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rs ssts t st sts r rtt s s rr
❋ré❞ér✐❝ ❇✐❤❛♥✱ P✐❡rr❡✲❏❡❛♥ ❙♣❛❡♥❧❡❤❛✉❡r
▲❆▼❆✱ ❯♥✐✈✳ ❙❛✈♦✐❡ ▼♦♥t ❇❧❛♥❝ ■♥r✐❛✴▲❖❘■❆✴❈◆❘❙✱ ♣r♦❥❡❝t ❈❆❘❆▼❊▲
❏♦✉r♥é❡s ◆❛t✐♦♥❛❧❡s ❞✉ ❈❛❧❝✉❧ ❋♦r♠❡❧ ✷✵✶✺✱ ❈❧✉♥②
✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
A ⊂ Zd✿ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts✳ α ∈ A ↔ X α✶
✶
· · · X αd
d ✳ ✶ ✶ ✶ ✶ ✇✐t❤ s✉♣♣♦rt
✳ ❆ s♦❧✉t✐♦♥
✶ ①
① ✵ ✐s ♣♦s✐t✐✈❡ ✐❢ ① ❛♥❞ ✵ ❢♦r ❛❧❧ ❀ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ t❤❡ ❥❛❝♦❜✐❛♥ ♠❛tr✐① ♦❢
✶
✐s ✐♥✈❡rt✐❜❧❡ ❛t ① Pr♦❜❧❡♠ st❛t❡♠❡♥ts
✶
✱ ❝♦♥str✉❝t
✶
s✉❝❤ t❤❛t
✶ ❳
❳ ✵ ❤❛s ♠❛♥② ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s❀
✷
❝♦♥str✉❝t ❛♥❞
✶
s✉❝❤ t❤❛t ❛♥❞
✶ ❳
❳ ✵ ❤❛s ♠❛♥② ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s✳ ▼♦t✐✈❛t✐♦♥ ✭ ✶✮✿ ❉❡s❝❛rt❡s✬ r✉❧❡ ♦❢ s✐❣♥s ✭✶✻✸✼✮ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ r♦♦ts ♦❢ ❛ ▲❛✉r❡♥t ♣♦❧②♥♦♠✐❛❧
✶ ✐s ❜♦✉♥❞❡❞
❜② t❤❡ ♥✉♠❜❡r ♦❢ s✐❣♥ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ❝♦♥s❡❝✉t✐✈❡ ❝♦❡✣❝✐❡♥ts✳ ❛❧❧ ♥♦♥③❡r♦ ❝♦♠♣❧❡① r♦♦ts ♦❢ ✭sq✉❛r❡❢r❡❡✮ ❛r❡ ♣♦s✐t✐✈❡ ✶ ✶ ❢♦r s♦♠❡ ✳
✷ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
A ⊂ Zd✿ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts✳ α ∈ A ↔ X α✶
✶
· · · X αd
d ✳
f✶, . . . , fd ∈ R[X ±✶
✶
, . . . , X ±✶
d
] ✇✐t❤ s✉♣♣♦rt A✳ ❆ s♦❧✉t✐♦♥ f✶(①) = · · · = fd(①) = ✵ ✐s ♣♦s✐t✐✈❡ ✐❢ ① ∈ Rd ❛♥❞ xi > ✵ ❢♦r ❛❧❧ i❀ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ t❤❡ ❥❛❝♦❜✐❛♥ ♠❛tr✐① ♦❢ (f✶, . . . , fd) ✐s ✐♥✈❡rt✐❜❧❡ ❛t ① Pr♦❜❧❡♠ st❛t❡♠❡♥ts
✶
✱ ❝♦♥str✉❝t
✶
s✉❝❤ t❤❛t
✶ ❳
❳ ✵ ❤❛s ♠❛♥② ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s❀
✷
❝♦♥str✉❝t ❛♥❞
✶
s✉❝❤ t❤❛t ❛♥❞
✶ ❳
❳ ✵ ❤❛s ♠❛♥② ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s✳ ▼♦t✐✈❛t✐♦♥ ✭ ✶✮✿ ❉❡s❝❛rt❡s✬ r✉❧❡ ♦❢ s✐❣♥s ✭✶✻✸✼✮ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ r♦♦ts ♦❢ ❛ ▲❛✉r❡♥t ♣♦❧②♥♦♠✐❛❧
✶ ✐s ❜♦✉♥❞❡❞
❜② t❤❡ ♥✉♠❜❡r ♦❢ s✐❣♥ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ❝♦♥s❡❝✉t✐✈❡ ❝♦❡✣❝✐❡♥ts✳ ❛❧❧ ♥♦♥③❡r♦ ❝♦♠♣❧❡① r♦♦ts ♦❢ ✭sq✉❛r❡❢r❡❡✮ ❛r❡ ♣♦s✐t✐✈❡ ✶ ✶ ❢♦r s♦♠❡ ✳
✷ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
A ⊂ Zd✿ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts✳ α ∈ A ↔ X α✶
✶
· · · X αd
d ✳
f✶, . . . , fd ∈ R[X ±✶
✶
, . . . , X ±✶
d
] ✇✐t❤ s✉♣♣♦rt A✳ ❆ s♦❧✉t✐♦♥ f✶(①) = · · · = fd(①) = ✵ ✐s ♣♦s✐t✐✈❡ ✐❢ ① ∈ Rd ❛♥❞ xi > ✵ ❢♦r ❛❧❧ i❀ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ t❤❡ ❥❛❝♦❜✐❛♥ ♠❛tr✐① ♦❢ (f✶, . . . , fd) ✐s ✐♥✈❡rt✐❜❧❡ ❛t ① Pr♦❜❧❡♠ st❛t❡♠❡♥ts
✶
♠❛♥② ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s❀
✷
f✶(❳) = · · · = fd(❳) = ✵ ❤❛s ♠❛♥② ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s✳ ▼♦t✐✈❛t✐♦♥ ✭ ✶✮✿ ❉❡s❝❛rt❡s✬ r✉❧❡ ♦❢ s✐❣♥s ✭✶✻✸✼✮ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ r♦♦ts ♦❢ ❛ ▲❛✉r❡♥t ♣♦❧②♥♦♠✐❛❧
✶ ✐s ❜♦✉♥❞❡❞
❜② t❤❡ ♥✉♠❜❡r ♦❢ s✐❣♥ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ❝♦♥s❡❝✉t✐✈❡ ❝♦❡✣❝✐❡♥ts✳ ❛❧❧ ♥♦♥③❡r♦ ❝♦♠♣❧❡① r♦♦ts ♦❢ ✭sq✉❛r❡❢r❡❡✮ ❛r❡ ♣♦s✐t✐✈❡ ✶ ✶ ❢♦r s♦♠❡ ✳
✷ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
A ⊂ Zd✿ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts✳ α ∈ A ↔ X α✶
✶
· · · X αd
d ✳
f✶, . . . , fd ∈ R[X ±✶
✶
, . . . , X ±✶
d
] ✇✐t❤ s✉♣♣♦rt A✳ ❆ s♦❧✉t✐♦♥ f✶(①) = · · · = fd(①) = ✵ ✐s ♣♦s✐t✐✈❡ ✐❢ ① ∈ Rd ❛♥❞ xi > ✵ ❢♦r ❛❧❧ i❀ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ t❤❡ ❥❛❝♦❜✐❛♥ ♠❛tr✐① ♦❢ (f✶, . . . , fd) ✐s ✐♥✈❡rt✐❜❧❡ ❛t ① Pr♦❜❧❡♠ st❛t❡♠❡♥ts
✶
♠❛♥② ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s❀
✷
f✶(❳) = · · · = fd(❳) = ✵ ❤❛s ♠❛♥② ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s✳ ▼♦t✐✈❛t✐♦♥ ✭d = ✶✮✿ ❉❡s❝❛rt❡s✬ r✉❧❡ ♦❢ s✐❣♥s ✭✶✻✸✼✮ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ r♦♦ts ♦❢ ❛ ▲❛✉r❡♥t ♣♦❧②♥♦♠✐❛❧ f ∈ R[X ±✶] ✐s ❜♦✉♥❞❡❞ ❜② t❤❡ ♥✉♠❜❡r ♦❢ s✐❣♥ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ❝♦♥s❡❝✉t✐✈❡ ❝♦❡✣❝✐❡♥ts✳ ❛❧❧ ♥♦♥③❡r♦ ❝♦♠♣❧❡① r♦♦ts ♦❢ ✭sq✉❛r❡❢r❡❡✮ f ❛r❡ ♣♦s✐t✐✈❡ ⇒ A = {a, a + ✶, . . . , a + s − ✶} ❢♦r s♦♠❡ a ∈ Z✳
✷ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❱✐r♦✬s ♠❡t❤♦❞ ✭✼✵s✮✿ ❡✛❡❝t✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ♣r❡s❝r✐❜❡❞ t♦♣♦❧♦❣② ❛♥❞ s✉♣♣♦rt✳ ❊①❛♠♣❧❡✿
