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◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❖♥ t❤❡ ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ❛ ◆❡❛r ❯♥❛♥✐♠✐t② ❖♣❡r❛t✐♦♥ ✐♥ ❛ ❈❧♦♥❡

❉♠✐tr✐② ❩❤✉❦ ③❤✉❦✳❞♠✐tr✐②❅❣♠❛✐❧✳❝♦♠

❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ▼❡❝❤❛♥✐❝s ▼♦s❝♦✇ ❙t❛t❡ ❯♥✐✈❡rs✐t②

❈♦♥❢❡r❡♥❝❡ ❆❆❆✽✽ ✿ ❲❛rs❛✇✱ ✷✵✶✹

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❖✉t❧✐♥❡

✶ ◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ✷ ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ✸ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ ✹ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

▼❛✐♥ ♥♦t❛t✐♦♥s✳ ▲❡t A ❜❡ ❛ ✜♥✐t❡ s❡t✳ ❊✈❡r②❜♦❞② ❦♥♦✇s ✇❤❛t ❝❧♦♥❡ ✐s✳✳✳ ❙♦♠❡ ◆♦t❛t✐♦♥s Om

A ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m✳

✐s t❤❡ s❡t ♦❢ ✜♥✐t❛r② r❡❧❛t✐♦♥s ♦♥ ♦❢ ❛r✐t② ❛t ♠♦st ✳ ✐s t❤❡ ❧❡❛st ❝❧♦♥❡ ❝♦♥t❛✐♥✐♥❣ ✳ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ t❤❡ r❡❧❛t✐♦♥ ✳ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ ❡✈❡r② r❡❧❛t✐♦♥ ❢r♦♠ ✳ ✐s t❤❡ ❧❡❛st r❡❧❛t✐♦♥ ♣r❡s❡r✈❡❞ ❜② ❛ ❝❧♦♥❡ ❛♥❞ ❝♦♥t❛✐♥✐♥❣ ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

▼❛✐♥ ♥♦t❛t✐♦♥s✳ ▲❡t A ❜❡ ❛ ✜♥✐t❡ s❡t✳ ❊✈❡r②❜♦❞② ❦♥♦✇s ✇❤❛t ❝❧♦♥❡ ✐s✳✳✳ ❙♦♠❡ ◆♦t❛t✐♦♥s Om

A ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m✳

Rm

A ✐s t❤❡ s❡t ♦❢ ✜♥✐t❛r② r❡❧❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m✳

✐s t❤❡ ❧❡❛st ❝❧♦♥❡ ❝♦♥t❛✐♥✐♥❣ ✳ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ t❤❡ r❡❧❛t✐♦♥ ✳ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ ❡✈❡r② r❡❧❛t✐♦♥ ❢r♦♠ ✳ ✐s t❤❡ ❧❡❛st r❡❧❛t✐♦♥ ♣r❡s❡r✈❡❞ ❜② ❛ ❝❧♦♥❡ ❛♥❞ ❝♦♥t❛✐♥✐♥❣ ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

▼❛✐♥ ♥♦t❛t✐♦♥s✳ ▲❡t A ❜❡ ❛ ✜♥✐t❡ s❡t✳ ❊✈❡r②❜♦❞② ❦♥♦✇s ✇❤❛t ❝❧♦♥❡ ✐s✳✳✳ ❙♦♠❡ ◆♦t❛t✐♦♥s Om

A ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m✳

Rm

A ✐s t❤❡ s❡t ♦❢ ✜♥✐t❛r② r❡❧❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m✳

[M] ✐s t❤❡ ❧❡❛st ❝❧♦♥❡ ❝♦♥t❛✐♥✐♥❣ M✳ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ t❤❡ r❡❧❛t✐♦♥ ✳ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ ❡✈❡r② r❡❧❛t✐♦♥ ❢r♦♠ ✳ ✐s t❤❡ ❧❡❛st r❡❧❛t✐♦♥ ♣r❡s❡r✈❡❞ ❜② ❛ ❝❧♦♥❡ ❛♥❞ ❝♦♥t❛✐♥✐♥❣ ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

▼❛✐♥ ♥♦t❛t✐♦♥s✳ ▲❡t A ❜❡ ❛ ✜♥✐t❡ s❡t✳ ❊✈❡r②❜♦❞② ❦♥♦✇s ✇❤❛t ❝❧♦♥❡ ✐s✳✳✳ ❙♦♠❡ ◆♦t❛t✐♦♥s Om

A ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m✳

Rm

A ✐s t❤❡ s❡t ♦❢ ✜♥✐t❛r② r❡❧❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m✳

[M] ✐s t❤❡ ❧❡❛st ❝❧♦♥❡ ❝♦♥t❛✐♥✐♥❣ M✳ Pol(ρ) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ t❤❡ r❡❧❛t✐♦♥ ρ✳ Pol(F) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ ❡✈❡r② r❡❧❛t✐♦♥ ❢r♦♠ F✳ ✐s t❤❡ ❧❡❛st r❡❧❛t✐♦♥ ♣r❡s❡r✈❡❞ ❜② ❛ ❝❧♦♥❡ ❛♥❞ ❝♦♥t❛✐♥✐♥❣ ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

▼❛✐♥ ♥♦t❛t✐♦♥s✳ ▲❡t A ❜❡ ❛ ✜♥✐t❡ s❡t✳ ❊✈❡r②❜♦❞② ❦♥♦✇s ✇❤❛t ❝❧♦♥❡ ✐s✳✳✳ ❙♦♠❡ ◆♦t❛t✐♦♥s Om

A ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m✳

Rm

A ✐s t❤❡ s❡t ♦❢ ✜♥✐t❛r② r❡❧❛t✐♦♥s ♦♥ A ♦❢ ❛r✐t② ❛t ♠♦st m✳

[M] ✐s t❤❡ ❧❡❛st ❝❧♦♥❡ ❝♦♥t❛✐♥✐♥❣ M✳ Pol(ρ) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ t❤❡ r❡❧❛t✐♦♥ ρ✳ Pol(F) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♣r❡s❡r✈✐♥❣ ❡✈❡r② r❡❧❛t✐♦♥ ❢r♦♠ F✳ ρC ✐s t❤❡ ❧❡❛st r❡❧❛t✐♦♥ ♣r❡s❡r✈❡❞ ❜② ❛ ❝❧♦♥❡ C ❛♥❞ ❝♦♥t❛✐♥✐♥❣ ρ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

◆❡❛r✲✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ❆ ♥❡❛r ✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ✭◆❯✮ ✐s ❛♥ ♦♣❡r❛t✐♦♥ f s❛t✐s❢②✐♥❣ f(x, . . . , x, y) = f(x, . . . , x, y, x) = · · · = f(y, x, . . . , x) = x. Pr♦♣❡rt✐❡s ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ♦❢ ❛r✐t② ✐s ❞❡✜♥❡❞ ❜② r❡❧❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st ✭❑✳ ❆✳ ❇❛❦❡r✱ ❆✳ ❋✳ P✐①❧❡②✮✳ ❋♦r ❛♥② ◆❯ t❤❡r❡ ❡①✐sts ✜♥✐t❡❧② ♠❛♥② ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ✐t✳ ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❍✳▲❛❦s❡r ❛♥❞ ❙✳ ❑❡r❦❤♦✛ s✉❝❝❡ss❢✉❧❧② st✉❞✐❡❞ t❤❡ ♦r❞❡r ♦❢ ❝❧♦♥❡s ✇✐t❤ ◆❯ ✭t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ❣❡♥❡r❛t✐♥❣ s❡t✮✳ ❙✉❜❧❛tt✐❝❡s ♦❢ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ◆❯ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ❛ ❝♦♠♣✉t❡r ✭✇❡ ❢♦✉♥❞ ✶✱✾✶✽✱✵✹✵ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ♠❛❥♦r✐t② ♦♣❡r❛t✐♦♥ ♦♥ ✸ ❡❧❡♠❡♥ts✱ ❛♥❞ s♦ ♦♥✮✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

◆❡❛r✲✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ❆ ♥❡❛r ✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ✭◆❯✮ ✐s ❛♥ ♦♣❡r❛t✐♦♥ f s❛t✐s❢②✐♥❣ f(x, . . . , x, y) = f(x, . . . , x, y, x) = · · · = f(y, x, . . . , x) = x. Pr♦♣❡rt✐❡s ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ♦❢ ❛r✐t② n ✐s ❞❡✜♥❡❞ ❜② r❡❧❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st n − 1 ✭❑✳ ❆✳ ❇❛❦❡r✱ ❆✳ ❋✳ P✐①❧❡②✮✳ ❋♦r ❛♥② ◆❯ t❤❡r❡ ❡①✐sts ✜♥✐t❡❧② ♠❛♥② ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ✐t✳ ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❍✳▲❛❦s❡r ❛♥❞ ❙✳ ❑❡r❦❤♦✛ s✉❝❝❡ss❢✉❧❧② st✉❞✐❡❞ t❤❡ ♦r❞❡r ♦❢ ❝❧♦♥❡s ✇✐t❤ ◆❯ ✭t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ❣❡♥❡r❛t✐♥❣ s❡t✮✳ ❙✉❜❧❛tt✐❝❡s ♦❢ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ◆❯ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ❛ ❝♦♠♣✉t❡r ✭✇❡ ❢♦✉♥❞ ✶✱✾✶✽✱✵✹✵ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ♠❛❥♦r✐t② ♦♣❡r❛t✐♦♥ ♦♥ ✸ ❡❧❡♠❡♥ts✱ ❛♥❞ s♦ ♦♥✮✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

◆❡❛r✲✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ❆ ♥❡❛r ✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ✭◆❯✮ ✐s ❛♥ ♦♣❡r❛t✐♦♥ f s❛t✐s❢②✐♥❣ f(x, . . . , x, y) = f(x, . . . , x, y, x) = · · · = f(y, x, . . . , x) = x. Pr♦♣❡rt✐❡s ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ♦❢ ❛r✐t② n ✐s ❞❡✜♥❡❞ ❜② r❡❧❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st n − 1 ✭❑✳ ❆✳ ❇❛❦❡r✱ ❆✳ ❋✳ P✐①❧❡②✮✳ ❋♦r ❛♥② ◆❯ t❤❡r❡ ❡①✐sts ✜♥✐t❡❧② ♠❛♥② ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ✐t✳ ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❍✳▲❛❦s❡r ❛♥❞ ❙✳ ❑❡r❦❤♦✛ s✉❝❝❡ss❢✉❧❧② st✉❞✐❡❞ t❤❡ ♦r❞❡r ♦❢ ❝❧♦♥❡s ✇✐t❤ ◆❯ ✭t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ❣❡♥❡r❛t✐♥❣ s❡t✮✳ ❙✉❜❧❛tt✐❝❡s ♦❢ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ◆❯ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ❛ ❝♦♠♣✉t❡r ✭✇❡ ❢♦✉♥❞ ✶✱✾✶✽✱✵✹✵ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ♠❛❥♦r✐t② ♦♣❡r❛t✐♦♥ ♦♥ ✸ ❡❧❡♠❡♥ts✱ ❛♥❞ s♦ ♦♥✮✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

