ ✷ ✷ ✶ ✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

■❢ k = ✷l + ✶ ✿ Carr✷l+✷q(X) = Carr✷l+✷q,✶(X✷l+✷q+✶)✱ ❢♦r ✵ ≤ q ≤ l − ✶✱ ❛♥❞ Carr✹l(X) = Carr✹l,✶(X✹l+✶) + Carr✹l,✷(X✷l)

• ❡♥❡r❛❧ ❢♦r♠✉❧❛s

Carr✷j,✶(X✷j+✶) = pj,j(xy)j(x∂x − y∂y)✱ Carr✷j,✷(Xj+✶) = ✶

✷

• n∈A(Xj+✶)

Carrn,ping(n)[Bn, Bping(n)]✱ ✇❤❡r❡ ping(n) = ping(n✶, n✷) = (n✷, n✶)✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

■❢ k = ✷l + ✶ ✿ Carr✷l+✷q(X) = Carr✷l+✷q,✶(X✷l+✷q+✶)✱ ❢♦r ✵ ≤ q ≤ l − ✶✱ ❛♥❞ Carr✹l(X) = Carr✹l,✶(X✹l+✶) + Carr✹l,✷(X✷l)

• ❡♥❡r❛❧ ❢♦r♠✉❧❛s

Carr✷j,✶(X✷j+✶) = pj,j(xy)j(x∂x − y∂y)✱ Carr✷j,✷(Xj+✶) = ✶

✷

• n∈A(Xj+✶)

Carrn,ping(n)[Bn, Bping(n)]✱ ✇❤❡r❡ ping(n) = ping(n✶, n✷) = (n✷, n✶)✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✻ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❲✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥s ❢♦r ❳ t♦ ❜❡ r❡❛❧ ❛♥❞ ❍❛♠✐❧t♦♥✐❛♥✱ ✇❡ ❤❛✈❡ ✿ Carr✷k(X) = F × (x∂x − y∂y) ✇✐t❤ ✿ F = pk,k + i  

✷l

• j=⌊ ✷l+✶

✷

⌋+✶ ✷l(✷l+✶) (✷l−j+✶)✷ |pj−✶,✷l−j|✷ + ✷l ✷l+✶|p−✶,✷l|✷

  ■❢

✷

✵✱ t❤❡r❡ ✐s ❛ ✧s♣❤❡r❡✧ ❧✐♥❦✐♥❣

✷ ❛♥❞ ✷ ✶ ✹ ✶

✭❈✶✮ ■❢ ✵✱ ✇❡ ❤❛✈❡ ❛♥ ♦❜str✉❝t✐♦♥ t♦ t❤❡ ✐s♦❝❤r♦♥✐❝✐t② ✦ ✭❈✷✮ ■❢ ✵✱ t❤❡ s♣❤❡r❡ ✐s tr✐✈✐❛❧

✷

✵✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✼ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❲✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥s ❢♦r ❳ t♦ ❜❡ r❡❛❧ ❛♥❞ ❍❛♠✐❧t♦♥✐❛♥✱ ✇❡ ❤❛✈❡ ✿ Carr✷k(X) = F × (x∂x − y∂y) ✇✐t❤ ✿ F = pk,k + i  

✷l

• j=⌊ ✷l+✶

✷

⌋+✶ ✷l(✷l+✶) (✷l−j+✶)✷ |pj−✶,✷l−j|✷ + ✷l ✷l+✶|p−✶,✷l|✷

  ■❢ Carr✷k(X) = ✵✱ t❤❡r❡ ✐s ❛ ✧s♣❤❡r❡✧ ❧✐♥❦✐♥❣ X✷l ❛♥❞ X✷k+✶ = X✹l−✶ ⇒ ✭❈✶✮ ■❢ ✵✱ ✇❡ ❤❛✈❡ ❛♥ ♦❜str✉❝t✐♦♥ t♦ t❤❡ ✐s♦❝❤r♦♥✐❝✐t② ✦ ✭❈✷✮ ■❢ ✵✱ t❤❡ s♣❤❡r❡ ✐s tr✐✈✐❛❧