✹ ✷ ✸ ✸ ✸ ✹
✱ ✜♥❞ s✳t✳
✷ ✵
✵ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡✳ ❋♦r ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✱ t❤❡ ❝✉r✈❡
✷ ✷ ✸ ✷ ✸ ✷ ✸ ✷ ✸ ✹ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡ ✐♥ ✷ ✵✳
❈♦♥s✐❞❡r❡❞ ❛s ♦♥❡ ♦❢ t❤❡ r♦♦ts ♦❢ tr♦♣✐❝❛❧ ❣❡♦♠❡tr②✳ ❊①t❡♥s✐♦♥s ❢♦r ❝♦♠♣❧❡t❡ ✐♥t❡rs❡❝t✐♦♥s ❜② ❇✐❤❛♥✱ ❙t✉r♠❢❡❧s✱ ■t❡♥❜❡r❣✴❘♦②✳
✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❱✐r♦✬s ♠❡t❤♦❞ ✭✼✵s✮✿ ❡✛❡❝t✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ♣r❡s❝r✐❜❡❞ t♦♣♦❧♦❣② ❛♥❞ s✉♣♣♦rt✳ ❊①❛♠♣❧❡✿ f = aX + bY ✹ + cX ✷Y ✸ + dX ✸Y + eX ✸Y ✹ ∈ R[X, Y ]✱ ✜♥❞ a, b, c, d, e ∈ R s✳t✳ {(x, y) ∈ R✷
>✵ | f (x, y) = ✵} ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡✳
❋♦r ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✱ t❤❡ ❝✉r✈❡
✷ ✷ ✸ ✷ ✸ ✷ ✸ ✷ ✸ ✹ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡ ✐♥ ✷ ✵✳
❈♦♥s✐❞❡r❡❞ ❛s ♦♥❡ ♦❢ t❤❡ r♦♦ts ♦❢ tr♦♣✐❝❛❧ ❣❡♦♠❡tr②✳ ❊①t❡♥s✐♦♥s ❢♦r ❝♦♠♣❧❡t❡ ✐♥t❡rs❡❝t✐♦♥s ❜② ❇✐❤❛♥✱ ❙t✉r♠❢❡❧s✱ ■t❡♥❜❡r❣✴❘♦②✳
✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❱✐r♦✬s ♠❡t❤♦❞ ✭✼✵s✮✿ ❡✛❡❝t✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ♣r❡s❝r✐❜❡❞ t♦♣♦❧♦❣② ❛♥❞ s✉♣♣♦rt✳ ❊①❛♠♣❧❡✿ f = aX + bY ✹ + cX ✷Y ✸ + dX ✸Y + eX ✸Y ✹ ∈ R[X, Y ]✱ ✜♥❞ a, b, c, d, e ∈ R s✳t✳ {(x, y) ∈ R✷
>✵ | f (x, y) = ✵} ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡✳
❋♦r ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✱ t❤❡ ❝✉r✈❡
✷ ✷ ✸ ✷ ✸ ✷ ✸ ✷ ✸ ✹ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡ ✐♥ ✷ ✵✳
❈♦♥s✐❞❡r❡❞ ❛s ♦♥❡ ♦❢ t❤❡ r♦♦ts ♦❢ tr♦♣✐❝❛❧ ❣❡♦♠❡tr②✳ ❊①t❡♥s✐♦♥s ❢♦r ❝♦♠♣❧❡t❡ ✐♥t❡rs❡❝t✐♦♥s ❜② ❇✐❤❛♥✱ ❙t✉r♠❢❡❧s✱ ■t❡♥❜❡r❣✴❘♦②✳
✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❱✐r♦✬s ♠❡t❤♦❞ ✭✼✵s✮✿ ❡✛❡❝t✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ♣r❡s❝r✐❜❡❞ t♦♣♦❧♦❣② ❛♥❞ s✉♣♣♦rt✳ ❊①❛♠♣❧❡✿ f = aX + bY ✹ + cX ✷Y ✸ + dX ✸Y + eX ✸Y ✹ ∈ R[X, Y ]✱ ✜♥❞ a, b, c, d, e ∈ R s✳t✳ {(x, y) ∈ R✷
>✵ | f (x, y) = ✵} ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡✳
❋♦r ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✱ t❤❡ ❝✉r✈❡
✷ ✷ ✸ ✷ ✸ ✷ ✸ ✷ ✸ ✹ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡ ✐♥ ✷ ✵✳
❈♦♥s✐❞❡r❡❞ ❛s ♦♥❡ ♦❢ t❤❡ r♦♦ts ♦❢ tr♦♣✐❝❛❧ ❣❡♦♠❡tr②✳ ❊①t❡♥s✐♦♥s ❢♦r ❝♦♠♣❧❡t❡ ✐♥t❡rs❡❝t✐♦♥s ❜② ❇✐❤❛♥✱ ❙t✉r♠❢❡❧s✱ ■t❡♥❜❡r❣✴❘♦②✳
✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❱✐r♦✬s ♠❡t❤♦❞ ✭✼✵s✮✿ ❡✛❡❝t✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ♣r❡s❝r✐❜❡❞ t♦♣♦❧♦❣② ❛♥❞ s✉♣♣♦rt✳ ❊①❛♠♣❧❡✿ f = aX + bY ✹ + cX ✷Y ✸ + dX ✸Y + eX ✸Y ✹ ∈ R[X, Y ]✱ ✜♥❞ a, b, c, d, e ∈ R s✳t✳ {(x, y) ∈ R✷
>✵ | f (x, y) = ✵} ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡✳
❋♦r ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✱ t❤❡ ❝✉r✈❡
✷ ✷ ✸ ✷ ✸ ✷ ✸ ✷ ✸ ✹ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡ ✐♥ ✷ ✵✳
❈♦♥s✐❞❡r❡❞ ❛s ♦♥❡ ♦❢ t❤❡ r♦♦ts ♦❢ tr♦♣✐❝❛❧ ❣❡♦♠❡tr②✳ ❊①t❡♥s✐♦♥s ❢♦r ❝♦♠♣❧❡t❡ ✐♥t❡rs❡❝t✐♦♥s ❜② ❇✐❤❛♥✱ ❙t✉r♠❢❡❧s✱ ■t❡♥❜❡r❣✴❘♦②✳
✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❱✐r♦✬s ♠❡t❤♦❞ ✭✼✵s✮✿ ❡✛❡❝t✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ♣r❡s❝r✐❜❡❞ t♦♣♦❧♦❣② ❛♥❞ s✉♣♣♦rt✳ ❊①❛♠♣❧❡✿ f = aX + bY ✹ + cX ✷Y ✸ + dX ✸Y + eX ✸Y ✹ ∈ R[X, Y ]✱ ✜♥❞ a, b, c, d, e ∈ R s✳t✳ {(x, y) ∈ R✷
>✵ | f (x, y) = ✵} ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡✳
✷ ✶ ✷ ✷ ✷ ❋♦r ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✱ t❤❡ ❝✉r✈❡
✷ ✷ ✸ ✷ ✸ ✷ ✸ ✷ ✸ ✹ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡ ✐♥ ✷ ✵✳
❈♦♥s✐❞❡r❡❞ ❛s ♦♥❡ ♦❢ t❤❡ r♦♦ts ♦❢ tr♦♣✐❝❛❧ ❣❡♦♠❡tr②✳ ❊①t❡♥s✐♦♥s ❢♦r ❝♦♠♣❧❡t❡ ✐♥t❡rs❡❝t✐♦♥s ❜② ❇✐❤❛♥✱ ❙t✉r♠❢❡❧s✱ ■t❡♥❜❡r❣✴❘♦②✳
✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❱✐r♦✬s ♠❡t❤♦❞ ✭✼✵s✮✿ ❡✛❡❝t✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ♣r❡s❝r✐❜❡❞ t♦♣♦❧♦❣② ❛♥❞ s✉♣♣♦rt✳ ❊①❛♠♣❧❡✿ f = aX + bY ✹ + cX ✷Y ✸ + dX ✸Y + eX ✸Y ✹ ∈ R[X, Y ]✱ ✜♥❞ a, b, c, d, e ∈ R s✳t✳ {(x, y) ∈ R✷
>✵ | f (x, y) = ✵} ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡✳
✷ ✶ ✷ ✷ ✷ ❋♦r ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✱ t❤❡ ❝✉r✈❡
✷ ✷ ✸ ✷ ✸ ✷ ✸ ✷ ✸ ✹ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡ ✐♥ ✷ ✵✳