◆❡❛r✲✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ❆ ♥❡❛r ✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ✭◆❯✮ ✐s ❛♥ ♦♣❡r❛t✐♦♥ f s❛t✐s❢②✐♥❣ f(x, . . . , x, y) = f(x, . . . , x, y, x) = · · · = f(y, x, . . . , x) = x. Pr♦♣❡rt✐❡s ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ♦❢ ❛r✐t② n ✐s ❞❡✜♥❡❞ ❜② r❡❧❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st n − 1 ✭❑✳ ❆✳ ❇❛❦❡r✱ ❆✳ ❋✳ P✐①❧❡②✮✳ ❋♦r ❛♥② ◆❯ t❤❡r❡ ❡①✐sts ✜♥✐t❡❧② ♠❛♥② ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ✐t✳ ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❍✳▲❛❦s❡r ❛♥❞ ❙✳ ❑❡r❦❤♦✛ s✉❝❝❡ss❢✉❧❧② st✉❞✐❡❞ t❤❡ ♦r❞❡r ♦❢ ❝❧♦♥❡s ✇✐t❤ ◆❯ ✭t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ❣❡♥❡r❛t✐♥❣ s❡t✮✳ ❙✉❜❧❛tt✐❝❡s ♦❢ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ◆❯ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ❛ ❝♦♠♣✉t❡r ✭✇❡ ❢♦✉♥❞ ✶✱✾✶✽✱✵✹✵ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ♠❛❥♦r✐t② ♦♣❡r❛t✐♦♥ ♦♥ ✸ ❡❧❡♠❡♥ts✱ ❛♥❞ s♦ ♦♥✮✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

◆❡❛r✲✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ❆ ♥❡❛r ✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ✭◆❯✮ ✐s ❛♥ ♦♣❡r❛t✐♦♥ f s❛t✐s❢②✐♥❣ f(x, . . . , x, y) = f(x, . . . , x, y, x) = · · · = f(y, x, . . . , x) = x. Pr♦♣❡rt✐❡s ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ♦❢ ❛r✐t② n ✐s ❞❡✜♥❡❞ ❜② r❡❧❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st n − 1 ✭❑✳ ❆✳ ❇❛❦❡r✱ ❆✳ ❋✳ P✐①❧❡②✮✳ ❋♦r ❛♥② ◆❯ t❤❡r❡ ❡①✐sts ✜♥✐t❡❧② ♠❛♥② ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ✐t✳ ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❍✳▲❛❦s❡r ❛♥❞ ❙✳ ❑❡r❦❤♦✛ s✉❝❝❡ss❢✉❧❧② st✉❞✐❡❞ t❤❡ ♦r❞❡r ♦❢ ❝❧♦♥❡s ✇✐t❤ ◆❯ ✭t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ❣❡♥❡r❛t✐♥❣ s❡t✮✳ ❙✉❜❧❛tt✐❝❡s ♦❢ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ◆❯ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ❛ ❝♦♠♣✉t❡r ✭✇❡ ❢♦✉♥❞ ✶✱✾✶✽✱✵✹✵ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ♠❛❥♦r✐t② ♦♣❡r❛t✐♦♥ ♦♥ ✸ ❡❧❡♠❡♥ts✱ ❛♥❞ s♦ ♦♥✮✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

◆❡❛r✲✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ❆ ♥❡❛r ✉♥❛♥✐♠✐t② ♦♣❡r❛t✐♦♥ ✭◆❯✮ ✐s ❛♥ ♦♣❡r❛t✐♦♥ f s❛t✐s❢②✐♥❣ f(x, . . . , x, y) = f(x, . . . , x, y, x) = · · · = f(y, x, . . . , x) = x. Pr♦♣❡rt✐❡s ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ♦❢ ❛r✐t② n ✐s ❞❡✜♥❡❞ ❜② r❡❧❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st n − 1 ✭❑✳ ❆✳ ❇❛❦❡r✱ ❆✳ ❋✳ P✐①❧❡②✮✳ ❋♦r ❛♥② ◆❯ t❤❡r❡ ❡①✐sts ✜♥✐t❡❧② ♠❛♥② ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ✐t✳ ❊✈❡r② ❝❧♦♥❡ ✇✐t❤ ❛ ◆❯ ♦♣❡r❛t✐♦♥ ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❍✳▲❛❦s❡r ❛♥❞ ❙✳ ❑❡r❦❤♦✛ s✉❝❝❡ss❢✉❧❧② st✉❞✐❡❞ t❤❡ ♦r❞❡r ♦❢ ❝❧♦♥❡s ✇✐t❤ ◆❯ ✭t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ❣❡♥❡r❛t✐♥❣ s❡t✮✳ ❙✉❜❧❛tt✐❝❡s ♦❢ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ◆❯ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ❛ ❝♦♠♣✉t❡r ✭✇❡ ❢♦✉♥❞ ✶✱✾✶✽✱✵✹✵ ❝❧♦♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ♠❛❥♦r✐t② ♦♣❡r❛t✐♦♥ ♦♥ ✸ ❡❧❡♠❡♥ts✱ ❛♥❞ s♦ ♦♥✮✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❚✇♦ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s ◗✉❡st✐♦♥ ❍♦✇ ❝❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ Pr♦❜❧❡♠ ✶

• ✐✈❡♥ ❛ ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s

✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ❝♦♥t❛✐♥s ❛ ◆❯ ♦♣❡r❛t✐♦♥✳ Pr♦❜❧❡♠ ✷

• ✐✈❡♥ ❛ ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s

✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ❝♦♥t❛✐♥s ❛ ◆❯ ♦♣❡r❛t✐♦♥✳ ◆♦t❡ t❤❛t ❢♦r ❛♥② ✜①❡❞ ✇❡ ❝❛♥ ❡❛s✐❧② ❝❤❡❝❦ ✐❢ ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯ ♦❢ ❛r✐t② ✳ ❚❤✉s✱ t♦ s♦❧✈❡ Pr♦❜❧❡♠ ✶ ❛♥❞ Pr♦❜❧❡♠ ✷ ✇❡ ❥✉st ♥❡❡❞ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ◆❯✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❚✇♦ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s ◗✉❡st✐♦♥ ❍♦✇ ❝❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ Pr♦❜❧❡♠ ✶

• ✐✈❡♥ ❛ ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s M✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r [M] ❝♦♥t❛✐♥s

❛ ◆❯ ♦♣❡r❛t✐♦♥✳ Pr♦❜❧❡♠ ✷

• ✐✈❡♥ ❛ ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s F✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r Pol(F) ❝♦♥t❛✐♥s

❛ ◆❯ ♦♣❡r❛t✐♦♥✳ ◆♦t❡ t❤❛t ❢♦r ❛♥② ✜①❡❞ ✇❡ ❝❛♥ ❡❛s✐❧② ❝❤❡❝❦ ✐❢ ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯ ♦❢ ❛r✐t② ✳ ❚❤✉s✱ t♦ s♦❧✈❡ Pr♦❜❧❡♠ ✶ ❛♥❞ Pr♦❜❧❡♠ ✷ ✇❡ ❥✉st ♥❡❡❞ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ◆❯✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❚✇♦ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s ◗✉❡st✐♦♥ ❍♦✇ ❝❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ Pr♦❜❧❡♠ ✶

• ✐✈❡♥ ❛ ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s M✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r [M] ❝♦♥t❛✐♥s

❛ ◆❯ ♦♣❡r❛t✐♦♥✳ Pr♦❜❧❡♠ ✷

• ✐✈❡♥ ❛ ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s F✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r Pol(F) ❝♦♥t❛✐♥s

❛ ◆❯ ♦♣❡r❛t✐♦♥✳ ◆♦t❡ t❤❛t ❢♦r ❛♥② ✜①❡❞ n ✇❡ ❝❛♥ ❡❛s✐❧② ❝❤❡❝❦ ✐❢ ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯ ♦❢ ❛r✐t② n✳ ❚❤✉s✱ t♦ s♦❧✈❡ Pr♦❜❧❡♠ ✶ ❛♥❞ Pr♦❜❧❡♠ ✷ ✇❡ ❥✉st ♥❡❡❞ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ◆❯✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❉❡✜♥✐t✐♦♥ NU(C) ❞❡♥♦t❡s t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ◆❯ ✐♥ ❛ ❝❧♦♥❡ C✳ P✉t NU(C) = ∞ ✐❢ C ❞♦❡s♥✬t ❝♦♥t❛✐♥ ❛ ◆❯✳ ❋❛❝t Pr♦❜❧❡♠ ✶ ✐s ❞❡❝✐❞❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s ❝♦♠♣✉t❛❜❧❡✳ Pr♦❜❧❡♠ ✷ ✐s ❞❡❝✐❞❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s ❝♦♠♣✉t❛❜❧❡✳ ❋♦r ✐❞❡♠♣♦t❡♥t ♦♣❡r❛t✐♦♥s ❛♥❞ ❝♦♥s❡r✈❛t✐✈❡ ♦♣❡r❛t✐♦♥s ✇❡ ♣✉t

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❉❡✜♥✐t✐♦♥ NU(C) ❞❡♥♦t❡s t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ◆❯ ✐♥ ❛ ❝❧♦♥❡ C✳ P✉t NU(C) = ∞ ✐❢ C ❞♦❡s♥✬t ❝♦♥t❛✐♥ ❛ ◆❯✳ NUOper(m) = max{NU(C) | C = [M], M ⊆ Om

A , NU(C) < ∞},

NURel(m) = max{NU(C) | C = Pol(F), F ⊆ Rm

A , NU(C) < ∞}.

❋❛❝t Pr♦❜❧❡♠ ✶ ✐s ❞❡❝✐❞❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s ❝♦♠♣✉t❛❜❧❡✳ Pr♦❜❧❡♠ ✷ ✐s ❞❡❝✐❞❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s ❝♦♠♣✉t❛❜❧❡✳ ❋♦r ✐❞❡♠♣♦t❡♥t ♦♣❡r❛t✐♦♥s ❛♥❞ ❝♦♥s❡r✈❛t✐✈❡ ♦♣❡r❛t✐♦♥s ✇❡ ♣✉t

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❉❡✜♥✐t✐♦♥ NU(C) ❞❡♥♦t❡s t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ◆❯ ✐♥ ❛ ❝❧♦♥❡ C✳ P✉t NU(C) = ∞ ✐❢ C ❞♦❡s♥✬t ❝♦♥t❛✐♥ ❛ ◆❯✳ NUOper(m) = max{NU(C) | C = [M], M ⊆ Om

A , NU(C) < ∞},

NURel(m) = max{NU(C) | C = Pol(F), F ⊆ Rm

A , NU(C) < ∞}.

❋❛❝t Pr♦❜❧❡♠ ✶ ✐s ❞❡❝✐❞❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ NUOper(m) ✐s ❝♦♠♣✉t❛❜❧❡✳ Pr♦❜❧❡♠ ✷ ✐s ❞❡❝✐❞❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ NURel(m) ✐s ❝♦♠♣✉t❛❜❧❡✳ ❋♦r ✐❞❡♠♣♦t❡♥t ♦♣❡r❛t✐♦♥s ❛♥❞ ❝♦♥s❡r✈❛t✐✈❡ ♦♣❡r❛t✐♦♥s ✇❡ ♣✉t

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❉❡✜♥✐t✐♦♥ NU(C) ❞❡♥♦t❡s t❤❡ ♠✐♥✐♠❛❧ ❛r✐t② ♦❢ ❛ ◆❯ ✐♥ ❛ ❝❧♦♥❡ C✳ P✉t NU(C) = ∞ ✐❢ C ❞♦❡s♥✬t ❝♦♥t❛✐♥ ❛ ◆❯✳ NUOper(m) = max{NU(C) | C = [M], M ⊆ Om

A , NU(C) < ∞},

NURel(m) = max{NU(C) | C = Pol(F), F ⊆ Rm

A , NU(C) < ∞}.