✷

✵✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✼ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❲✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥s ❢♦r ❳ t♦ ❜❡ r❡❛❧ ❛♥❞ ❍❛♠✐❧t♦♥✐❛♥✱ ✇❡ ❤❛✈❡ ✿ Carr✷k(X) = F × (x∂x − y∂y) ✇✐t❤ ✿ F = pk,k + i  

✷l

• j=⌊ ✷l+✶

✷

⌋+✶ ✷l(✷l+✶) (✷l−j+✶)✷ |pj−✶,✷l−j|✷ + ✷l ✷l+✶|p−✶,✷l|✷

  ■❢ Carr✷k(X) = ✵✱ t❤❡r❡ ✐s ❛ ✧s♣❤❡r❡✧ ❧✐♥❦✐♥❣ X✷l ❛♥❞ X✷k+✶ = X✹l−✶ ⇒ ✭❈✶✮ ■❢ Im(pk,k) > ✵✱ ✇❡ ❤❛✈❡ ❛♥ ♦❜str✉❝t✐♦♥ t♦ t❤❡ ✐s♦❝❤r♦♥✐❝✐t② ✦ ✭❈✷✮ ■❢ ✵✱ t❤❡ s♣❤❡r❡ ✐s tr✐✈✐❛❧

✷

✵✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✼ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❲✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥s ❢♦r ❳ t♦ ❜❡ r❡❛❧ ❛♥❞ ❍❛♠✐❧t♦♥✐❛♥✱ ✇❡ ❤❛✈❡ ✿ Carr✷k(X) = F × (x∂x − y∂y) ✇✐t❤ ✿ F = pk,k + i  

✷l

• j=⌊ ✷l+✶

✷

⌋+✶ ✷l(✷l+✶) (✷l−j+✶)✷ |pj−✶,✷l−j|✷ + ✷l ✷l+✶|p−✶,✷l|✷

  ■❢ Carr✷k(X) = ✵✱ t❤❡r❡ ✐s ❛ ✧s♣❤❡r❡✧ ❧✐♥❦✐♥❣ X✷l ❛♥❞ X✷k+✶ = X✹l−✶ ⇒ ✭❈✶✮ ■❢ Im(pk,k) > ✵✱ ✇❡ ❤❛✈❡ ❛♥ ♦❜str✉❝t✐♦♥ t♦ t❤❡ ✐s♦❝❤r♦♥✐❝✐t② ✦ ✭❈✷✮ ■❢ pk,k = ✵✱ t❤❡ s♣❤❡r❡ ✐s tr✐✈✐❛❧ ⇒ X✷l = ✵✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✼ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶ ❲❡ ❝♦♥s✐❞❡r X = Xlin +

✷n

• r=✷

Xr ✿

✶ ■❢ t❤❡r❡ ❡①✐sts ✶

✶ s✳t ✵ ❢♦r ✵ ✶ ❛♥❞ ✵✱ ❜② ✭❈✶✮✱ ❝❛♥✬t ❜❡ ✐s♦❝❤r♦♥♦✉s✱

✷ ■❢

✵ ❢♦r ✶ ✶✱ ❜② t❤❡ ❝♦♥❞✐t✐♦♥ ✭❈✷ ✱ ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s ♦r ✐s tr✐✈✐❛❧✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✽ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶ ❲❡ ❝♦♥s✐❞❡r X = Xlin +