❈♦♥s✐❞❡r❡❞ ❛s ♦♥❡ ♦❢ t❤❡ r♦♦ts ♦❢ tr♦♣✐❝❛❧ ❣❡♦♠❡tr②✳ ❊①t❡♥s✐♦♥s ❢♦r ❝♦♠♣❧❡t❡ ✐♥t❡rs❡❝t✐♦♥s ❜② ❇✐❤❛♥✱ ❙t✉r♠❢❡❧s✱ ■t❡♥❜❡r❣✴❘♦②✳
✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❱✐r♦✬s ♠❡t❤♦❞ ✭✼✵s✮✿ ❡✛❡❝t✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ♣r❡s❝r✐❜❡❞ t♦♣♦❧♦❣② ❛♥❞ s✉♣♣♦rt✳ ❊①❛♠♣❧❡✿ f = aX + bY ✹ + cX ✷Y ✸ + dX ✸Y + eX ✸Y ✹ ∈ R[X, Y ]✱ ✜♥❞ a, b, c, d, e ∈ R s✳t✳ {(x, y) ∈ R✷
>✵ | f (x, y) = ✵} ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡✳
✷, + ✶, − ✷, + ✷, + ✷, + ❋♦r ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✱ t❤❡ ❝✉r✈❡
✷ ✷ ✸ ✷ ✸ ✷ ✸ ✷ ✸ ✹ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡ ✐♥ ✷ ✵✳
❈♦♥s✐❞❡r❡❞ ❛s ♦♥❡ ♦❢ t❤❡ r♦♦ts ♦❢ tr♦♣✐❝❛❧ ❣❡♦♠❡tr②✳ ❊①t❡♥s✐♦♥s ❢♦r ❝♦♠♣❧❡t❡ ✐♥t❡rs❡❝t✐♦♥s ❜② ❇✐❤❛♥✱ ❙t✉r♠❢❡❧s✱ ■t❡♥❜❡r❣✴❘♦②✳
✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❱✐r♦✬s ♠❡t❤♦❞ ✭✼✵s✮✿ ❡✛❡❝t✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ♣r❡s❝r✐❜❡❞ t♦♣♦❧♦❣② ❛♥❞ s✉♣♣♦rt✳ ❊①❛♠♣❧❡✿ f = aX + bY ✹ + cX ✷Y ✸ + dX ✸Y + eX ✸Y ✹ ∈ R[X, Y ]✱ ✜♥❞ a, b, c, d, e ∈ R s✳t✳ {(x, y) ∈ R✷
>✵ | f (x, y) = ✵} ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡✳
✷, + ✶, − ✷, + ✷, + ✷, + ❋♦r t > ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✱ t❤❡ ❝✉r✈❡ t✷ X − t X ✷Y ✸ + t✷ Y ✸ + t✷X ✸Y + t✷X ✸Y ✹ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡ ✐♥ R✷
>✵✳
❈♦♥s✐❞❡r❡❞ ❛s ♦♥❡ ♦❢ t❤❡ r♦♦ts ♦❢ tr♦♣✐❝❛❧ ❣❡♦♠❡tr②✳ ❊①t❡♥s✐♦♥s ❢♦r ❝♦♠♣❧❡t❡ ✐♥t❡rs❡❝t✐♦♥s ❜② ❇✐❤❛♥✱ ❙t✉r♠❢❡❧s✱ ■t❡♥❜❡r❣✴❘♦②✳
✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❱✐r♦✬s ♠❡t❤♦❞ ✭✼✵s✮✿ ❡✛❡❝t✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ♣r❡s❝r✐❜❡❞ t♦♣♦❧♦❣② ❛♥❞ s✉♣♣♦rt✳ ❊①❛♠♣❧❡✿ f = aX + bY ✹ + cX ✷Y ✸ + dX ✸Y + eX ✸Y ✹ ∈ R[X, Y ]✱ ✜♥❞ a, b, c, d, e ∈ R s✳t✳ {(x, y) ∈ R✷
>✵ | f (x, y) = ✵} ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡✳
✷, + ✶, − ✷, + ✷, + ✷, + ❋♦r t > ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✱ t❤❡ ❝✉r✈❡ t✷ X − t X ✷Y ✸ + t✷ Y ✸ + t✷X ✸Y + t✷X ✸Y ✹ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡ ✐♥ R✷
>✵✳
❈♦♥s✐❞❡r❡❞ ❛s ♦♥❡ ♦❢ t❤❡ r♦♦ts ♦❢ tr♦♣✐❝❛❧ ❣❡♦♠❡tr②✳ ❊①t❡♥s✐♦♥s ❢♦r ❝♦♠♣❧❡t❡ ✐♥t❡rs❡❝t✐♦♥s ❜② ❇✐❤❛♥✱ ❙t✉r♠❢❡❧s✱ ■t❡♥❜❡r❣✴❘♦②✳
✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❆ ✈❛r✐❛♥t ♦❢ ❱✐r♦✬s ❝♦♥str✉❝t✐♦♥ ❢♦r ✐s♦❧❛t❡❞ s♦❧✉t✐♦♥s✿ ❞❡♣❡♥❞s ♦♥ t❤❡ s✐❣♥s ♦❢ ♠✐♥♦rs ♦❢ ❛ ❝♦❡✣❝✐❡♥t ♠❛tr✐①✳ ❚❤❡♦r❡♠ ■❢ ❛ ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ ✐♥ ❛❞♠✐ts ❛ r❡❣✉❧❛r✱ ❜❛❧❛♥❝❡❞✱ ❛♥❞ ✉♥✐♠♦❞✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♠❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ s②st❡♠ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ s✉♣♣♦rt ✭✰ ❝♦♥str✉❝t✐♦♥✮✳ ▼❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ s②st❡♠✿ ❛❧❧ t♦r✐❝ ❝♦♠♣❧❡① s♦❧✉t✐♦♥s ❛r❡ r❡❛❧✱ ♣♦s✐t✐✈❡✱ ❛♥❞ ♥♦♥✲❞❡❣❡♥❡r❛t❡✳ ❋❡✇♥♦♠✐❛❧ s②st❡♠s ❚❤❡r❡ ❡①✐sts ❛ s②st❡♠ ♦❢ ✺ ❡q✉❛t✐♦♥s ✐♥ ✺ ✈❛r✐❛❜❧❡s✱ ✐♥✈♦❧✈✐♥❣ ✶✶ ♠♦♥♦♠✐❛❧s✱ ✇✐t❤ ❛t ❧❡❛st ✸✽ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ✭✰ ❝♦♥str✉❝t✐♦♥✮✳ ✉♥❞❡r s♦♠❡ ❛ss✉♠♣t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐st s②st❡♠s ♦❢ ❡q✉❛t✐♦♥s ✐♥ ✈❛r✐❛❜❧❡s✱ ✐♥✈♦❧✈✐♥❣ ❛t ♠♦st ✷ ✶ ♠♦♥♦♠✐❛❧s ❛♥❞ ❤❛✈✐♥❣ ❛s②♠♣t♦t✐❝❛❧❧② ✷ ✶ ✷✶ ✹ ✹ ✷ ✷ ✶✷ ✽ ✷ ♣♦s✐t✐✈❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s✳
✹ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❆ ✈❛r✐❛♥t ♦❢ ❱✐r♦✬s ❝♦♥str✉❝t✐♦♥ ❢♦r ✐s♦❧❛t❡❞ s♦❧✉t✐♦♥s✿ ❞❡♣❡♥❞s ♦♥ t❤❡ s✐❣♥s ♦❢ ♠✐♥♦rs ♦❢ ❛ ❝♦❡✣❝✐❡♥t ♠❛tr✐①✳ ❚❤❡♦r❡♠ ■❢ ❛ ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ ✐♥ Zd ❛❞♠✐ts ❛ r❡❣✉❧❛r✱ ❜❛❧❛♥❝❡❞✱ ❛♥❞ ✉♥✐♠♦❞✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♠❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ s②st❡♠ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ s✉♣♣♦rt ✭✰ ❝♦♥str✉❝t✐♦♥✮✳ ▼❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ s②st❡♠✿ ❛❧❧ t♦r✐❝ ❝♦♠♣❧❡① s♦❧✉t✐♦♥s ❛r❡ r❡❛❧✱ ♣♦s✐t✐✈❡✱ ❛♥❞ ♥♦♥✲❞❡❣❡♥❡r❛t❡✳ ❋❡✇♥♦♠✐❛❧ s②st❡♠s ❚❤❡r❡ ❡①✐sts ❛ s②st❡♠ ♦❢ ✺ ❡q✉❛t✐♦♥s ✐♥ ✺ ✈❛r✐❛❜❧❡s✱ ✐♥✈♦❧✈✐♥❣ ✶✶ ♠♦♥♦♠✐❛❧s✱ ✇✐t❤ ❛t ❧❡❛st ✸✽ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ✭✰ ❝♦♥str✉❝t✐♦♥✮✳ ✉♥❞❡r s♦♠❡ ❛ss✉♠♣t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐st s②st❡♠s ♦❢ ❡q✉❛t✐♦♥s ✐♥ ✈❛r✐❛❜❧❡s✱ ✐♥✈♦❧✈✐♥❣ ❛t ♠♦st ✷ ✶ ♠♦♥♦♠✐❛❧s ❛♥❞ ❤❛✈✐♥❣ ❛s②♠♣t♦t✐❝❛❧❧② ✷ ✶ ✷✶ ✹ ✹ ✷ ✷ ✶✷ ✽ ✷ ♣♦s✐t✐✈❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s✳