❋❛❝t Pr♦❜❧❡♠ ✶ ✐s ❞❡❝✐❞❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ NUOper(m) ✐s ❝♦♠♣✉t❛❜❧❡✳ Pr♦❜❧❡♠ ✷ ✐s ❞❡❝✐❞❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ NURel(m) ✐s ❝♦♠♣✉t❛❜❧❡✳ ❋♦r ✐❞❡♠♣♦t❡♥t ♦♣❡r❛t✐♦♥s IOA ❛♥❞ ❝♦♥s❡r✈❛t✐✈❡ ♦♣❡r❛t✐♦♥s COA ✇❡ ♣✉t NUIdemOper(m) = max{NU(C) | C = [M], M ⊆ IOm

A , NU(C) < ∞},

NUConsOper(m) = max{NU(C) | C = [M], M ⊆ COm

A , NU(C) < ∞}.

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

Pr♦❜❧❡♠ ✶✿ ●✐✈❡♥ ❛ ✜♥✐t❡ ❛❧❣❡❜r❛✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❝♦♥t❛✐♥s ❛ ♥❡❛r✲✉♥❛♥✐♠✐t② t❡r♠ ❲❤❛t ✇❛s ❦♥♦✇♥ ▼✳▼❛r♦t✐ ♣r♦✈❡❞ t❤❛t Pr♦❜❧❡♠ ✶ ✐s ❛❧❣♦r✐t❤♠✐❝❛❧❧② ❞❡❝✐❞❛❜❧❡✳ ◆♦ ✉♣♣❡r ❜♦✉♥❞ ♦♥ ✇❛s ❦♥♦✇♥✳ ❚❤❡♦r❡♠

✶

✳

✷

✳

✸

✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

Pr♦❜❧❡♠ ✶✿ ●✐✈❡♥ ❛ ✜♥✐t❡ ❛❧❣❡❜r❛✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❝♦♥t❛✐♥s ❛ ♥❡❛r✲✉♥❛♥✐♠✐t② t❡r♠ ❲❤❛t ✇❛s ❦♥♦✇♥ ▼✳▼❛r♦t✐ ♣r♦✈❡❞ t❤❛t Pr♦❜❧❡♠ ✶ ✐s ❛❧❣♦r✐t❤♠✐❝❛❧❧② ❞❡❝✐❞❛❜❧❡✳ ◆♦ ✉♣♣❡r ❜♦✉♥❞ ♦♥ ✇❛s ❦♥♦✇♥✳ ❚❤❡♦r❡♠

✶

✳

✷

✳

✸

✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

Pr♦❜❧❡♠ ✶✿ ●✐✈❡♥ ❛ ✜♥✐t❡ ❛❧❣❡❜r❛✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❝♦♥t❛✐♥s ❛ ♥❡❛r✲✉♥❛♥✐♠✐t② t❡r♠ ❲❤❛t ✇❛s ❦♥♦✇♥ ▼✳▼❛r♦t✐ ♣r♦✈❡❞ t❤❛t Pr♦❜❧❡♠ ✶ ✐s ❛❧❣♦r✐t❤♠✐❝❛❧❧② ❞❡❝✐❞❛❜❧❡✳ ◆♦ ✉♣♣❡r ❜♦✉♥❞ ♦♥ NUOper(m) ✇❛s ❦♥♦✇♥✳ ❚❤❡♦r❡♠

✶

✳

✷

✳

✸

✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

Pr♦❜❧❡♠ ✶✿ ●✐✈❡♥ ❛ ✜♥✐t❡ ❛❧❣❡❜r❛✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❝♦♥t❛✐♥s ❛ ♥❡❛r✲✉♥❛♥✐♠✐t② t❡r♠ ❲❤❛t ✇❛s ❦♥♦✇♥ ▼✳▼❛r♦t✐ ♣r♦✈❡❞ t❤❛t Pr♦❜❧❡♠ ✶ ✐s ❛❧❣♦r✐t❤♠✐❝❛❧❧② ❞❡❝✐❞❛❜❧❡✳ ◆♦ ✉♣♣❡r ❜♦✉♥❞ ♦♥ NUOper(m) ✇❛s ❦♥♦✇♥✳ ❚❤❡♦r❡♠

✶ NUOper(m) ≤ |A|2 · (|A| · m)(3|A|)|A|✳ ✷

✳

✸

✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

Pr♦❜❧❡♠ ✶✿ ●✐✈❡♥ ❛ ✜♥✐t❡ ❛❧❣❡❜r❛✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❝♦♥t❛✐♥s ❛ ♥❡❛r✲✉♥❛♥✐♠✐t② t❡r♠ ❲❤❛t ✇❛s ❦♥♦✇♥ ▼✳▼❛r♦t✐ ♣r♦✈❡❞ t❤❛t Pr♦❜❧❡♠ ✶ ✐s ❛❧❣♦r✐t❤♠✐❝❛❧❧② ❞❡❝✐❞❛❜❧❡✳ ◆♦ ✉♣♣❡r ❜♦✉♥❞ ♦♥ NUOper(m) ✇❛s ❦♥♦✇♥✳ ❚❤❡♦r❡♠

✶ NUOper(m) ≤ |A|2 · (|A| · m)(3|A|)|A|✳ ✷ NUIdemOper(m) ≤ m · |A|3✳ ✸

✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

Pr♦❜❧❡♠ ✶✿ ●✐✈❡♥ ❛ ✜♥✐t❡ ❛❧❣❡❜r❛✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❝♦♥t❛✐♥s ❛ ♥❡❛r✲✉♥❛♥✐♠✐t② t❡r♠ ❲❤❛t ✇❛s ❦♥♦✇♥ ▼✳▼❛r♦t✐ ♣r♦✈❡❞ t❤❛t Pr♦❜❧❡♠ ✶ ✐s ❛❧❣♦r✐t❤♠✐❝❛❧❧② ❞❡❝✐❞❛❜❧❡✳ ◆♦ ✉♣♣❡r ❜♦✉♥❞ ♦♥ NUOper(m) ✇❛s ❦♥♦✇♥✳ ❚❤❡♦r❡♠

✶ NUOper(m) ≤ |A|2 · (|A| · m)(3|A|)|A|✳ ✷ NUIdemOper(m) ≤ m · |A|3✳ ✸ NUConsOper(m) ≤ m · |A|2✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

Pr♦❜❧❡♠ ✷✿ ●✐✈❡♥ ❛ r❡❧❛t✐♦♥✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❛❞♠✐ts ❛ ◆❯ ♦♣❡r❛t✐♦♥ ❲❤❛t ✇❛s ❦♥♦✇♥ ▲✳❇❛rt♦ ♣r♦✈❡❞ ❩❛❞♦r✐ ❈♦♥❥❡❝t✉r❡ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❡ ❞❡❝✐❞❛❜✐❧✐t② ♦❢ Pr♦❜❧❡♠ ✷✳ ▲✳❇❛rt♦ s❤♦✇❡❞ t❤❛t ✳ ❚❤❡♦r❡♠ ✳ ❚❤❡♦r❡♠

✶

✳

✷

❢♦r ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

Pr♦❜❧❡♠ ✷✿ ●✐✈❡♥ ❛ r❡❧❛t✐♦♥✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❛❞♠✐ts ❛ ◆❯ ♦♣❡r❛t✐♦♥ ❲❤❛t ✇❛s ❦♥♦✇♥ ▲✳❇❛rt♦ ♣r♦✈❡❞ ❩❛❞♦r✐ ❈♦♥❥❡❝t✉r❡ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❡ ❞❡❝✐❞❛❜✐❧✐t② ♦❢ Pr♦❜❧❡♠ ✷✳ ▲✳❇❛rt♦ s❤♦✇❡❞ t❤❛t ✳ ❚❤❡♦r❡♠ ✳ ❚❤❡♦r❡♠

✶

✳

✷

❢♦r ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

Pr♦❜❧❡♠ ✷✿ ●✐✈❡♥ ❛ r❡❧❛t✐♦♥✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❛❞♠✐ts ❛ ◆❯ ♦♣❡r❛t✐♦♥ ❲❤❛t ✇❛s ❦♥♦✇♥ ▲✳❇❛rt♦ ♣r♦✈❡❞ ❩❛❞♦r✐ ❈♦♥❥❡❝t✉r❡ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❡ ❞❡❝✐❞❛❜✐❧✐t② ♦❢ Pr♦❜❧❡♠ ✷✳ ▲✳❇❛rt♦ s❤♦✇❡❞ t❤❛t NURel(m) ≤ 48|A|m ✳ ❚❤❡♦r❡♠ ✳ ❚❤❡♦r❡♠

✶

✳

✷

❢♦r ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

Pr♦❜❧❡♠ ✷✿ ●✐✈❡♥ ❛ r❡❧❛t✐♦♥✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❛❞♠✐ts ❛ ◆❯ ♦♣❡r❛t✐♦♥ ❲❤❛t ✇❛s ❦♥♦✇♥ ▲✳❇❛rt♦ ♣r♦✈❡❞ ❩❛❞♦r✐ ❈♦♥❥❡❝t✉r❡ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❡ ❞❡❝✐❞❛❜✐❧✐t② ♦❢ Pr♦❜❧❡♠ ✷✳ ▲✳❇❛rt♦ s❤♦✇❡❞ t❤❛t NURel(m) ≤ 48|A|m ✳ ❚❤❡♦r❡♠ NURel(m) ≤ ((|A| − 1)(m − 1))3|A| + 1✳ ❚❤❡♦r❡♠

✶

✳

✷

❢♦r ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

Pr♦❜❧❡♠ ✷✿ ●✐✈❡♥ ❛ r❡❧❛t✐♦♥✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❛❞♠✐ts ❛ ◆❯ ♦♣❡r❛t✐♦♥ ❲❤❛t ✇❛s ❦♥♦✇♥ ▲✳❇❛rt♦ ♣r♦✈❡❞ ❩❛❞♦r✐ ❈♦♥❥❡❝t✉r❡ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❡ ❞❡❝✐❞❛❜✐❧✐t② ♦❢ Pr♦❜❧❡♠ ✷✳ ▲✳❇❛rt♦ s❤♦✇❡❞ t❤❛t NURel(m) ≤ 48|A|m ✳ ❚❤❡♦r❡♠ NURel(m) ≤ ((|A| − 1)(m − 1))3|A| + 1✳ ❚❤❡♦r❡♠

✶ NURel(m) ≥ (m − 1)2|A|−2✳ ✷

❢♦r ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

Pr♦❜❧❡♠ ✷✿ ●✐✈❡♥ ❛ r❡❧❛t✐♦♥✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❛❞♠✐ts ❛ ◆❯ ♦♣❡r❛t✐♦♥ ❲❤❛t ✇❛s ❦♥♦✇♥ ▲✳❇❛rt♦ ♣r♦✈❡❞ ❩❛❞♦r✐ ❈♦♥❥❡❝t✉r❡ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❡ ❞❡❝✐❞❛❜✐❧✐t② ♦❢ Pr♦❜❧❡♠ ✷✳ ▲✳❇❛rt♦ s❤♦✇❡❞ t❤❛t NURel(m) ≤ 48|A|m ✳ ❚❤❡♦r❡♠ NURel(m) ≤ ((|A| − 1)(m − 1))3|A| + 1✳ ❚❤❡♦r❡♠

✶ NURel(m) ≥ (m − 1)2|A|−2✳ ✷ NURel(2) ≥ 22|A|−3 ❢♦r |A| ≥ 4✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ❚❤❡♦r❡♠

✶ ??? ≤ NUOper(m) ≤ |A|2 · (|A| · m)(3|A|)|A|✳ ✷ ??? ≤ NUIdemOper(m) ≤ m · |A|3✳ ✸ ??? ≤ NUConsOper(m) ≤ m · |A|2✳