✷n

• r=✷

Xr ✿

✶ ■❢ t❤❡r❡ ❡①✐sts ✶ ≤ k < n − ✶ s✳t pj,j = ✵ ❢♦r j = ✵, ..., k − ✶

❛♥❞ Im(pk,k) > ✵✱ ⇒ ❜② ✭❈✶✮✱ X ❝❛♥✬t ❜❡ ✐s♦❝❤r♦♥♦✉s✱

✷ ■❢

✵ ❢♦r ✶ ✶✱ ❜② t❤❡ ❝♦♥❞✐t✐♦♥ ✭❈✷ ✱ ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s ♦r ✐s tr✐✈✐❛❧✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✽ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶ ❲❡ ❝♦♥s✐❞❡r X = Xlin +

✷n

• r=✷

Xr ✿

✶ ■❢ t❤❡r❡ ❡①✐sts ✶ ≤ k < n − ✶ s✳t pj,j = ✵ ❢♦r j = ✵, ..., k − ✶

❛♥❞ Im(pk,k) > ✵✱ ⇒ ❜② ✭❈✶✮✱ X ❝❛♥✬t ❜❡ ✐s♦❝❤r♦♥♦✉s✱

✷ ■❢ pk,k = ✵ ❢♦r ✶ ≤ k ≤ n − ✶✱

⇒ ❜② t❤❡ ❝♦♥❞✐t✐♦♥ ✭❈✷)✱ X ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s ♦r Xr ✐s tr✐✈✐❛❧✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✽ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷ ❲❡ ❝♦♥s✐❞❡r X = Xlin + Xk + ... + X✷n ❢♦r k ≥ ✷ ❛♥❞ n ≤ k − ✶✳ ■❢ ✐s ❡✈❡♥✱ ❛s ✶ ✇❡ ❤❛✈❡ ✿ ❱❡❝t♦r ✜❡❧❞

✶

✳✳✳

✷

❉❡♣t❤ ✶ ✳✳✳ ✷ ✶ ❲❡ ❤❛✈❡ ✿ ✷ ✶ ✷ ✷ ✶ ✱ ◆♦ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❧❡♥❣t❤ ✶ ❛♥❞ ✷ ✐♥ ❛ s❛♠❡ ❞❡♣t❤✱ ❡❛❝❤ ✐s tr✐✈✐❛❧ ♦r ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ■❢ ✐s ♦❞❞✱ ✇❡ ❤❛✈❡ ❛♥ ❛♥❛❧♦❣♦✉s r❡s✉❧t✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷ ❲❡ ❝♦♥s✐❞❡r X = Xlin + Xk + ... + X✷n ❢♦r k ≥ ✷ ❛♥❞ n ≤ k − ✶✳ ■❢ k ✐s ❡✈❡♥✱ ❛s n ≤ k − ✶ ✇❡ ❤❛✈❡ ✿ ❱❡❝t♦r ✜❡❧❞ Xk Xk+✶ ✳✳✳ X✷n ❉❡♣t❤ k − ✶ k ✳✳✳ ✷n − ✶ ❲❡ ❤❛✈❡ ✿ ✷ ✶ ✷ ✷ ✶ ✱ ◆♦ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❧❡♥❣t❤ ✶ ❛♥❞ ✷ ✐♥ ❛ s❛♠❡ ❞❡♣t❤✱ ❡❛❝❤ ✐s tr✐✈✐❛❧ ♦r ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ■❢ ✐s ♦❞❞✱ ✇❡ ❤❛✈❡ ❛♥ ❛♥❛❧♦❣♦✉s r❡s✉❧t✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷ ❲❡ ❝♦♥s✐❞❡r X = Xlin + Xk + ... + X✷n ❢♦r k ≥ ✷ ❛♥❞ n ≤ k − ✶✳ ■❢ k ✐s ❡✈❡♥✱ ❛s n ≤ k − ✶ ✇❡ ❤❛✈❡ ✿ ❱❡❝t♦r ✜❡❧❞ Xk Xk+✶ ✳✳✳ X✷n ❉❡♣t❤ k − ✶ k ✳✳✳ ✷n − ✶ ❲❡ ❤❛✈❡ ✿ ✷(k − ✶) ≥ ✷n > ✷n − ✶ ✱ ⇒ ◆♦ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❧❡♥❣t❤ ✶ ❛♥❞ ✷ ✐♥ ❛ s❛♠❡ ❞❡♣t❤✱ ⇒ ❡❛❝❤ Xr ✐s tr✐✈✐❛❧ ♦r X ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ■❢ k ✐s ♦❞❞✱ ✇❡ ❤❛✈❡ ❛♥ ❛♥❛❧♦❣♦✉s r❡s✉❧t✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷✾ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❆ ❧❛st t❤❡♦r❡♠ ❬P✳✱❈r❡ss♦♥❪ ▲❡t X ❜❡ ❛ ♥♦♥ tr✐✈✐❛❧ r❡❛❧ ♣♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ ✈❡❝t♦r ✜❡❧❞ ♦♥ t❤❡ ❢♦r♠ ✿ X = Xlin + Xk + ... + X✷l +