✹ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❆ ✈❛r✐❛♥t ♦❢ ❱✐r♦✬s ❝♦♥str✉❝t✐♦♥ ❢♦r ✐s♦❧❛t❡❞ s♦❧✉t✐♦♥s✿ ❞❡♣❡♥❞s ♦♥ t❤❡ s✐❣♥s ♦❢ ♠✐♥♦rs ♦❢ ❛ ❝♦❡✣❝✐❡♥t ♠❛tr✐①✳ ❚❤❡♦r❡♠ ■❢ ❛ ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ ✐♥ Zd ❛❞♠✐ts ❛ r❡❣✉❧❛r✱ ❜❛❧❛♥❝❡❞✱ ❛♥❞ ✉♥✐♠♦❞✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♠❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ s②st❡♠ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ s✉♣♣♦rt ✭✰ ❝♦♥str✉❝t✐♦♥✮✳ ▼❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ s②st❡♠✿ ❛❧❧ t♦r✐❝ ❝♦♠♣❧❡① s♦❧✉t✐♦♥s ❛r❡ r❡❛❧✱ ♣♦s✐t✐✈❡✱ ❛♥❞ ♥♦♥✲❞❡❣❡♥❡r❛t❡✳ ❋❡✇♥♦♠✐❛❧ s②st❡♠s ❚❤❡r❡ ❡①✐sts ❛ s②st❡♠ ♦❢ ✺ ❡q✉❛t✐♦♥s ✐♥ ✺ ✈❛r✐❛❜❧❡s✱ ✐♥✈♦❧✈✐♥❣ ✶✶ ♠♦♥♦♠✐❛❧s✱ ✇✐t❤ ❛t ❧❡❛st ✸✽ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ✭✰ ❝♦♥str✉❝t✐♦♥✮✳ ✉♥❞❡r s♦♠❡ ❛ss✉♠♣t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐st s②st❡♠s ♦❢ d ❡q✉❛t✐♦♥s ✐♥ d ✈❛r✐❛❜❧❡s✱ ✐♥✈♦❧✈✐♥❣ ❛t ♠♦st ✷d + ✶ ♠♦♥♦♠✐❛❧s ❛♥❞ ❤❛✈✐♥❣ ❛s②♠♣t♦t✐❝❛❧❧② ( √ ✷ + ✶)d √ d · ✷✶/✹(✹ + ✷ √ ✷) (✶✷ − ✽ √ ✷)√π ♣♦s✐t✐✈❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s✳
✹ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❙t✉r♠❢❡❧s ✬✾✹ ■❢ ❛ ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ A ✐♥ Zd ❛❞♠✐ts ❛ r❡❣✉❧❛r ✉♥✐♠♦❞✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐st s②st❡♠s ✇✐t❤ s✉♣♣♦rt A s✉❝❤ t❤❛t ❛❧❧ t♦r✐❝ ❝♦♠♣❧❡① s♦❧✉t✐♦♥s ❛r❡ r❡❛❧✳ ❚❤✐s ✇♦r❦✿ ✐❢ ❜❛❧❛♥❝❡❞✱ t❤❡♥ t❤❡ s♦❧✉t✐♦♥s ❝❛♥ ❜❡ ♠❛❞❡ ♣♦s✐t✐✈❡✳ ■t❡♥❜❡r❣✴❘♦② ❝♦♥str✉❝t✐♦♥✿ ❜❛s❡❞ ♦♥ s✐❣♥❡❞ ◆❡✇t♦♥ ♣♦❧②t♦♣❡s✱ ♠✐①❡❞ s②st❡♠s✳ ❙♦♣r✉♥♦✈❛✴❙♦tt✐❧❡✿ ❝♦♥str✉❝t✐♦♥s ♦♥ ❲r♦♥s❦✐ s②st❡♠s ✇✐t❤ ❧♦✇❡r ❜♦✉♥❞s ♦♥ t❤❡✐r ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s✳ ❘❡❛❧ s♦❧✉t✐♦♥s ♦❢ ❢❡✇♥♦♠✐❛❧ s②st❡♠s✿ ❇✐❤❛♥✱ ●r❡♥❡t✱ ❑♦✐r❛♥✱ P❤✐❧❧✐♣s♦♥✱ P♦rt✐❡r✱ ❘♦❥❛s✱ ❘♦②✱ ❙♦tt✐❧❡✱ ❙t✉r♠❢❡❧s✱ ❚❛✈❡♥❛s✱✳ ✳ ✳ ❇❛❧❛♥❝❡❞ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①❡s✿ ■③♠❡st✐❡✈✱ ❏♦s✇✐❣✱ ❙t❛♥❧❡②✱ ❲✐tt❡✱ ❩✐❡❣❧❡r✱✳ ✳ ✳ ❇✐❤❛♥✬s ❝♦♥❥❡❝t✉r❡ ■❢ ✐s t❤❡ s✉♣♣♦rt ♦❢ ❛ ♠❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ ♣♦❧②♥♦♠✐❛❧ s②st❡♠✱ t❤❡♥ ✐t ❤❛s ❛ ❜❛s✐s ♦❢ ❛✣♥❡ r❡❧❛t✐♦♥s ✇❤♦s❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ✐♥ ✷ ✶ ✵ ✶ ✷ ✳ ❆✣♥❡ r❡❧❛t✐♦♥✿ s✳t✳ ✵ ❛♥❞ ✵
✺ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❙t✉r♠❢❡❧s ✬✾✹ ■❢ ❛ ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ A ✐♥ Zd ❛❞♠✐ts ❛ r❡❣✉❧❛r ✉♥✐♠♦❞✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐st s②st❡♠s ✇✐t❤ s✉♣♣♦rt A s✉❝❤ t❤❛t ❛❧❧ t♦r✐❝ ❝♦♠♣❧❡① s♦❧✉t✐♦♥s ❛r❡ r❡❛❧✳ ❚❤✐s ✇♦r❦✿ ✐❢ ❜❛❧❛♥❝❡❞✱ t❤❡♥ t❤❡ s♦❧✉t✐♦♥s ❝❛♥ ❜❡ ♠❛❞❡ ♣♦s✐t✐✈❡✳ ■t❡♥❜❡r❣✴❘♦② ❝♦♥str✉❝t✐♦♥✿ ❜❛s❡❞ ♦♥ s✐❣♥❡❞ ◆❡✇t♦♥ ♣♦❧②t♦♣❡s✱ ♠✐①❡❞ s②st❡♠s✳ ❙♦♣r✉♥♦✈❛✴❙♦tt✐❧❡✿ ❝♦♥str✉❝t✐♦♥s ♦♥ ❲r♦♥s❦✐ s②st❡♠s ✇✐t❤ ❧♦✇❡r ❜♦✉♥❞s ♦♥ t❤❡✐r ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s✳ ❘❡❛❧ s♦❧✉t✐♦♥s ♦❢ ❢❡✇♥♦♠✐❛❧ s②st❡♠s✿ ❇✐❤❛♥✱ ●r❡♥❡t✱ ❑♦✐r❛♥✱ P❤✐❧❧✐♣s♦♥✱ P♦rt✐❡r✱ ❘♦❥❛s✱ ❘♦②✱ ❙♦tt✐❧❡✱ ❙t✉r♠❢❡❧s✱ ❚❛✈❡♥❛s✱✳ ✳ ✳ ❇❛❧❛♥❝❡❞ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①❡s✿ ■③♠❡st✐❡✈✱ ❏♦s✇✐❣✱ ❙t❛♥❧❡②✱ ❲✐tt❡✱ ❩✐❡❣❧❡r✱✳ ✳ ✳ ❇✐❤❛♥✬s ❝♦♥❥❡❝t✉r❡ ■❢ ✐s t❤❡ s✉♣♣♦rt ♦❢ ❛ ♠❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ ♣♦❧②♥♦♠✐❛❧ s②st❡♠✱ t❤❡♥ ✐t ❤❛s ❛ ❜❛s✐s ♦❢ ❛✣♥❡ r❡❧❛t✐♦♥s ✇❤♦s❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ✐♥ ✷ ✶ ✵ ✶ ✷ ✳ ❆✣♥❡ r❡❧❛t✐♦♥✿ s✳t✳ ✵ ❛♥❞ ✵
✺ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❙t✉r♠❢❡❧s ✬✾✹ ■❢ ❛ ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ A ✐♥ Zd ❛❞♠✐ts ❛ r❡❣✉❧❛r ✉♥✐♠♦❞✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐st s②st❡♠s ✇✐t❤ s✉♣♣♦rt A s✉❝❤ t❤❛t ❛❧❧ t♦r✐❝ ❝♦♠♣❧❡① s♦❧✉t✐♦♥s ❛r❡ r❡❛❧✳ ❚❤✐s ✇♦r❦✿ ✐❢ ❜❛❧❛♥❝❡❞✱ t❤❡♥ t❤❡ s♦❧✉t✐♦♥s ❝❛♥ ❜❡ ♠❛❞❡ ♣♦s✐t✐✈❡✳ ■t❡♥❜❡r❣✴❘♦② ❝♦♥str✉❝t✐♦♥✿ ❜❛s❡❞ ♦♥ s✐❣♥❡❞ ◆❡✇t♦♥ ♣♦❧②t♦♣❡s✱ ♠✐①❡❞ s②st❡♠s✳ ❙♦♣r✉♥♦✈❛✴❙♦tt✐❧❡✿ ❝♦♥str✉❝t✐♦♥s ♦♥ ❲r♦♥s❦✐ s②st❡♠s ✇✐t❤ ❧♦✇❡r ❜♦✉♥❞s ♦♥ t❤❡✐r ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s✳ ❘❡❛❧ s♦❧✉t✐♦♥s ♦❢ ❢❡✇♥♦♠✐❛❧ s②st❡♠s✿ ❇✐❤❛♥✱ ●r❡♥❡t✱ ❑♦✐r❛♥✱ P❤✐❧❧✐♣s♦♥✱ P♦rt✐❡r✱ ❘♦❥❛s✱ ❘♦②✱ ❙♦tt✐❧❡✱ ❙t✉r♠❢❡❧s✱ ❚❛✈❡♥❛s✱✳ ✳ ✳ ❇❛❧❛♥❝❡❞ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①❡s✿ ■③♠❡st✐❡✈✱ ❏♦s✇✐❣✱ ❙t❛♥❧❡②✱ ❲✐tt❡✱ ❩✐❡❣❧❡r✱✳ ✳ ✳ ❇✐❤❛♥✬s ❝♦♥❥❡❝t✉r❡ ■❢ A ⊂ Zd ✐s t❤❡ s✉♣♣♦rt ♦❢ ❛ ♠❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ ♣♦❧②♥♦♠✐❛❧ s②st❡♠✱ t❤❡♥ ✐t ❤❛s ❛ ❜❛s✐s ♦❢ ❛✣♥❡ r❡❧❛t✐♦♥s ✇❤♦s❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ✐♥ {−✷, −✶, ✵, ✶, ✷}✳ ❆✣♥❡ r❡❧❛t✐♦♥✿ (bα)α∈A ∈ Z|A| s✳t✳