❚❤❡♦r❡♠

✶

✳

✷

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ❚❤❡♦r❡♠

✶ ??? ≤ NUOper(m) ≤ |A|2 · (|A| · m)(3|A|)|A|✳ ✷ ??? ≤ NUIdemOper(m) ≤ m · |A|3✳ ✸ ??? ≤ NUConsOper(m) ≤ m · |A|2✳

❚❤❡♦r❡♠

✶

✳

✷

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ❚❤❡♦r❡♠

✶ ??? ≤ NUOper(m) ≤ |A|2 · (|A| · m)(3|A|)|A|✳ ✷ ??? ≤ NUIdemOper(m) ≤ m · |A|3✳ ✸ ??? ≤ NUConsOper(m) ≤ m · |A|2✳

❚❤❡♦r❡♠

✶ (m − 1)2|A|−2 ≤ NURel(m) ≤ ((|A| − 1)(m − 1))3|A| + 1✳ ✷ 22|A|−3 ≤ NURel(2) ≤ (|A| − 1)3|A| + 1

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ❚❤❡♦r❡♠

✶ ??? ≤ NUOper(m) ≤ |A|2 · (|A| · m)(3|A|)|A|✳ ✷ ??? ≤ NUIdemOper(m) ≤ m · |A|3✳ ✸ ??? ≤ NUConsOper(m) ≤ m · |A|2✳

❚❤❡♦r❡♠

✶ (m − 1)2|A|−2 ≤ NURel(m) ≤ ((|A| − 1)(m − 1))3|A| + 1✳ ✷ 22|A|−3 ≤ NURel(2) ≤ (|A| − 1)3|A| + 1

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❛❧❧② ✉s❡ t❤❡s❡ ❡st✐♠❛t❡s t♦ s♦❧✈❡ Pr♦❜❧❡♠ ✶ ❛♥❞ Pr♦❜❧❡♠ ✷❄ ❊①❛♠♣❧❡s ❚♦ ❝❤❡❝❦ t❤❛t ❜✐♥❛r② r❡❧❛t✐♦♥s ♦♥ ✸ ❡❧❡♠❡♥ts ❛❞♠✐t ❛ ◆❯ ✇❡ ♥❡❡❞ t♦ ❝❤❡❝❦ ❛❧❧ ◆❯ ♦❢ ❛r✐t② ✳ ❚♦ ❝❤❡❝❦ t❤❛t ❜✐♥❛r② ♦♣❡r❛t✐♦♥s ♦♥ ✸ ❡❧❡♠❡♥ts ❣❡♥❡r❛t❡ ❛ ◆❯ ✇❡ ♥❡❡❞ t♦ ❝❤❡❝❦ ❛❧❧ ◆❯ ♦❢ ❛r✐t②

❈❛♥ ✇❡ ❞♦ ❜❡tt❡r❄

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❛❧❧② ✉s❡ t❤❡s❡ ❡st✐♠❛t❡s t♦ s♦❧✈❡ Pr♦❜❧❡♠ ✶ ❛♥❞ Pr♦❜❧❡♠ ✷❄ ❊①❛♠♣❧❡s ❚♦ ❝❤❡❝❦ t❤❛t ❜✐♥❛r② r❡❧❛t✐♦♥s ♦♥ ✸ ❡❧❡♠❡♥ts ❛❞♠✐t ❛ ◆❯ ✇❡ ♥❡❡❞ t♦ ❝❤❡❝❦ ❛❧❧ ◆❯ ♦❢ ❛r✐t② 433✳ ❚♦ ❝❤❡❝❦ t❤❛t ❜✐♥❛r② ♦♣❡r❛t✐♦♥s ♦♥ ✸ ❡❧❡♠❡♥ts ❣❡♥❡r❛t❡ ❛ ◆❯ ✇❡ ♥❡❡❞ t♦ ❝❤❡❝❦ ❛❧❧ ◆❯ ♦❢ ❛r✐t②

❈❛♥ ✇❡ ❞♦ ❜❡tt❡r❄

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❛❧❧② ✉s❡ t❤❡s❡ ❡st✐♠❛t❡s t♦ s♦❧✈❡ Pr♦❜❧❡♠ ✶ ❛♥❞ Pr♦❜❧❡♠ ✷❄ ❊①❛♠♣❧❡s ❚♦ ❝❤❡❝❦ t❤❛t ❜✐♥❛r② r❡❧❛t✐♦♥s ♦♥ ✸ ❡❧❡♠❡♥ts ❛❞♠✐t ❛ ◆❯ ✇❡ ♥❡❡❞ t♦ ❝❤❡❝❦ ❛❧❧ ◆❯ ♦❢ ❛r✐t② 433✳ ❚♦ ❝❤❡❝❦ t❤❛t ❜✐♥❛r② ♦♣❡r❛t✐♦♥s ♦♥ ✸ ❡❧❡♠❡♥ts ❣❡♥❡r❛t❡ ❛ ◆❯ ✇❡ ♥❡❡❞ t♦ ❝❤❡❝❦ ❛❧❧ ◆❯ ♦❢ ❛r✐t② 32 · (3 · 2)93

❈❛♥ ✇❡ ❞♦ ❜❡tt❡r❄

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❛❧❧② ✉s❡ t❤❡s❡ ❡st✐♠❛t❡s t♦ s♦❧✈❡ Pr♦❜❧❡♠ ✶ ❛♥❞ Pr♦❜❧❡♠ ✷❄ ❊①❛♠♣❧❡s ❚♦ ❝❤❡❝❦ t❤❛t ❜✐♥❛r② r❡❧❛t✐♦♥s ♦♥ ✸ ❡❧❡♠❡♥ts ❛❞♠✐t ❛ ◆❯ ✇❡ ♥❡❡❞ t♦ ❝❤❡❝❦ ❛❧❧ ◆❯ ♦❢ ❛r✐t② 433✳ ❚♦ ❝❤❡❝❦ t❤❛t ❜✐♥❛r② ♦♣❡r❛t✐♦♥s ♦♥ ✸ ❡❧❡♠❡♥ts ❣❡♥❡r❛t❡ ❛ ◆❯ ✇❡ ♥❡❡❞ t♦ ❝❤❡❝❦ ❛❧❧ ◆❯ ♦❢ ❛r✐t② 32 · (3 · 2)93

❈❛♥ ✇❡ ❞♦ ❜❡tt❡r❄

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈r✐t❡r✐❛ ♦❢ ❡①✐st❡♥❝❡ ◆❯ ❢♦r ✐❞❡♠♣♦t❡♥t ❝❛s❡ ❉❡✜♥✐t✐♦♥ Block(D, B) =

∞

• n=1

Pol(Dn \ (D \ B)n) ❢♦r B ⊂ D ⊆ A✳ Lin(D, ϕ) = Pol{(x1, x2, x3, x4) | ϕ(x1) + ϕ(x2) = ϕ(x3) + ϕ(x4)} ❢♦r D ⊆ A✱ ❛ ✜♥✐t❡ ✜❡❧❞ F ❛♥❞ ❛ s✉r❥❡❝t✐✈❡ ♠❛♣♣✐♥❣ ϕ : D → F✳ Lin(D, ϕ) ✐s t❤❡ ❝❧♦♥❡ ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s t❤❛t ❛r❡ ❧✐♥❡❛r ♦♥ D ✇✐t❤ r❡s♣❡❝t t♦ ϕ✳ ❚❤❡♦r❡♠ ❆♥ ✐❞❡♠♣♦t❡♥t ❝❧♦♥❡ ❞♦❡s♥✬t ❝♦♥t❛✐♥ ❛ ◆❯ ✐✛

✶

❢♦r s♦♠❡ ✱ ♦r

✷

❢♦r s♦♠❡ ❛♥❞ ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈r✐t❡r✐❛ ♦❢ ❡①✐st❡♥❝❡ ◆❯ ❢♦r ✐❞❡♠♣♦t❡♥t ❝❛s❡ ❉❡✜♥✐t✐♦♥ Block(D, B) =

∞

• n=1

Pol(Dn \ (D \ B)n) ❢♦r B ⊂ D ⊆ A✳ Lin(D, ϕ) = Pol{(x1, x2, x3, x4) | ϕ(x1) + ϕ(x2) = ϕ(x3) + ϕ(x4)} ❢♦r D ⊆ A✱ ❛ ✜♥✐t❡ ✜❡❧❞ F ❛♥❞ ❛ s✉r❥❡❝t✐✈❡ ♠❛♣♣✐♥❣ ϕ : D → F✳ Lin(D, ϕ) ✐s t❤❡ ❝❧♦♥❡ ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s t❤❛t ❛r❡ ❧✐♥❡❛r ♦♥ D ✇✐t❤ r❡s♣❡❝t t♦ ϕ✳ ❚❤❡♦r❡♠ ❆♥ ✐❞❡♠♣♦t❡♥t ❝❧♦♥❡ C ❞♦❡s♥✬t ❝♦♥t❛✐♥ ❛ ◆❯ ✐✛

✶ C ⊆ Block(B, D) ❢♦r s♦♠❡ B, D✱ ♦r ✷ C ⊆ Lin(D, ϕ) ❢♦r s♦♠❡ D ❛♥❞ ϕ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈r✐t❡r✐❛ ♦❢ ❡①✐st❡♥❝❡ ◆❯ ❢♦r ✐❞❡♠♣♦t❡♥t ❝❛s❡ ❈♦r♦❧❧❛r② Block(B, D) ❛♥❞ Lin(D, ϕ) ❛r❡ ♠❛①✐♠❛❧ ✐❞❡♠♣♦t❡♥t ❝❧♦♥❡s t❤❛t ❞♦ ♥♦t ❝♦♥t❛✐♥ ❛ ◆❯✳ ❍♦✇ ❝❛♥ ✇❡ ❝❤❡❝❦ ✇❤❡t❤❡r ❛ ✜♥✐t❡ s❡t ♦❢ ✐❞❡♠♣♦t❡♥t ♦♣❡r❛t✐♦♥s ❣❡♥❡r❛t❡ ❛ ◆❯❄ ❲❡ ❝❤❡❝❦ ✇❤❡t❤❡r ❢♦r s♦♠❡ ✳ ◆♦t❡ t❤❛t ✐✛ ♣r❡s❡r✈❡s ❛♥❞ ❢♦r s♦♠❡ ✈❛r✐❛❜❧❡ ✇❡ ❤❛✈❡ ✳ ❙♦✱ ✐t ✐s ❡❛s②✦ ❲❡ ❝❤❡❝❦ t❤❛t ✳ ❊❛s②✦ ❚❤✐s ✐❞❡❛ ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞ ❢♦r ♥♦♥✐❞❡♠♣♦t❡♥t ❝❛s❡✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈r✐t❡r✐❛ ♦❢ ❡①✐st❡♥❝❡ ◆❯ ❢♦r ✐❞❡♠♣♦t❡♥t ❝❛s❡ ❈♦r♦❧❧❛r② Block(B, D) ❛♥❞ Lin(D, ϕ) ❛r❡ ♠❛①✐♠❛❧ ✐❞❡♠♣♦t❡♥t ❝❧♦♥❡s t❤❛t ❞♦ ♥♦t ❝♦♥t❛✐♥ ❛ ◆❯✳ ❍♦✇ ❝❛♥ ✇❡ ❝❤❡❝❦ ✇❤❡t❤❡r ❛ ✜♥✐t❡ s❡t ♦❢ ✐❞❡♠♣♦t❡♥t ♦♣❡r❛t✐♦♥s ❣❡♥❡r❛t❡ ❛ ◆❯❄ ❲❡ ❝❤❡❝❦ ✇❤❡t❤❡r ❢♦r s♦♠❡ ✳ ◆♦t❡ t❤❛t ✐✛ ♣r❡s❡r✈❡s ❛♥❞ ❢♦r s♦♠❡ ✈❛r✐❛❜❧❡ ✇❡ ❤❛✈❡ ✳ ❙♦✱ ✐t ✐s ❡❛s②✦ ❲❡ ❝❤❡❝❦ t❤❛t ✳ ❊❛s②✦ ❚❤✐s ✐❞❡❛ ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞ ❢♦r ♥♦♥✐❞❡♠♣♦t❡♥t ❝❛s❡✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈r✐t❡r✐❛ ♦❢ ❡①✐st❡♥❝❡ ◆❯ ❢♦r ✐❞❡♠♣♦t❡♥t ❝❛s❡ ❈♦r♦❧❧❛r② Block(B, D) ❛♥❞ Lin(D, ϕ) ❛r❡ ♠❛①✐♠❛❧ ✐❞❡♠♣♦t❡♥t ❝❧♦♥❡s t❤❛t ❞♦ ♥♦t ❝♦♥t❛✐♥ ❛ ◆❯✳ ❍♦✇ ❝❛♥ ✇❡ ❝❤❡❝❦ ✇❤❡t❤❡r ❛ ✜♥✐t❡ s❡t ♦❢ ✐❞❡♠♣♦t❡♥t ♦♣❡r❛t✐♦♥s ❣❡♥❡r❛t❡ ❛ ◆❯❄ ❲❡ ❝❤❡❝❦ ✇❤❡t❤❡r M ⊆ Block(D, B) ❢♦r s♦♠❡ B ⊂ D ⊆ A✳ ◆♦t❡ t❤❛t f ∈ Block(D, B) ✐✛ f ♣r❡s❡r✈❡s D ❛♥❞ ❢♦r s♦♠❡ ✈❛r✐❛❜❧❡ i ✇❡ ❤❛✈❡ f(D, D, . . . , D, B