m

• n=✶

✷(cn−✶)

• j=cn

Xj ✇❤❡r❡ k ≥ ✷✱ l ≤ k − ✶ ❛♥❞ t❤❡ s❡q✉❡♥❝❡ cn ✐s ❞❡✜♥❡❞ ❜② ✿ c✶ = l ❛♥❞ ∀n ≥ ✷✱ cn = ✹(cn−✶ − ✶)✳ ❚❤❡♥ X ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ❙♦♠❡ ❡①❛♠♣❧❡s ✿

✷ ✹ ✺ ✻✱ ✷ ✹ ✺ ✻ ✷✷ ✶✷ ✽✻ ✹✹ ✸✹✷ ✶✼✷

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✸✵ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❆ ❧❛st t❤❡♦r❡♠ ❬P✳✱❈r❡ss♦♥❪ ▲❡t X ❜❡ ❛ ♥♦♥ tr✐✈✐❛❧ r❡❛❧ ♣♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ ✈❡❝t♦r ✜❡❧❞ ♦♥ t❤❡ ❢♦r♠ ✿ X = Xlin + Xk + ... + X✷l +

m

• n=✶

✷(cn−✶)

• j=cn

Xj ✇❤❡r❡ k ≥ ✷✱ l ≤ k − ✶ ❛♥❞ t❤❡ s❡q✉❡♥❝❡ cn ✐s ❞❡✜♥❡❞ ❜② ✿ c✶ = l ❛♥❞ ∀n ≥ ✷✱ cn = ✹(cn−✶ − ✶)✳ ❚❤❡♥ X ✐s ♥♦♥✐s♦❝❤r♦♥♦✉s✳ ❙♦♠❡ ❡①❛♠♣❧❡s ✿ X = Xlin + X✷ + X✹ + X✺ + X✻✱ X = Xlin + X✷ + X✹ + X✺ + X✻ +

✷✷

• j=✶✷

Xj +

✽✻

• j=✹✹

Xj +

✸✹✷

• j=✶✼✷

Xj

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✸✵ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

P❡rs♣❡❝t✐✈❡s ❚♦ ❝♦♠♣❧❡t❡ ♦✉r ▼❛♣❧❡ ♣r♦❣r❛♠✱ ❚♦ tr② t♦ ❣❡♥❡r❛❧✐③❡ t❤❡ ❚❤❡♦r❡♠ ✷ ❢♦r n > k − ✶✱ ❚♦ ❡①t❡♥❞ ♦✉r st✉❞② t♦ t❤❡ ✐s♦❝❤r♦♥♦✉s ❝♦♠♣❧❡① ❍❛♠✐❧t♦♥✐❛♥ ❝❛s❡✳

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✸✶ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s

❚❤❛♥❦ ②♦✉r ❢♦r ②♦✉r ❛tt❡♥t✐♦♥ ✦

❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✸✷ ✴ ✸✷