α∈A bαα = ✵ ❛♥❞ α∈A bα = ✵. ✺ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
C✿ ❢✉❧❧ r❛♥❦ d × (d + ✶) r❡❛❧ ♠❛tr✐①✳ ❲❤❛t ❛r❡ t❤❡ ❝♦♥❞✐t✐♦♥s ♦♥ A s✉❝❤ t❤❛t C · X✶ X✷ ✳ ✳ ✳ Xd ✶ = ✵ ❤❛s ♦♥❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥❄ ❈r❛♠❡r✬s r✉❧❡✿ s✐❣♥s ♦❢ ♠❛①✐♠❛❧ ♠✐♥♦rs ♠✉st ❛❧t❡r♥❛t❡✳ Pr♦♣❡rt② ✐♥✈❛r✐❛♥t ❜② ♣❡r♠✉t❛t✐♦♥ ♦❢ t❤❡ ❝♦❧✉♠♥s✳ ❯♣ t♦ ❛♥ ✐♥✈❡rt✐❜❧❡ ♠♦♥♦♠✐❛❧ ♠❛♣✱ ❡①t❡♥❞s t♦ ❛♥② ✈❡❝t♦r ♦❢ ✶ ♠♦♥♦♠✐❛❧s s✳t✳ t❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ t❤❡ ❡①♣♦♥❡♥t ✈❡❝t♦rs ✐s ❛ ✲s✐♠♣❧❡①✳
✻ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
C✿ ❢✉❧❧ r❛♥❦ d × (d + ✶) r❡❛❧ ♠❛tr✐①✳ ❲❤❛t ❛r❡ t❤❡ ❝♦♥❞✐t✐♦♥s ♦♥ A s✉❝❤ t❤❛t C · X✶ X✷ ✳ ✳ ✳ Xd ✶ = ✵ ❤❛s ♦♥❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥❄ ❈r❛♠❡r✬s r✉❧❡✿ s✐❣♥s ♦❢ ♠❛①✐♠❛❧ ♠✐♥♦rs ♠✉st ❛❧t❡r♥❛t❡✳ Pr♦♣❡rt② ✐♥✈❛r✐❛♥t ❜② ♣❡r♠✉t❛t✐♦♥ ♦❢ t❤❡ ❝♦❧✉♠♥s✳ ❯♣ t♦ ❛♥ ✐♥✈❡rt✐❜❧❡ ♠♦♥♦♠✐❛❧ ♠❛♣✱ ❡①t❡♥❞s t♦ ❛♥② ✈❡❝t♦r ♦❢ ✶ ♠♦♥♦♠✐❛❧s s✳t✳ t❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ t❤❡ ❡①♣♦♥❡♥t ✈❡❝t♦rs ✐s ❛ ✲s✐♠♣❧❡①✳
✻ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
C✿ ❢✉❧❧ r❛♥❦ d × (d + ✶) r❡❛❧ ♠❛tr✐①✳ ❲❤❛t ❛r❡ t❤❡ ❝♦♥❞✐t✐♦♥s ♦♥ A s✉❝❤ t❤❛t C · X✶ X✷ ✳ ✳ ✳ Xd ✶ = ✵ ❤❛s ♦♥❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥❄ ❈r❛♠❡r✬s r✉❧❡✿ s✐❣♥s ♦❢ ♠❛①✐♠❛❧ ♠✐♥♦rs ♠✉st ❛❧t❡r♥❛t❡✳ Pr♦♣❡rt② ✐♥✈❛r✐❛♥t ❜② ♣❡r♠✉t❛t✐♦♥ ♦❢ t❤❡ ❝♦❧✉♠♥s✳ ❯♣ t♦ ❛♥ ✐♥✈❡rt✐❜❧❡ ♠♦♥♦♠✐❛❧ ♠❛♣✱ ❡①t❡♥❞s t♦ ❛♥② ✈❡❝t♦r ♦❢ d + ✶ ♠♦♥♦♠✐❛❧s s✳t✳ t❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ t❤❡ ❡①♣♦♥❡♥t ✈❡❝t♦rs ✐s ❛ d✲s✐♠♣❧❡①✳
✻ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
✷ ✶ ✷ ✷ ✷ ✵ ✶ −✶ −✶ ✵ ✶ ✵ −✶ −✶ ✶
t✷X tX ✷Y ✸ t✷Y ✹ t✷X ✸Y t✷X ✸Y ✹ = ✵ ❤❛s ✹ ♣♦s✐t✐✈❡ s♦❧s ❢♦r t > ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✳ P♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ❆ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① Γ ⊂ Rd ♦♥ s ✈❡rt✐❝❡s ✐s ❝❛❧❧❡❞ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡✱ ✐❢ t❤❡r❡ ❡①✐sts ❛ d × s ♠❛tr✐① C ✭✇✐t❤ ❝♦❧✉♠♥s ✐♥❞❡①❡❞ ❜② ✈❡rt✐❝❡s ♦❢ Γ✮ s✉❝❤ ❡✈❡r② d × (d + ✶) s✉❜♠❛tr✐① ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛ d✲s✐♠♣❧❡① ❤❛s ❢✉❧❧ r❛♥❦ ❛♥❞ ❤❛s ❛ ♣♦s✐t✐✈❡ ❦❡r♥❡❧ ✈❡❝t♦r✳ ❚❤❡♦r❡♠ ▲❡t ❜❡ ❛ ✜♥✐t❡ ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥✱ ❛ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ✐♥❝❧✉❞❡❞ ✐♥ ❛ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ ✳ ■❢ ✐s ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛t❡❞✱ t❤❡♥ ❢♦r ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✱ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ s♦❧s ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ s②st❡♠ ✐s ❜♦✉♥❞❡❞ ❜❡❧♦✇ ❜② t❤❡ ♥✉♠❜❡r ♦❢ ✲s✐♠♣❧✐❝❡s✳
✼ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
✷ ✶ ✷ ✷ ✷ ✵ ✶ −✶ −✶ ✵ ✶ ✵ −✶ −✶ ✶
t✷X tX ✷Y ✸ t✷Y ✹ t✷X ✸Y t✷X ✸Y ✹ = ✵ ❤❛s ✹ ♣♦s✐t✐✈❡ s♦❧s ❢♦r t > ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✳ P♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ❆ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① Γ ⊂ Rd ♦♥ s ✈❡rt✐❝❡s ✐s ❝❛❧❧❡❞ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡✱ ✐❢ t❤❡r❡ ❡①✐sts ❛ d × s ♠❛tr✐① C ✭✇✐t❤ ❝♦❧✉♠♥s ✐♥❞❡①❡❞ ❜② ✈❡rt✐❝❡s ♦❢ Γ✮ s✉❝❤ ❡✈❡r② d × (d + ✶) s✉❜♠❛tr✐① ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛ d✲s✐♠♣❧❡① ❤❛s ❢✉❧❧ r❛♥❦ ❛♥❞ ❤❛s ❛ ♣♦s✐t✐✈❡ ❦❡r♥❡❧ ✈❡❝t♦r✳ ❚❤❡♦r❡♠ ▲❡t A ⊂ Zn ❜❡ ❛ ✜♥✐t❡ ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥✱ Γ ⊂ Rd ❛ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ✐♥❝❧✉❞❡❞ ✐♥ ❛ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ Γ✳ ■❢ Γ ✐s ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛t❡❞✱ t❤❡♥ ❢♦r t > ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✱ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ s♦❧s ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ s②st❡♠ ✐s ❜♦✉♥❞❡❞ ❜❡❧♦✇ ❜② t❤❡ ♥✉♠❜❡r ♦❢ d✲s✐♠♣❧✐❝❡s✳