• i

, D, . . . , D) ⊆ B✳ ❙♦✱ ✐t ✐s ❡❛s②✦ ❲❡ ❝❤❡❝❦ t❤❛t ✳ ❊❛s②✦ ❚❤✐s ✐❞❡❛ ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞ ❢♦r ♥♦♥✐❞❡♠♣♦t❡♥t ❝❛s❡✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈r✐t❡r✐❛ ♦❢ ❡①✐st❡♥❝❡ ◆❯ ❢♦r ✐❞❡♠♣♦t❡♥t ❝❛s❡ ❈♦r♦❧❧❛r② Block(B, D) ❛♥❞ Lin(D, ϕ) ❛r❡ ♠❛①✐♠❛❧ ✐❞❡♠♣♦t❡♥t ❝❧♦♥❡s t❤❛t ❞♦ ♥♦t ❝♦♥t❛✐♥ ❛ ◆❯✳ ❍♦✇ ❝❛♥ ✇❡ ❝❤❡❝❦ ✇❤❡t❤❡r ❛ ✜♥✐t❡ s❡t ♦❢ ✐❞❡♠♣♦t❡♥t ♦♣❡r❛t✐♦♥s ❣❡♥❡r❛t❡ ❛ ◆❯❄ ❲❡ ❝❤❡❝❦ ✇❤❡t❤❡r M ⊆ Block(D, B) ❢♦r s♦♠❡ B ⊂ D ⊆ A✳ ◆♦t❡ t❤❛t f ∈ Block(D, B) ✐✛ f ♣r❡s❡r✈❡s D ❛♥❞ ❢♦r s♦♠❡ ✈❛r✐❛❜❧❡ i ✇❡ ❤❛✈❡ f(D, D, . . . , D, B

• i

, D, . . . , D) ⊆ B✳ ❙♦✱ ✐t ✐s ❡❛s②✦ ❲❡ ❝❤❡❝❦ t❤❛t ✳ ❊❛s②✦ ❚❤✐s ✐❞❡❛ ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞ ❢♦r ♥♦♥✐❞❡♠♣♦t❡♥t ❝❛s❡✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈r✐t❡r✐❛ ♦❢ ❡①✐st❡♥❝❡ ◆❯ ❢♦r ✐❞❡♠♣♦t❡♥t ❝❛s❡ ❈♦r♦❧❧❛r② Block(B, D) ❛♥❞ Lin(D, ϕ) ❛r❡ ♠❛①✐♠❛❧ ✐❞❡♠♣♦t❡♥t ❝❧♦♥❡s t❤❛t ❞♦ ♥♦t ❝♦♥t❛✐♥ ❛ ◆❯✳ ❍♦✇ ❝❛♥ ✇❡ ❝❤❡❝❦ ✇❤❡t❤❡r ❛ ✜♥✐t❡ s❡t ♦❢ ✐❞❡♠♣♦t❡♥t ♦♣❡r❛t✐♦♥s ❣❡♥❡r❛t❡ ❛ ◆❯❄ ❲❡ ❝❤❡❝❦ ✇❤❡t❤❡r M ⊆ Block(D, B) ❢♦r s♦♠❡ B ⊂ D ⊆ A✳ ◆♦t❡ t❤❛t f ∈ Block(D, B) ✐✛ f ♣r❡s❡r✈❡s D ❛♥❞ ❢♦r s♦♠❡ ✈❛r✐❛❜❧❡ i ✇❡ ❤❛✈❡ f(D, D, . . . , D, B

• i

, D, . . . , D) ⊆ B✳ ❙♦✱ ✐t ✐s ❡❛s②✦ ❲❡ ❝❤❡❝❦ t❤❛t M ⊆ Lin(D, ϕ)✳ ❊❛s②✦ ❚❤✐s ✐❞❡❛ ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞ ❢♦r ♥♦♥✐❞❡♠♣♦t❡♥t ❝❛s❡✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈r✐t❡r✐❛ ♦❢ ❡①✐st❡♥❝❡ ◆❯ ❢♦r ✐❞❡♠♣♦t❡♥t ❝❛s❡ ❈♦r♦❧❧❛r② Block(B, D) ❛♥❞ Lin(D, ϕ) ❛r❡ ♠❛①✐♠❛❧ ✐❞❡♠♣♦t❡♥t ❝❧♦♥❡s t❤❛t ❞♦ ♥♦t ❝♦♥t❛✐♥ ❛ ◆❯✳ ❍♦✇ ❝❛♥ ✇❡ ❝❤❡❝❦ ✇❤❡t❤❡r ❛ ✜♥✐t❡ s❡t ♦❢ ✐❞❡♠♣♦t❡♥t ♦♣❡r❛t✐♦♥s ❣❡♥❡r❛t❡ ❛ ◆❯❄ ❲❡ ❝❤❡❝❦ ✇❤❡t❤❡r M ⊆ Block(D, B) ❢♦r s♦♠❡ B ⊂ D ⊆ A✳ ◆♦t❡ t❤❛t f ∈ Block(D, B) ✐✛ f ♣r❡s❡r✈❡s D ❛♥❞ ❢♦r s♦♠❡ ✈❛r✐❛❜❧❡ i ✇❡ ❤❛✈❡ f(D, D, . . . , D, B

• i

, D, . . . , D) ⊆ B✳ ❙♦✱ ✐t ✐s ❡❛s②✦ ❲❡ ❝❤❡❝❦ t❤❛t M ⊆ Lin(D, ϕ)✳ ❊❛s②✦ ❚❤✐s ✐❞❡❛ ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞ ❢♦r ♥♦♥✐❞❡♠♣♦t❡♥t ❝❛s❡✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❈r✐t❡r✐❛ ♦❢ ❡①✐st❡♥❝❡ ◆❯ ❢♦r ✐❞❡♠♣♦t❡♥t ❝❛s❡ ❈♦r♦❧❧❛r② Block(B, D) ❛♥❞ Lin(D, ϕ) ❛r❡ ♠❛①✐♠❛❧ ✐❞❡♠♣♦t❡♥t ❝❧♦♥❡s t❤❛t ❞♦ ♥♦t ❝♦♥t❛✐♥ ❛ ◆❯✳ ❍♦✇ ❝❛♥ ✇❡ ❝❤❡❝❦ ✇❤❡t❤❡r ❛ ✜♥✐t❡ s❡t ♦❢ ✐❞❡♠♣♦t❡♥t ♦♣❡r❛t✐♦♥s ❣❡♥❡r❛t❡ ❛ ◆❯❄ ❲❡ ❝❤❡❝❦ ✇❤❡t❤❡r M ⊆ Block(D, B) ❢♦r s♦♠❡ B ⊂ D ⊆ A✳ ◆♦t❡ t❤❛t f ∈ Block(D, B) ✐✛ f ♣r❡s❡r✈❡s D ❛♥❞ ❢♦r s♦♠❡ ✈❛r✐❛❜❧❡ i ✇❡ ❤❛✈❡ f(D, D, . . . , D, B

• i

, D, . . . , D) ⊆ B✳ ❙♦✱ ✐t ✐s ❡❛s②✦ ❲❡ ❝❤❡❝❦ t❤❛t M ⊆ Lin(D, ϕ)✳ ❊❛s②✦ ❚❤✐s ✐❞❡❛ ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞ ❢♦r ♥♦♥✐❞❡♠♣♦t❡♥t ❝❛s❡✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❚❤❡♦r❡♠ NUIdemOper(m) ≤ m · |A|3✳ ❲❡ ♥❡❡❞ t♦ ♣r♦✈❡ t❤❛t ❛♥② ❝❧♦♥❡ ❣❡♥❡r❛t❡❞ ❜② ♦♣❡r❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st m ❡✐t❤❡r ❝♦♥t❛✐♥s ◆❯ ♦❢ ❛r✐t② m · |A|3✱ ♦r ❞♦❡s♥✬t ❝♦♥t❛✐♥ ◆❯ ❛t ❛❧❧✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙❦❡t❝❤ ♣r♦♦❢ ▲❡t C ❜❡ ❛ ❝❧♦♥❡ ❣❡♥❡r❛t❡❞ ❜② ♦♣❡r❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st m ❛♥❞ NU(C) = n + 1✳

✶ ✇❡ ✜♥❞

❛♥❞ s✉❝❤ t❤❛t ✳ ❲❡ ♣✉t ❛♥❞ ✳

✷ ❲❡ ❜✉✐❧❞ ❛ s❡q✉❡♥❝❡

s❛t✐s❢②✐♥❣ ❛♥❞ ✱ ✳

✸ ❲❡ ✜♥✐s❤ t❤❡ s❡q✉❡♥❝❡ ✐❢

♣r❡s❡r✈❡s ✳ ❙✐♥❝❡ t❤❡ s❡q✉❡♥❝❡ ❤❛s ❛t ♠♦st ❡❧❡♠❡♥ts✱ ✇❡ ❣❡t ✳ ■❢ t❤❡♥ ♣r❡s❡r✈❡s ❢♦r ❛♥② ✳ ❈♦♥tr❛❞✐❝t✐♦♥✦✦✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙❦❡t❝❤ ♣r♦♦❢ ▲❡t C ❜❡ ❛ ❝❧♦♥❡ ❣❡♥❡r❛t❡❞ ❜② ♦♣❡r❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st m ❛♥❞ NU(C) = n + 1✳

✶ ✇❡ ✜♥❞ a, b ∈ A ❛♥❞ n1 ≥ n/|A|2 s✉❝❤ t❤❛t

{a}n1 / ∈ {a, b}n1 \ {a}n1C✳ ❲❡ ♣✉t ❛♥❞ ✳

✷ ❲❡ ❜✉✐❧❞ ❛ s❡q✉❡♥❝❡

s❛t✐s❢②✐♥❣ ❛♥❞ ✱ ✳

✸ ❲❡ ✜♥✐s❤ t❤❡ s❡q✉❡♥❝❡ ✐❢

♣r❡s❡r✈❡s ✳ ❙✐♥❝❡ t❤❡ s❡q✉❡♥❝❡ ❤❛s ❛t ♠♦st ❡❧❡♠❡♥ts✱ ✇❡ ❣❡t ✳ ■❢ t❤❡♥ ♣r❡s❡r✈❡s ❢♦r ❛♥② ✳ ❈♦♥tr❛❞✐❝t✐♦♥✦✦✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙❦❡t❝❤ ♣r♦♦❢ ▲❡t C ❜❡ ❛ ❝❧♦♥❡ ❣❡♥❡r❛t❡❞ ❜② ♦♣❡r❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st m ❛♥❞ NU(C) = n + 1✳

✶ ✇❡ ✜♥❞ a, b ∈ A ❛♥❞ n1 ≥ n/|A|2 s✉❝❤ t❤❛t

{a}n1 / ∈ {a, b}n1 \ {a}n1C✳ ❲❡ ♣✉t D = {a, b}C ❛♥❞ B1 = {b}✳

✷ ❲❡ ❜✉✐❧❞ ❛ s❡q✉❡♥❝❡

s❛t✐s❢②✐♥❣ ❛♥❞ ✱ ✳

✸ ❲❡ ✜♥✐s❤ t❤❡ s❡q✉❡♥❝❡ ✐❢

♣r❡s❡r✈❡s ✳ ❙✐♥❝❡ t❤❡ s❡q✉❡♥❝❡ ❤❛s ❛t ♠♦st ❡❧❡♠❡♥ts✱ ✇❡ ❣❡t ✳ ■❢ t❤❡♥ ♣r❡s❡r✈❡s ❢♦r ❛♥② ✳ ❈♦♥tr❛❞✐❝t✐♦♥✦✦✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙❦❡t❝❤ ♣r♦♦❢ ▲❡t C ❜❡ ❛ ❝❧♦♥❡ ❣❡♥❡r❛t❡❞ ❜② ♦♣❡r❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st m ❛♥❞ NU(C) = n + 1✳

✶ ✇❡ ✜♥❞ a, b ∈ A ❛♥❞ n1 ≥ n/|A|2 s✉❝❤ t❤❛t

{a}n1 / ∈ {a, b}n1 \ {a}n1C✳ ❲❡ ♣✉t D = {a, b}C ❛♥❞ B1 = {b}✳

✷ ❲❡ ❜✉✐❧❞ ❛ s❡q✉❡♥❝❡ (n1, B1), (n2, B2), (n3, B3), . . .

s❛t✐s❢②✐♥❣ {a}ni / ∈ Dni \ (D \ Bi)niC ❛♥❞ ni+1 ≥ ni − (m − 1)✱ Bi ⊂ Bi+1✳

✸ ❲❡ ✜♥✐s❤ t❤❡ s❡q✉❡♥❝❡ ✐❢

♣r❡s❡r✈❡s ✳ ❙✐♥❝❡ t❤❡ s❡q✉❡♥❝❡ ❤❛s ❛t ♠♦st ❡❧❡♠❡♥ts✱ ✇❡ ❣❡t ✳ ■❢ t❤❡♥ ♣r❡s❡r✈❡s ❢♦r ❛♥② ✳ ❈♦♥tr❛❞✐❝t✐♦♥✦✦✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙❦❡t❝❤ ♣r♦♦❢ ▲❡t C ❜❡ ❛ ❝❧♦♥❡ ❣❡♥❡r❛t❡❞ ❜② ♦♣❡r❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st m ❛♥❞ NU(C) = n + 1✳

✶ ✇❡ ✜♥❞ a, b ∈ A ❛♥❞ n1 ≥ n/|A|2 s✉❝❤ t❤❛t

{a}n1 / ∈ {a, b}n1 \ {a}n1C✳ ❲❡ ♣✉t D = {a, b}C ❛♥❞ B1 = {b}✳

✷ ❲❡ ❜✉✐❧❞ ❛ s❡q✉❡♥❝❡ (n1, B1), (n2, B2), (n3, B3), . . .

s❛t✐s❢②✐♥❣ {a}ni / ∈ Dni \ (D \ Bi)niC ❛♥❞ ni+1 ≥ ni − (m − 1)✱ Bi ⊂ Bi+1✳

✸ ❲❡ ✜♥✐s❤ t❤❡ s❡q✉❡♥❝❡ ✐❢ C ♣r❡s❡r✈❡s Dni \ (D \ Bi)ni✳ ❙✐♥❝❡

t❤❡ s❡q✉❡♥❝❡ ❤❛s ❛t ♠♦st |A| − 1 ❡❧❡♠❡♥ts✱ ✇❡ ❣❡t ni > n/k2 − (|A| − 1) · (m − 1)✳ ■❢ t❤❡♥ ♣r❡s❡r✈❡s ❢♦r ❛♥② ✳ ❈♦♥tr❛❞✐❝t✐♦♥✦✦✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙❦❡t❝❤ ♣r♦♦❢ ▲❡t C ❜❡ ❛ ❝❧♦♥❡ ❣❡♥❡r❛t❡❞ ❜② ♦♣❡r❛t✐♦♥s ♦❢ ❛r✐t② ❛t ♠♦st m ❛♥❞ NU(C) = n + 1✳

✶ ✇❡ ✜♥❞ a, b ∈ A ❛♥❞ n1 ≥ n/|A|2 s✉❝❤ t❤❛t

{a}n1 / ∈ {a, b}n1 \ {a}n1C✳ ❲❡ ♣✉t D = {a, b}C ❛♥❞ B1 = {b}✳

✷ ❲❡ ❜✉✐❧❞ ❛ s❡q✉❡♥❝❡ (n1, B1), (n2, B2), (n3, B3), . . .

s❛t✐s❢②✐♥❣ {a}ni / ∈ Dni \ (D \ Bi)niC ❛♥❞ ni+1 ≥ ni − (m − 1)✱ Bi ⊂ Bi+1✳

✸ ❲❡ ✜♥✐s❤ t❤❡ s❡q✉❡♥❝❡ ✐❢ C ♣r❡s❡r✈❡s Dni \ (D \ Bi)ni✳ ❙✐♥❝❡

t❤❡ s❡q✉❡♥❝❡ ❤❛s ❛t ♠♦st |A| − 1 ❡❧❡♠❡♥ts✱ ✇❡ ❣❡t ni > n/k2 − (|A| − 1) · (m − 1)✳ ■❢ ni ≥ m t❤❡♥ C ♣r❡s❡r✈❡s Dp \ (D \ Bi)p ❢♦r ❛♥② p✳ ❈♦♥tr❛❞✐❝t✐♦♥✦✦✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✶ C ❝♦♥t❛✐♥s ❛ ◆❯ ♦❢ ❛r✐t② n + 1 ❜✉t ♥♦t n✳ ❲❡ ❦♥♦✇ ❢r♦♠ ❆❆❆✽✼

✶ ❆ ❝❧♦♥❡ C ❞♦❡s♥✬t ❝♦♥t❛✐♥ ❛ ◆❯ ♦❢ ❛r✐t② n ✐✛ t❤❡r❡ ❡①✐sts ❛

❝♦♠♣❛t✐❜❧❡ ❦❡② ✭❝r✐t✐❝❛❧✮ r❡❧❛t✐♦♥ ♦❢ ❛r✐t② n✳

✷ ❋♦r ❛♥② ❦❡② r❡❧❛t✐♦♥ ρ ♣r❡s❡r✈❡❞ ❜② ❛ ◆❯ ✇❡ ❝❛♥ ✜♥❞

(a1, a2, . . . , an) / ∈ ρ s✉❝❤ t❤❛t ({a1, b1} × {a2, b2} × · · · × {an, bn}) \ {(a1, a2, . . . , an)} ⊆ ρ. ❚❤✉s✱ ✇❡ ❤❛✈❡ ♣r❡s❡r✈❡❞ ❜② ✱ ❛♥❞ ❲❡ ❝♦♥s✐❞❡r ✐❞❡♠♣♦t❡♥t ❝❛s❡✦ ❲❡ ❝❤♦♦s❡ t❤❡ ♠♦st ♣♦♣✉❧❛r ♣❛✐r t♦ ❣❡t ❢♦r ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✶ C ❝♦♥t❛✐♥s ❛ ◆❯ ♦❢ ❛r✐t② n + 1 ❜✉t ♥♦t n✳ ❲❡ ❦♥♦✇ ❢r♦♠ ❆❆❆✽✼

✶ ❆ ❝❧♦♥❡ C ❞♦❡s♥✬t ❝♦♥t❛✐♥ ❛ ◆❯ ♦❢ ❛r✐t② n ✐✛ t❤❡r❡ ❡①✐sts ❛

❝♦♠♣❛t✐❜❧❡ ❦❡② ✭❝r✐t✐❝❛❧✮ r❡❧❛t✐♦♥ ♦❢ ❛r✐t② n✳

✷ ❋♦r ❛♥② ❦❡② r❡❧❛t✐♦♥ ρ ♣r❡s❡r✈❡❞ ❜② ❛ ◆❯ ✇❡ ❝❛♥ ✜♥❞

(a1, a2, . . . , an) / ∈ ρ s✉❝❤ t❤❛t ({a1, b1} × {a2, b2} × · · · × {an, bn}) \ {(a1, a2, . . . , an)} ⊆ ρ. ❚❤✉s✱ ✇❡ ❤❛✈❡ ρ ♣r❡s❡r✈❡❞ ❜② C✱ (a1, a2, . . . , an) / ∈ ρ ❛♥❞ ({a1, b1} × {a2, b2} × · · · × {an, bn}) \ {(a1, a2, . . . , an)} ⊆ ρ. ❲❡ ❝♦♥s✐❞❡r ✐❞❡♠♣♦t❡♥t ❝❛s❡✦ ❲❡ ❝❤♦♦s❡ t❤❡ ♠♦st ♣♦♣✉❧❛r ♣❛✐r t♦ ❣❡t ❢♦r ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✶ C ❝♦♥t❛✐♥s ❛ ◆❯ ♦❢ ❛r✐t② n + 1 ❜✉t ♥♦t n✳ ❲❡ ❦♥♦✇ ❢r♦♠ ❆❆❆✽✼