✼ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
✷ ✶ ✷ ✷ ✷ ❇❛❧❛♥❝❡❞ ❝♦♠♣❧❡①✿ ✶✲s❦❡❧❡t♦♥ ✐s (d + ✶)✲❝♦❧♦r✐❛❜❧❡ ❇✐♣❛rt✐t❡ ❝♦♠♣❧❡①✿ d✲s✐♠♣❧✐❝❡s ❛r❡ ✷✲❝♦❧♦r✐❛❜❧❡ Pr♦♣♦s✐t✐♦♥ ❜❛❧❛♥❝❡❞ ❡❛s② t♦ ❝❤❡❝❦ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡ ♥♦t ❡❛s② t♦ ❝❤❡❝❦ ❜✐♣❛rt✐t❡ ❡❛s② t♦ ❝❤❡❝❦ ▼✳ ❏♦s✇✐❣✿ ❢♦r tr✐❛♥❣✉❧❛t✐♦♥s✱ ❜❛❧❛♥❝❡❞ ❜✐♣❛rt✐t❡✳ ✭❜✉t ♥♦t t❤❡ ❝❛s❡ ❢♦r ❣❡♥❡r❛❧ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①✮✳ ◗✉❡st✐♦♥ ❜✐♣❛rt✐t❡ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡❄
✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
✷ ✶ ✷ ✷ ✷ ❇❛❧❛♥❝❡❞ ❝♦♠♣❧❡①✿ ✶✲s❦❡❧❡t♦♥ ✐s (d + ✶)✲❝♦❧♦r✐❛❜❧❡ ❇✐♣❛rt✐t❡ ❝♦♠♣❧❡①✿ d✲s✐♠♣❧✐❝❡s ❛r❡ ✷✲❝♦❧♦r✐❛❜❧❡ Pr♦♣♦s✐t✐♦♥ ❜❛❧❛♥❝❡❞ ❡❛s② t♦ ❝❤❡❝❦ ⇒ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡ ♥♦t ❡❛s② t♦ ❝❤❡❝❦ ⇒ ❜✐♣❛rt✐t❡ ❡❛s② t♦ ❝❤❡❝❦ ▼✳ ❏♦s✇✐❣✿ ❢♦r tr✐❛♥❣✉❧❛t✐♦♥s✱ ❜❛❧❛♥❝❡❞ ⇔ ❜✐♣❛rt✐t❡✳ ✭❜✉t ♥♦t t❤❡ ❝❛s❡ ❢♦r ❣❡♥❡r❛❧ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①✮✳ ◗✉❡st✐♦♥ ❜✐♣❛rt✐t❡ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡❄
✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
✷ ✶ ✷ ✷ ✷ ❇❛❧❛♥❝❡❞ ❝♦♠♣❧❡①✿ ✶✲s❦❡❧❡t♦♥ ✐s (d + ✶)✲❝♦❧♦r✐❛❜❧❡ ❇✐♣❛rt✐t❡ ❝♦♠♣❧❡①✿ d✲s✐♠♣❧✐❝❡s ❛r❡ ✷✲❝♦❧♦r✐❛❜❧❡ Pr♦♣♦s✐t✐♦♥ ❜❛❧❛♥❝❡❞ ❡❛s② t♦ ❝❤❡❝❦ ⇒ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡ ♥♦t ❡❛s② t♦ ❝❤❡❝❦ ⇒ ❜✐♣❛rt✐t❡ ❡❛s② t♦ ❝❤❡❝❦ ▼✳ ❏♦s✇✐❣✿ ❢♦r tr✐❛♥❣✉❧❛t✐♦♥s✱ ❜❛❧❛♥❝❡❞ ⇔ ❜✐♣❛rt✐t❡✳ ✭❜✉t ♥♦t t❤❡ ❝❛s❡ ❢♦r ❣❡♥❡r❛❧ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①✮✳ ◗✉❡st✐♦♥ ❜✐♣❛rt✐t❡ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡❄
✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
✷ ✶ ✷ ✷ ✷ ❇❛❧❛♥❝❡❞ ❝♦♠♣❧❡①✿ ✶✲s❦❡❧❡t♦♥ ✐s (d + ✶)✲❝♦❧♦r✐❛❜❧❡ ❇✐♣❛rt✐t❡ ❝♦♠♣❧❡①✿ d✲s✐♠♣❧✐❝❡s ❛r❡ ✷✲❝♦❧♦r✐❛❜❧❡ Pr♦♣♦s✐t✐♦♥ ❜❛❧❛♥❝❡❞ ❡❛s② t♦ ❝❤❡❝❦ ⇒ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡ ♥♦t ❡❛s② t♦ ❝❤❡❝❦ ⇒ ❜✐♣❛rt✐t❡ ❡❛s② t♦ ❝❤❡❝❦ ▼✳ ❏♦s✇✐❣✿ ❢♦r tr✐❛♥❣✉❧❛t✐♦♥s✱ ❜❛❧❛♥❝❡❞ ⇔ ❜✐♣❛rt✐t❡✳ ✭❜✉t ♥♦t t❤❡ ❝❛s❡ ❢♦r ❣❡♥❡r❛❧ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①✮✳ ◗✉❡st✐♦♥ ❜✐♣❛rt✐t❡
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⇒ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡❄
✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❚❤❡♦r❡♠ ■❢ ❛ ✜♥✐t❡ ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ A ⊂ Rd ❛❞♠✐ts ❛ r❡❣✉❧❛r ❜❛❧❛♥❝❡❞ ✉♥✐♠♦❞✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♠❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ s②st❡♠ ✇✐t❤ s✉♣♣♦rt A✳ Pr♦♦❢✳ ❑♦✉❝❤♥✐r❡♥❦♦✬s t❤❡♦r❡♠✳ ❍♦❧❞s tr✉❡ ❢♦r s❡✈❡r❛❧ ❝❧❛ss✐❝❛❧ ❢❛♠✐❧✐❡s ♦❢ ✿ ♦r❞❡r ♣♦❧②t♦♣❡s ✭❡✳❣✳ ♠✉❧t✐❧✐♥❡❛r s②st❡♠s✮ ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s t❤❡ ❤②♣❡rs✐♠♣❧❡① ❆❧❧ ♦❢ t❤❡♠ s❛t✐s❢② ❇✐❤❛♥✬s ❝♦♥❥❡❝t✉r❡✳
✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❚❤❡♦r❡♠ ■❢ ❛ ✜♥✐t❡ ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ A ⊂ Rd ❛❞♠✐ts ❛ r❡❣✉❧❛r ❜❛❧❛♥❝❡❞ ✉♥✐♠♦❞✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♠❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ s②st❡♠ ✇✐t❤ s✉♣♣♦rt A✳ Pr♦♦❢✳ ❑♦✉❝❤♥✐r❡♥❦♦✬s t❤❡♦r❡♠✳ ❍♦❧❞s tr✉❡ ❢♦r s❡✈❡r❛❧ ❝❧❛ss✐❝❛❧ ❢❛♠✐❧✐❡s ♦❢ ✿ ♦r❞❡r ♣♦❧②t♦♣❡s ✭❡✳❣✳ ♠✉❧t✐❧✐♥❡❛r s②st❡♠s✮ ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s t❤❡ ❤②♣❡rs✐♠♣❧❡① ❆❧❧ ♦❢ t❤❡♠ s❛t✐s❢② ❇✐❤❛♥✬s ❝♦♥❥❡❝t✉r❡✳
✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❚❤❡♦r❡♠ ■❢ ❛ ✜♥✐t❡ ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ A ⊂ Rd ❛❞♠✐ts ❛ r❡❣✉❧❛r ❜❛❧❛♥❝❡❞ ✉♥✐♠♦❞✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♠❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ s②st❡♠ ✇✐t❤ s✉♣♣♦rt A✳ Pr♦♦❢✳ ❑♦✉❝❤♥✐r❡♥❦♦✬s t❤❡♦r❡♠✳ ❍♦❧❞s tr✉❡ ❢♦r s❡✈❡r❛❧ ❝❧❛ss✐❝❛❧ ❢❛♠✐❧✐❡s ♦❢ ✿ ♦r❞❡r ♣♦❧②t♦♣❡s ✭❡✳❣✳ ♠✉❧t✐❧✐♥❡❛r s②st❡♠s✮ ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s t❤❡ ❤②♣❡rs✐♠♣❧❡① ❆❧❧ ♦❢ t❤❡♠ s❛t✐s❢② ❇✐❤❛♥✬s ❝♦♥❥❡❝t✉r❡✳
✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❚❤❡♦r❡♠ ■❢ ❛ ✜♥✐t❡ ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ A ⊂ Rd ❛❞♠✐ts ❛ r❡❣✉❧❛r ❜❛❧❛♥❝❡❞ ✉♥✐♠♦❞✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♠❛①✐♠❛❧❧② ♣♦s✐t✐✈❡ s②st❡♠ ✇✐t❤ s✉♣♣♦rt A✳ Pr♦♦❢✳ ❑♦✉❝❤♥✐r❡♥❦♦✬s t❤❡♦r❡♠✳ ❍♦❧❞s tr✉❡ ❢♦r s❡✈❡r❛❧ ❝❧❛ss✐❝❛❧ ❢❛♠✐❧✐❡s ♦❢ A✿ ♦r❞❡r ♣♦❧②t♦♣❡s ✭❡✳❣✳ ♠✉❧t✐❧✐♥❡❛r s②st❡♠s✮ ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s t❤❡ ❤②♣❡rs✐♠♣❧❡① ❆❧❧ ♦❢ t❤❡♠ s❛t✐s❢② ❇✐❤❛♥✬s ❝♦♥❥❡❝t✉r❡✳
✾ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❑❤♦✈❛♥s❦✐✐✬s ♣r♦❜❧❡♠
❡q✉❛t✐♦♥s✱ d ✉♥❦♥♦✇♥s ✐♥✈♦❧✈✐♥❣ ❛t ♠♦st d + k + ✶ ♠♦♥♦♠✐❛❧s❄ Ξd,k✿ ♠❛① ♦❢ ♥❜✳ ♦❢ ♣♦s✐t✐✈❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧s ♦✈❡r ❛❧❧ s✉❝❤ s②st❡♠s✳ ❇✐❤❛♥✴❙♦tt✐❧❡✿
✷
✸ ✹ ✷ ✷
✳ ❇✐❤❛♥✴❘♦❥❛s✴❙♦tt✐❧❡✿ ✶ ❯♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧s ✇✐t❤ ❞✐s❥♦✐♥t ✈❛r✐❛❜❧❡s✿ ✶ ✳ Pr♦❜❧❡♠✿ ❉♦❡s t❤❡r❡ ❡①✐st ❛ s②st❡♠ ✇✐t❤ ❛♥❞ ♠♦r❡ t❤❛♥ ✷ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s❄
✶✵ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❑❤♦✈❛♥s❦✐✐✬s ♣r♦❜❧❡♠
❡q✉❛t✐♦♥s✱ d ✉♥❦♥♦✇♥s ✐♥✈♦❧✈✐♥❣ ❛t ♠♦st d + k + ✶ ♠♦♥♦♠✐❛❧s❄ Ξd,k✿ ♠❛① ♦❢ ♥❜✳ ♦❢ ♣♦s✐t✐✈❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧s ♦✈❡r ❛❧❧ s✉❝❤ s②st❡♠s✳ ❇✐❤❛♥✴❙♦tt✐❧❡✿ Ξd,k ≤ e✷+✸
✹ ✷(k
✷)dk✳
❇✐❤❛♥✴❘♦❥❛s✴❙♦tt✐❧❡✿ Ξd,k ≥ (⌊d/k⌋ + ✶)k ❯♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧s ✇✐t❤ ❞✐s❥♦✐♥t ✈❛r✐❛❜❧❡s✿ Ξd,k ≥ (⌊k/d⌋ + ✶)d✳ Pr♦❜❧❡♠✿ ❉♦❡s t❤❡r❡ ❡①✐st ❛ s②st❡♠ ✇✐t❤ d = k ❛♥❞ ♠♦r❡ t❤❛♥ ✷d ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s❄
✶✵ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❈②❝❧✐❝ ♣♦❧②t♦♣❡ C(s, d)✿ ❝♦♥✈❡① ❤✉❧❧ ♦❢ s ♣♦✐♥ts (ai, a✷
i , . . . , ad i ) ∈ Rd✳
❆ ❜✐♣❛rt✐t❡ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ✇✐t❤ ♠❛♥② s✐♠♣❧✐❝❡s ❲❡ ♣r♦♣♦s❡ t♦ ✉s❡ ❛ ❜✐♣❛rt✐t❡ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ✐♥❝❧✉❞❡❞ ✐♥ ❛ tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ t❤❡ ❝②❝❧✐❝ ♣♦❧②t♦♣❡ C(✷d + ✶, d)✳ ❆s d ❣r♦✇s✱ t❤✐s s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ❤❛s Θ √ ✷ + ✶ d √ d
❜✉t ♥♦t ❜❛❧❛♥❝❡❞✦ ◆❡❡❞s ❝♦♠♣✉t❛t✐♦♥❛❧ ♠❡t❤♦❞s t♦ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛t❡ ✐t✳
✶✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❍♦✇ t♦ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛t❡ ❛ ✭♥♦♥✲❜❛❧❛♥❝❡❞✮ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① Γ ⊂ Rd❄ P♦s✐t✐✈❡ ♠❛tr✐① ❝♦♠♣❧❡t✐♦♥ ❆ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ✐s ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡ ✐✛ t❤❡r❡ ❡①✐sts ❛ ♠❛tr✐① ♦❢ r❛♥❦ s✉❝❤ t❤❛t ✵ ✐❢ s ✵ ♦t❤❡r✇✐s❡ ✇❤❡r❡ s✶ s ❛r❡ t❤❡ ✈❡rt✐❝❡s ♦❢ ✱ ❛♥❞
✶
❛r❡ ✐ts ✲s✐♠♣❧✐❝❡s✳ ■❢ s✉❝❤ ❛ ♠❛tr✐① ❡①✐sts✱ t❤❡♥ ❛ ❜❛s✐s ♦❢ ✐ts ❧❡❢t ❦❡r♥❡❧ ✐s ❛ ♠❛tr✐① ✇❤✐❝❤ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛t❡s ✳ ❈②❝❧✐❝ ♣♦❧②t♦♣❡ ✰ ◆❡✇t♦♥❙▲❘❆ ✭❙❝❤♦st✴❙✳✮ ❛ s②st❡♠ ♦❢ ✺ ❡qs✱ ✺ ✉♥❦♥♦✇♥s✱ ✶✶ ♠♦♥♦♠✐❛❧s✱ ❛♥❞ ✸✽ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s✦
✺ ✺
✸✽ ✭♣r❡✈✐♦✉s❧②✱ ✸✷
✺ ✺
✽✸✶✶✷✹✹✮✳
✶✷ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❍♦✇ t♦ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛t❡ ❛ ✭♥♦♥✲❜❛❧❛♥❝❡❞✮ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① Γ ⊂ Rd❄ P♦s✐t✐✈❡ ♠❛tr✐① ❝♦♠♣❧❡t✐♦♥ ❆ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① Γ ⊂ Rd ✐s ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡ ✐✛ t❤❡r❡ ❡①✐sts ❛ k × ℓ ♠❛tr✐① M ♦❢ r❛♥❦ k − d s✉❝❤ t❤❛t Mi,j
= ✵ ♦t❤❡r✇✐s❡, ✇❤❡r❡ s✶, . . . , sk ❛r❡ t❤❡ ✈❡rt✐❝❡s ♦❢ Γ✱ ❛♥❞ ∆✶, . . . , ∆ℓ ❛r❡ ✐ts d✲s✐♠♣❧✐❝❡s✳ ■❢ s✉❝❤ ❛ ♠❛tr✐① ❡①✐sts✱ t❤❡♥ ❛ ❜❛s✐s ♦❢ ✐ts ❧❡❢t ❦❡r♥❡❧ ✐s ❛ d × k ♠❛tr✐① ✇❤✐❝❤ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛t❡s Γ✳ ❈②❝❧✐❝ ♣♦❧②t♦♣❡ ✰ ◆❡✇t♦♥❙▲❘❆ ✭❙❝❤♦st✴❙✳✮ ❛ s②st❡♠ ♦❢ ✺ ❡qs✱ ✺ ✉♥❦♥♦✇♥s✱ ✶✶ ♠♦♥♦♠✐❛❧s✱ ❛♥❞ ✸✽ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s✦
✺ ✺
✸✽ ✭♣r❡✈✐♦✉s❧②✱ ✸✷
✺ ✺
✽✸✶✶✷✹✹✮✳
✶✷ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❍♦✇ t♦ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛t❡ ❛ ✭♥♦♥✲❜❛❧❛♥❝❡❞✮ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① Γ ⊂ Rd❄ P♦s✐t✐✈❡ ♠❛tr✐① ❝♦♠♣❧❡t✐♦♥ ❆ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① Γ ⊂ Rd ✐s ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛❜❧❡ ✐✛ t❤❡r❡ ❡①✐sts ❛ k × ℓ ♠❛tr✐① M ♦❢ r❛♥❦ k − d s✉❝❤ t❤❛t Mi,j