✶ ❆ ❝❧♦♥❡ C ❞♦❡s♥✬t ❝♦♥t❛✐♥ ❛ ◆❯ ♦❢ ❛r✐t② n ✐✛ t❤❡r❡ ❡①✐sts ❛

❝♦♠♣❛t✐❜❧❡ ❦❡② ✭❝r✐t✐❝❛❧✮ r❡❧❛t✐♦♥ ♦❢ ❛r✐t② n✳

✷ ❋♦r ❛♥② ❦❡② r❡❧❛t✐♦♥ ρ ♣r❡s❡r✈❡❞ ❜② ❛ ◆❯ ✇❡ ❝❛♥ ✜♥❞

(a1, a2, . . . , an) / ∈ ρ s✉❝❤ t❤❛t ({a1, b1} × {a2, b2} × · · · × {an, bn}) \ {(a1, a2, . . . , an)} ⊆ ρ. ❚❤✉s✱ ✇❡ ❤❛✈❡ ρ ♣r❡s❡r✈❡❞ ❜② C✱ (a1, a2, . . . , an) / ∈ ρ ❛♥❞ ({a1, b1} × {a2, b2} × · · · × {an, bn}) \ {(a1, a2, . . . , an)} ⊆ ρ. (a1, . . . , an) / ∈ ({a1, b1} × · · · × {an, bn}) \ {(a1, . . . , an)}C. ❲❡ ❝♦♥s✐❞❡r ✐❞❡♠♣♦t❡♥t ❝❛s❡✦ ❲❡ ❝❤♦♦s❡ t❤❡ ♠♦st ♣♦♣✉❧❛r ♣❛✐r t♦ ❣❡t ❢♦r ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✶ C ❝♦♥t❛✐♥s ❛ ◆❯ ♦❢ ❛r✐t② n + 1 ❜✉t ♥♦t n✳ ❲❡ ❦♥♦✇ ❢r♦♠ ❆❆❆✽✼

✶ ❆ ❝❧♦♥❡ C ❞♦❡s♥✬t ❝♦♥t❛✐♥ ❛ ◆❯ ♦❢ ❛r✐t② n ✐✛ t❤❡r❡ ❡①✐sts ❛

❝♦♠♣❛t✐❜❧❡ ❦❡② ✭❝r✐t✐❝❛❧✮ r❡❧❛t✐♦♥ ♦❢ ❛r✐t② n✳

✷ ❋♦r ❛♥② ❦❡② r❡❧❛t✐♦♥ ρ ♣r❡s❡r✈❡❞ ❜② ❛ ◆❯ ✇❡ ❝❛♥ ✜♥❞

(a1, a2, . . . , an) / ∈ ρ s✉❝❤ t❤❛t ({a1, b1} × {a2, b2} × · · · × {an, bn}) \ {(a1, a2, . . . , an)} ⊆ ρ. ❚❤✉s✱ ✇❡ ❤❛✈❡ ρ ♣r❡s❡r✈❡❞ ❜② C✱ (a1, a2, . . . , an) / ∈ ρ ❛♥❞ ({a1, b1} × {a2, b2} × · · · × {an, bn}) \ {(a1, a2, . . . , an)} ⊆ ρ. (a1, . . . , an) / ∈ ({a1, b1} × · · · × {an, bn}) \ {(a1, . . . , an)}C. ❲❡ ❝♦♥s✐❞❡r ✐❞❡♠♣♦t❡♥t ❝❛s❡✦ ❲❡ ❝❤♦♦s❡ t❤❡ ♠♦st ♣♦♣✉❧❛r ♣❛✐r t♦ ❣❡t ❢♦r ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✶ C ❝♦♥t❛✐♥s ❛ ◆❯ ♦❢ ❛r✐t② n + 1 ❜✉t ♥♦t n✳ ❲❡ ❦♥♦✇ ❢r♦♠ ❆❆❆✽✼

✶ ❆ ❝❧♦♥❡ C ❞♦❡s♥✬t ❝♦♥t❛✐♥ ❛ ◆❯ ♦❢ ❛r✐t② n ✐✛ t❤❡r❡ ❡①✐sts ❛

❝♦♠♣❛t✐❜❧❡ ❦❡② ✭❝r✐t✐❝❛❧✮ r❡❧❛t✐♦♥ ♦❢ ❛r✐t② n✳

✷ ❋♦r ❛♥② ❦❡② r❡❧❛t✐♦♥ ρ ♣r❡s❡r✈❡❞ ❜② ❛ ◆❯ ✇❡ ❝❛♥ ✜♥❞

(a1, a2, . . . , an) / ∈ ρ s✉❝❤ t❤❛t ({a1, b1} × {a2, b2} × · · · × {an, bn}) \ {(a1, a2, . . . , an)} ⊆ ρ. ❚❤✉s✱ ✇❡ ❤❛✈❡ ρ ♣r❡s❡r✈❡❞ ❜② C✱ (a1, a2, . . . , an) / ∈ ρ ❛♥❞ ({a1, b1} × {a2, b2} × · · · × {an, bn}) \ {(a1, a2, . . . , an)} ⊆ ρ. (a1, . . . , an) / ∈ ({a1, b1} × · · · × {an, bn}) \ {(a1, . . . , an)}C. ❲❡ ❝♦♥s✐❞❡r ✐❞❡♠♣♦t❡♥t ❝❛s❡✦ ❲❡ ❝❤♦♦s❡ t❤❡ ♠♦st ♣♦♣✉❧❛r ♣❛✐r (ai, bi) t♦ ❣❡t {a}n1 / ∈ {a, b}n1 \ {a}n1C ❢♦r n1 ≥ n/|A|2✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✷ ❲❡ ❤❛✈❡ n1 ❛♥❞ {a}n1 / ∈ {a, b}n1 \ {a}n1C. ❋♦r ❛♥❞ ✇❡ ❤❛✈❡ ❲❡ ❤❛✈❡ t❤❡ ✜rst ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✿ ✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ✳ ■❢ ♣r❡s❡r✈❡s ✇❡ ❛r❡ ❞♦♥❡✳ ■❢ ♥♦t✱ t❤❡♥ ❛♥ ♦♣❡r❛t✐♦♥ ❞♦❡s♥✬t ♣r❡s❡r✈❡ ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✷ ❲❡ ❤❛✈❡ n1 ❛♥❞ {a}n1 / ∈ {a, b}n1 \ {a}n1C. ❋♦r D = {a, b}C ❛♥❞ B1 = {b} ✇❡ ❤❛✈❡ {a}n1 / ∈ Dn1 \ (D \ B1)n1C ⊆ {a, b}n1 \ {b}n1C. ❲❡ ❤❛✈❡ t❤❡ ✜rst ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✿ (n1, B1)✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ✳ ■❢ ♣r❡s❡r✈❡s ✇❡ ❛r❡ ❞♦♥❡✳ ■❢ ♥♦t✱ t❤❡♥ ❛♥ ♦♣❡r❛t✐♦♥ ❞♦❡s♥✬t ♣r❡s❡r✈❡ ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✷ ❲❡ ❤❛✈❡ n1 ❛♥❞ {a}n1 / ∈ {a, b}n1 \ {a}n1C. ❋♦r D = {a, b}C ❛♥❞ B1 = {b} ✇❡ ❤❛✈❡ {a}n1 / ∈ Dn1 \ (D \ B1)n1C ⊆ {a, b}n1 \ {b}n1C. ❲❡ ❤❛✈❡ t❤❡ ✜rst ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✿ (n1, B1)✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ (ni, Bi)✳ ■❢ C ♣r❡s❡r✈❡s Dni \ (D \ Bi)ni ✇❡ ❛r❡ ❞♦♥❡✳ ■❢ ♥♦t✱ t❤❡♥ ❛♥ ♦♣❡r❛t✐♦♥ ❞♦❡s♥✬t ♣r❡s❡r✈❡ ✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✷ ❲❡ ❤❛✈❡ n1 ❛♥❞ {a}n1 / ∈ {a, b}n1 \ {a}n1C. ❋♦r D = {a, b}C ❛♥❞ B1 = {b} ✇❡ ❤❛✈❡ {a}n1 / ∈ Dn1 \ (D \ B1)n1C ⊆ {a, b}n1 \ {b}n1C. ❲❡ ❤❛✈❡ t❤❡ ✜rst ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✿ (n1, B1)✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ (ni, Bi)✳ ■❢ C ♣r❡s❡r✈❡s Dni \ (D \ Bi)ni ✇❡ ❛r❡ ❞♦♥❡✳ ■❢ ♥♦t✱ t❤❡♥ ❛♥ ♦♣❡r❛t✐♦♥ f ∈ C ❞♦❡s♥✬t ♣r❡s❡r✈❡ Dni \ (D \ Bi)ni✳ f                         . . . . . . . . . . . . . . . . . . . . . . . .                         . . . . . . . . . . . . . . . . . . . . . . . .                         . . . . . . . . . . . . . . . . . . . . . . . .                         . . . . . . . . . . . . . . . . . . . . . . . .                         / ∈ Dni \ (D \ Bi)ni.

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✷ ❲❡ ❤❛✈❡ n1 ❛♥❞ {a}n1 / ∈ {a, b}n1 \ {a}n1C. ❋♦r D = {a, b}C ❛♥❞ B1 = {b} ✇❡ ❤❛✈❡ {a}n1 / ∈ Dn1 \ (D \ B1)n1C ⊆ {a, b}n1 \ {b}n1C. ❲❡ ❤❛✈❡ t❤❡ ✜rst ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✿ (n1, B1)✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ (ni, Bi)✳ ■❢ C ♣r❡s❡r✈❡s Dni \ (D \ Bi)ni ✇❡ ❛r❡ ❞♦♥❡✳ ■❢ ♥♦t✱ t❤❡♥ ❛♥ ♦♣❡r❛t✐♦♥ f ∈ C ❞♦❡s♥✬t ♣r❡s❡r✈❡ Dni \ (D \ Bi)ni✳ f                         b . . . . . . . . . . . . . . . . . . . . .                         . . . b . . . . . . . . . . . . . . . . . .                         . . . . . . b . . . . . . . . . . . . . . .                         . . . . . . . . . b . . . . . . . . . . . .                         / ∈ Dni \ (D \ Bi)ni.

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✷ ❲❡ ❤❛✈❡ n1 ❛♥❞ {a}n1 / ∈ {a, b}n1 \ {a}n1C. ❋♦r D = {a, b}C ❛♥❞ B1 = {b} ✇❡ ❤❛✈❡ {a}n1 / ∈ Dn1 \ (D \ B1)n1C ⊆ {a, b}n1 \ {b}n1C. ❲❡ ❤❛✈❡ t❤❡ ✜rst ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✿ (n1, B1)✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ (ni, Bi)✳ ■❢ C ♣r❡s❡r✈❡s Dni \ (D \ Bi)ni ✇❡ ❛r❡ ❞♦♥❡✳ ■❢ ♥♦t✱ t❤❡♥ ❛♥ ♦♣❡r❛t✐♦♥ f ∈ C ❞♦❡s♥✬t ♣r❡s❡r✈❡ Dni \ (D \ Bi)ni✳ f                         b . . . . . . . . . . . . . . . . . . . . .                         . . . b . . . . . . . . . . . . . . . . . .                         . . . . . . b . . . . . . . . . . . . . . .                         . . . . . . . . . b . . . . . . . . . . . .                         =             c1 c2 c3 c4 . . . . . . . . . . . .             / ∈ Dni \ (D \ Bi)ni.

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✷ ❲❡ ❤❛✈❡ n1 ❛♥❞ {a}n1 / ∈ {a, b}n1 \ {a}n1C. ❋♦r D = {a, b}C ❛♥❞ B1 = {b} ✇❡ ❤❛✈❡ {a}n1 / ∈ Dn1 \ (D \ B1)n1C ⊆ {a, b}n1 \ {b}n1C. ❲❡ ❤❛✈❡ t❤❡ ✜rst ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✿ (n1, B1)✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ (ni, Bi)✳ ■❢ C ♣r❡s❡r✈❡s Dni \ (D \ Bi)ni ✇❡ ❛r❡ ❞♦♥❡✳ ■❢ ♥♦t✱ t❤❡♥ ❛♥ ♦♣❡r❛t✐♦♥ f ∈ C ❞♦❡s♥✬t ♣r❡s❡r✈❡ Dni \ (D \ Bi)ni✳ f                         b . . . . . . . . . a a a a                         . . . b . . . . . . a a a a                         . . . . . . b . . . a a a a                         . . . . . . . . . b a a a a                         =             c1 c2 c3 c4 a a a a             / ∈ Dni \ (D \ Bi)ni.

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✷ ❲❡ ❣❡t k ≤ m s✉❝❤ t❤❛t (c1, . . . , ck, a, . . . , a) ∈ Dni \ (D \ Bi)niC ❛♥❞ (c1, . . . , ck, a, . . . , a) / ∈ Dni \ (D \ Bi)ni. ❲▲❖●✱ ❧❡t ❜❡ t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ✇✐t❤ t❤✐s ♣r♦♣❡rt②✳ ❚❤❡♥ ❙✐♥❝❡ ✇❡ ❝♦♥s✐❞❡r ✐❞❡♠♣♦t❡♥t ❝❛s❡✱ ✇❡ ❤❛✈❡ P✉t ✱ ✳ ❚❤❡♥ ▲❡t ❜❡ t❤❡ ♥❡①t ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✷ ❲❡ ❣❡t k ≤ m s✉❝❤ t❤❛t (c1, . . . , ck, a, . . . , a) ∈ Dni \ (D \ Bi)niC ❛♥❞ (c1, . . . , ck, a, . . . , a) / ∈ Dni \ (D \ Bi)ni. ❲▲❖●✱ ❧❡t k ❜❡ t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ✇✐t❤ t❤✐s ♣r♦♣❡rt②✳ ❚❤❡♥ (c1, . . . , ck−1, a, a, . . . , a) / ∈ Dni \ (D \ Bi)niC ❙✐♥❝❡ ✇❡ ❝♦♥s✐❞❡r ✐❞❡♠♣♦t❡♥t ❝❛s❡✱ ✇❡ ❤❛✈❡ P✉t ✱ ✳ ❚❤❡♥ ▲❡t ❜❡ t❤❡ ♥❡①t ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✷ ❲❡ ❣❡t k ≤ m s✉❝❤ t❤❛t (c1, . . . , ck, a, . . . , a) ∈ Dni \ (D \ Bi)niC ❛♥❞ (c1, . . . , ck, a, . . . , a) / ∈ Dni \ (D \ Bi)ni. ❲▲❖●✱ ❧❡t k ❜❡ t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ✇✐t❤ t❤✐s ♣r♦♣❡rt②✳ ❚❤❡♥ (c1, . . . , ck−1, a, a, . . . , a) / ∈ Dni \ (D \ Bi)niC ❙✐♥❝❡ ✇❡ ❝♦♥s✐❞❡r ✐❞❡♠♣♦t❡♥t ❝❛s❡✱ ✇❡ ❤❛✈❡ (c1, . . . , ck−1, ck, D, . . . , D) ⊆ Dni \ (D \ Bi)niC P✉t ✱ ✳ ❚❤❡♥ ▲❡t ❜❡ t❤❡ ♥❡①t ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✷ ❲❡ ❣❡t k ≤ m s✉❝❤ t❤❛t (c1, . . . , ck, a, . . . , a) ∈ Dni \ (D \ Bi)niC ❛♥❞ (c1, . . . , ck, a, . . . , a) / ∈ Dni \ (D \ Bi)ni. ❲▲❖●✱ ❧❡t k ❜❡ t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ✇✐t❤ t❤✐s ♣r♦♣❡rt②✳ ❚❤❡♥ (c1, . . . , ck−1, a, a, . . . , a) / ∈ Dni \ (D \ Bi)niC ❙✐♥❝❡ ✇❡ ❝♦♥s✐❞❡r ✐❞❡♠♣♦t❡♥t ❝❛s❡✱ ✇❡ ❤❛✈❡ (c1, . . . , ck−1, ck, D, . . . , D) ⊆ Dni \ (D \ Bi)niC P✉t ni+1 = ni − (k − 1)✱ Bi+1 = Bi ∪ {ck}✳ ❚❤❡♥ {a}ni+1 / ∈ Dni+1 \ (D \ Bi+1)ni+1C. ▲❡t (ni+1, Bi+1) ❜❡ t❤❡ ♥❡①t ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✸ ❲❡ ✜♥✐s❤ t❤❡ s❡q✉❡♥❝❡ ✇✐t❤ Dni \ (D \ Bi)ni ♣r❡s❡r✈❡❞ ❜② C. ■❢ ni ≥ m t❤❡♥ C ♣r❡s❡r✈❡s Dp \ (D \ Bi)p ❢♦r ❛♥② p✳ ❇✉t ❛ ◆❯ ❝❛♥♥♦t ♣r❡s❡r✈❡ Dp \ (D \ Bi)p ❢♦r ❛♥② p✳ ❈♦♥tr❛❞✐❝t✐♦♥✦✦✦ ❚❤✉s✱ ❚❤❡r❡❢♦r❡✱ ✳

❲❡ ❝♦♠♣❧❡t❡ t❤❡ ♣r♦♦❢

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❙t❡♣ ✸ ❲❡ ✜♥✐s❤ t❤❡ s❡q✉❡♥❝❡ ✇✐t❤ Dni \ (D \ Bi)ni ♣r❡s❡r✈❡❞ ❜② C. ■❢ ni ≥ m t❤❡♥ C ♣r❡s❡r✈❡s Dp \ (D \ Bi)p ❢♦r ❛♥② p✳ ❇✉t ❛ ◆❯ ❝❛♥♥♦t ♣r❡s❡r✈❡ Dp \ (D \ Bi)p ❢♦r ❛♥② p✳ ❈♦♥tr❛❞✐❝t✐♦♥✦✦✦ ❚❤✉s✱ m > ni > n/|A|2 − (|A| − 1) · (m − 1) ❚❤❡r❡❢♦r❡✱ n < |A|3 · m✳

❲❡ ❝♦♠♣❧❡t❡ t❤❡ ♣r♦♦❢

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ ✇❤❡r❡ ❛♥❞ ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t ❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❄ ■ ❝❛♥✬t✦

✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ ✇❤❡r❡ ❛♥❞ ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t ❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❄ ■ ❝❛♥✬t✦

✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ ✇❤❡r❡ ❛♥❞ ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t ❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❄ ■ ❝❛♥✬t✦

✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ ✇❤❡r❡ ❛♥❞ ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t ❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❄ ■ ❝❛♥✬t✦

✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ ✇❤❡r❡ ❛♥❞ ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t ❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❄ ■ ❝❛♥✬t✦

✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t ❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❄ ■ ❝❛♥✬t✦

✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t [A] ∩ [B]❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❄ ■ ❝❛♥✬t✦

✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t [A] ∩ [B]❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄

■ ❝❛♥✬t✦

✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❄ ■ ❝❛♥✬t✦

✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t [A] ∩ [B]❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❄ ■ ❝❛♥✬t✦

✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t [A] ∩ [B]❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄

■ ❝❛♥✬t✦

✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❄ ■ ❝❛♥✬t✦

✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t [A] ∩ [B]❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❄ ■ ❝❛♥✬t✦

✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t [A] ∩ [B]❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t [A] ∩ [B] = [C]❄

■ ❝❛♥✬t✦

✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t [A] ∩ [B]❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t [A] ∩ [B] = [C]❄ ■ ❝❛♥✬t✦ ✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t

❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t [A] ∩ [B]❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t [A] ∩ [B] = [C]❄ ■ ❝❛♥✬t✦ ✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t [A] ∩ [B] ❝♦♥t❛✐♥s ◆❯❄

■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❲❤❛t ❞♦ ②♦✉ ❦♥♦✇ ❛❜♦✉t [A] ∩ [B]❄

✶ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞❄ ■ ❝❛♥✬t✦ ✷ ❈❛♥ ②♦✉ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✜♥✐t❡❧② r❡❧❛t❡❞❄ ■ ❝❛♥✬t✦ ✸ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t [A] ∩ [B] = [C]❄ ■ ❝❛♥✬t✦ ✹ ❈❛♥ ②♦✉ ❝❤❡❝❦ t❤❛t [A] ∩ [B] ❝♦♥t❛✐♥s ◆❯❄ ■ ❝❛♥✦

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ ✇❤❡r❡ ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ ✳ ■ ❞♦♥✬t ❦♥♦✇✦ ❖♣❡♥ ♣r♦❜❧❡♠✿ ✐s t❤✐s ♣r♦❜❧❡♠ ❞❡❝✐❞❛❜❧❡❄

• ✐✈❡♥ r❡❧❛t✐♦♥s

❛♥❞ ✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ❝♦♥t❛✐♥s ❛ ◆❯✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A1] ∩ [A2] ∩ · · · ∩ [An] ✇❤❡r❡ A1, A2, . . . , An ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ ✳ ■ ❞♦♥✬t ❦♥♦✇✦ ❖♣❡♥ ♣r♦❜❧❡♠✿ ✐s t❤✐s ♣r♦❜❧❡♠ ❞❡❝✐❞❛❜❧❡❄

• ✐✈❡♥ r❡❧❛t✐♦♥s

❛♥❞ ✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ❝♦♥t❛✐♥s ❛ ◆❯✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A1] ∩ [A2] ∩ · · · ∩ [An] ✇❤❡r❡ A1, A2, . . . , An ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [Pol(ρ1) ∪ Pol(ρ2)]✳ ■ ❞♦♥✬t ❦♥♦✇✦ ❖♣❡♥ ♣r♦❜❧❡♠✿ ✐s t❤✐s ♣r♦❜❧❡♠ ❞❡❝✐❞❛❜❧❡❄

• ✐✈❡♥ r❡❧❛t✐♦♥s

❛♥❞ ✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r ❝♦♥t❛✐♥s ❛ ◆❯✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❋✉rt❤❡r ■♥✈❡st✐❣❛t✐♦♥s ◗✉❡st✐♦♥ ❈❛♥ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❛ ❝❧♦♥❡ ❝♦♥t❛✐♥s ❛ ◆❯❄ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ s❡t ♦❢ r❡❧❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A] ∩ [B] ✇❤❡r❡ A ❛♥❞ B ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [A1] ∩ [A2] ∩ · · · ∩ [An] ✇❤❡r❡ A1, A2, . . . , An ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ♦♣❡r❛t✐♦♥s✳ ❉❡❝✐❞❛❜❧❡✦ ❋♦r ❝❧♦♥❡s ♦❢ t❤❡ ❢♦r♠ [Pol(ρ1) ∪ Pol(ρ2)]✳ ■ ❞♦♥✬t ❦♥♦✇✦ ❖♣❡♥ ♣r♦❜❧❡♠✿ ✐s t❤✐s ♣r♦❜❧❡♠ ❞❡❝✐❞❛❜❧❡❄

• ✐✈❡♥ r❡❧❛t✐♦♥s ρ1 ❛♥❞ ρ2✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r [Pol(ρ1) ∪ Pol(ρ2)]

❝♦♥t❛✐♥s ❛ ◆❯✳

◆♦t❛t✐♦♥s ❛♥❞ ❉❡✜♥✐t✐♦♥s ▼✐♥✐♠❛❧ ❆r✐t② ♦❢ ◆❯ ❈r✐t❡r✐❛ ♦❢ ❊①✐st❡♥❝❡ ◆❯ Pr♦♦❢ ❢♦r t❤❡ ■❞❡♠♣♦t❡♥t ❈❛s❡

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