= ✵ ♦t❤❡r✇✐s❡, ✇❤❡r❡ s✶, . . . , sk ❛r❡ t❤❡ ✈❡rt✐❝❡s ♦❢ Γ✱ ❛♥❞ ∆✶, . . . , ∆ℓ ❛r❡ ✐ts d✲s✐♠♣❧✐❝❡s✳ ■❢ s✉❝❤ ❛ ♠❛tr✐① ❡①✐sts✱ t❤❡♥ ❛ ❜❛s✐s ♦❢ ✐ts ❧❡❢t ❦❡r♥❡❧ ✐s ❛ d × k ♠❛tr✐① ✇❤✐❝❤ ♣♦s✐t✐✈❡❧② ❞❡❝♦r❛t❡s Γ✳ ❈②❝❧✐❝ ♣♦❧②t♦♣❡ ✰ ◆❡✇t♦♥❙▲❘❆ ✭❙❝❤♦st✴❙✳✮ ❛ s②st❡♠ ♦❢ ✺ ❡qs✱ ✺ ✉♥❦♥♦✇♥s✱ ✶✶ ♠♦♥♦♠✐❛❧s✱ ❛♥❞ ✸✽ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s✦ ⇒ Ξ✺,✺ ≥ ✸✽. ✭♣r❡✈✐♦✉s❧②✱ ✸✷ ≤ Ξ✺,✺ ≤ ✽✸✶✶✷✹✹✮✳
✶✷ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
C =
✶✹✵✸✻ ✷✻✵✸✶ −✷✾✵✹✼ ✹✺✽✹✺ ✷✷✹✽✺ ✶✸✹✷✶✽ −✷✵✻✹✼ ✽✵✹✾✻ ✶✹✸✶✷ ✻✾✺✶✺ −✸✾✵✶✺ ✶✷✼✷✹✸ −✻✼✸✾ ✹✷✵✾✽ ✶✾✸✺✾ ✸✻✵✻✷✸ ✶✻✵✵✵ ✽✸✺✷✾ ✶✽✵✹ ✶✸✶✹✻✾ ✹✽✻✷ ✹✹✵✻✶ ✶✾✾✸✼ ✻✶✶✹✾ −✽✸✼✾ ✼✼✾✹✷ −✷✶✵✺ ✶✽✾✹✾ ✺✻✸✺ ✶✷✷✸✼✾ ✾✷✷✾ ✺✾✾✽✾ ✺✸✾✶ ✶✶✸✻✼✶ ✶✼✺✾✸ ✸✸✺✹✼ −✺✵✺✷✺ ✶✶✷✽✵✽ −✶✸✽✹✸ ✸✸✹✺✽ ✶✽✸✺✼ ✶✶✻✽✽✷ −✺✹✻✽✻ ✶✸✷✺✷✶ ✻✸✾✶ ✾✹✷✾✻ −✸✸✷✾ ✶✹✹✶✵✵ ✼✾✺✼ ✶✺✻✵✼✽ −✺✻✽✺ ✹✽✹✺✶ −✶✹✹✺✾ ✼✹✻✺✸ ✸✵✷✶✽ ✷✹✺✻✶✺ −✶✷✷✷✼ ✷✺✾✷✼ ✹✾✶✷✼ ✶✹✺✷✵✹ −✶✹✶✶✼ ✹✼✻✵✾ ✷✾✺✶✺ ✺✾✻✺✽ −✹✷✸✷✽ ✽✸✻✵✾ −✶✷✷✹✾ ✶✹✺✷✶✾ −✶✸✻✻✸ ✾✼✽✼✸ −✷✺✽✸✶ ✾✵✺✽✷ ✷✻✷✽✼ ✸✸✼✸✾ ✻✽✶✽ ✷✸✹✵✼ −✶✹✺✼✾ ✹✹✼✻✺ −✶✶✶✷✻ ✺✽✽✽✾ ✷✷✹✼ ✶✷✷✼✼✵ ✶✶✶✸✾ ✶✵✵✺✸✼ ✶✹✹✷✶ ✼✹✽✶✽ −✻✵✵✶✻ ✻✹✹✻✵✼ ✶✺✾✽✹ ✹✼✾✹✺ −✷✷✺✷✸ ✼✷✽✸✹ −✶✵✼✸✹ ✹✶✶✻✺ ✽✺✸✶ ✷✹✽✸✼ −✷✶✷✺✼ ✹✼✺✾✶ ✷✷✵✶✼ ✸✼✵✼✺ ✺✸✹✻ ✷✽✹✸✺✸ ✶✾✼✺✼ ✶✾✹✶✼✸ ✺✼✹✵ ✽✸✵✷✾ −✻✷✷✼✶ ✹✻✻✶✶✶ ✺✺✾✶ ✸✼✾✵✷
C · ✶ t X✶X✷X✸X✹X✺ t✷✻X ✷
✶ X ✷✷ ✷ X ✷✸ ✸ X ✷✹ ✹ X ✷✺ ✺
t✸✻X ✸
✶ X ✸✷ ✷ X ✸✸ ✸ X ✸✹ ✹ X ✸✺ ✺
✳ ✳ ✳ t✶✵✻X ✶✵
✶ X ✶✵✷ ✷
X ✶✵✸
✸
X ✶✵✹
✹
X ✶✵✺
✺
= ✵ ✵ ✵ ✵ ✵
✶✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
▲✐♠✐ts✿ ❚❤❡r❡ ❡①✐st A s✳t✳ t❤❡ ♠❛① ♥❜✳ ♦❢ ♣♦s✳ s♦❧s ❝❛♥♥♦t ❜❡ r❡❛❝❤❡❞ ❜② t❤✐s ♠❡t❤♦❞✳ ❘❡str✐❝t❡❞ ❛t t❤❡ ♠♦♠❡♥t t♦ ✉♥♠✐①❡❞ s②st❡♠s✳ ❚❤❡♦r②✿ ■s t❤❡ ❜✐♣❛rt✐t❡ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ❢r♦♠ t❤❡ ❝②❝❧✐❝ ♣♦❧②t♦♣❡ ❛❧✇❛②s ❞❡❝♦r❛❜❧❡❄ ✐❢ ②❡s✱ t❤❡♥ ❧✐♠ s✉♣
✶
✷ ✶ ■♥ ❣❡♥❡r❛❧✱ ❡①✐st❡♥❝❡ ♦❢ ❛ ❜✐♣❛rt✐t❡ s✐♠♣❧✳ ❝♦♠♣❧❡① ✇❤✐❝❤ ✐s ♥♦t ❞❡❝♦r❛❜❧❡❄ ✐❢ ♥♦✱ s✐♠♣❧❡r ♣r♦♦❢s ❢♦r ❧♦✇❡r ❜♦✉♥❞s ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s✳ ✏❢♦r ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✑✿ ❡①♣❧✐❝✐t
✵❄
❇✐❤❛♥✬s ❝♦♥❥❡❝t✉r❡✳ ❈♦♠♣✉t❛t✐♦♥s✿
✐♥ ✱ ❝♦♠♣✉t❡ ✭✐❢ ✐t ❡①✐sts✮ ❛ r❡❣✉❧❛r ✉♥✐♠♦❞✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ ✐ts ❝♦♥✈❡① ❤✉❧❧✳ ■❢ ✉♥✐♠♦❞✉❧❛r ✐s ♥♦t ♣♦ss✐❜❧❡✱ ✜♥❞ ❛ ❜✐♣❛rt✐t❡ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ✇✐t❤ ✈❡rt✐❝❡s ✇✐t❤ ❛s ♠❛♥② ✲s✐♠♣❧✐❝❡s ❛ ♣♦ss✐❜❧❡✳ ✭❍②❜r✐❞ s②♠❜♦❧✐❝✲♥✉♠❡r✐❝✮ ❝♦♠♣✉t❛t✐♦♥❛❧ t♦♦❧s ❢♦r t❤❡ ♣♦s✐t✐✈❡ ♠❛tr✐① ❝♦♠♣❧❡t✐♦♥ ♣r♦❜❧❡♠✳
✶✹ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
▲✐♠✐ts✿ ❚❤❡r❡ ❡①✐st A s✳t✳ t❤❡ ♠❛① ♥❜✳ ♦❢ ♣♦s✳ s♦❧s ❝❛♥♥♦t ❜❡ r❡❛❝❤❡❞ ❜② t❤✐s ♠❡t❤♦❞✳ ❘❡str✐❝t❡❞ ❛t t❤❡ ♠♦♠❡♥t t♦ ✉♥♠✐①❡❞ s②st❡♠s✳ ❚❤❡♦r②✿ ■s t❤❡ ❜✐♣❛rt✐t❡ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ❢r♦♠ t❤❡ ❝②❝❧✐❝ ♣♦❧②t♦♣❡ ❛❧✇❛②s ❞❡❝♦r❛❜❧❡❄ ✐❢ ②❡s✱ t❤❡♥ ❧✐♠ s✉♣(Ξd,d)✶/d ≥ √ ✷ + ✶ ■♥ ❣❡♥❡r❛❧✱ ❡①✐st❡♥❝❡ ♦❢ ❛ ❜✐♣❛rt✐t❡ s✐♠♣❧✳ ❝♦♠♣❧❡① ✇❤✐❝❤ ✐s ♥♦t ❞❡❝♦r❛❜❧❡❄ ✐❢ ♥♦✱ s✐♠♣❧❡r ♣r♦♦❢s ❢♦r ❧♦✇❡r ❜♦✉♥❞s ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s✳ ✏❢♦r t > ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✑✿ ❡①♣❧✐❝✐t t✵❄ ❇✐❤❛♥✬s ❝♦♥❥❡❝t✉r❡✳ ❈♦♠♣✉t❛t✐♦♥s✿
✐♥ ✱ ❝♦♠♣✉t❡ ✭✐❢ ✐t ❡①✐sts✮ ❛ r❡❣✉❧❛r ✉♥✐♠♦❞✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ ✐ts ❝♦♥✈❡① ❤✉❧❧✳ ■❢ ✉♥✐♠♦❞✉❧❛r ✐s ♥♦t ♣♦ss✐❜❧❡✱ ✜♥❞ ❛ ❜✐♣❛rt✐t❡ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ✇✐t❤ ✈❡rt✐❝❡s ✇✐t❤ ❛s ♠❛♥② ✲s✐♠♣❧✐❝❡s ❛ ♣♦ss✐❜❧❡✳ ✭❍②❜r✐❞ s②♠❜♦❧✐❝✲♥✉♠❡r✐❝✮ ❝♦♠♣✉t❛t✐♦♥❛❧ t♦♦❧s ❢♦r t❤❡ ♣♦s✐t✐✈❡ ♠❛tr✐① ❝♦♠♣❧❡t✐♦♥ ♣r♦❜❧❡♠✳
✶✹ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
▲✐♠✐ts✿ ❚❤❡r❡ ❡①✐st A s✳t✳ t❤❡ ♠❛① ♥❜✳ ♦❢ ♣♦s✳ s♦❧s ❝❛♥♥♦t ❜❡ r❡❛❝❤❡❞ ❜② t❤✐s ♠❡t❤♦❞✳ ❘❡str✐❝t❡❞ ❛t t❤❡ ♠♦♠❡♥t t♦ ✉♥♠✐①❡❞ s②st❡♠s✳ ❚❤❡♦r②✿ ■s t❤❡ ❜✐♣❛rt✐t❡ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ❢r♦♠ t❤❡ ❝②❝❧✐❝ ♣♦❧②t♦♣❡ ❛❧✇❛②s ❞❡❝♦r❛❜❧❡❄ ✐❢ ②❡s✱ t❤❡♥ ❧✐♠ s✉♣(Ξd,d)✶/d ≥ √ ✷ + ✶ ■♥ ❣❡♥❡r❛❧✱ ❡①✐st❡♥❝❡ ♦❢ ❛ ❜✐♣❛rt✐t❡ s✐♠♣❧✳ ❝♦♠♣❧❡① ✇❤✐❝❤ ✐s ♥♦t ❞❡❝♦r❛❜❧❡❄ ✐❢ ♥♦✱ s✐♠♣❧❡r ♣r♦♦❢s ❢♦r ❧♦✇❡r ❜♦✉♥❞s ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s✳ ✏❢♦r t > ✵ s✉✣❝✐❡♥t❧② s♠❛❧❧✑✿ ❡①♣❧✐❝✐t t✵❄ ❇✐❤❛♥✬s ❝♦♥❥❡❝t✉r❡✳ ❈♦♠♣✉t❛t✐♦♥s✿
tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ ✐ts ❝♦♥✈❡① ❤✉❧❧✳ ■❢ ✉♥✐♠♦❞✉❧❛r ✐s ♥♦t ♣♦ss✐❜❧❡✱ ✜♥❞ ❛ ❜✐♣❛rt✐t❡ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ✇✐t❤ ✈❡rt✐❝❡s A ✇✐t❤ ❛s ♠❛♥② d✲s✐♠♣❧✐❝❡s ❛ ♣♦ss✐❜❧❡✳ ✭❍②❜r✐❞ s②♠❜♦❧✐❝✲♥✉♠❡r✐❝✮ ❝♦♠♣✉t❛t✐♦♥❛❧ t♦♦❧s ❢♦r t❤❡ ♣♦s✐t✐✈❡ ♠❛tr✐① ❝♦♠♣❧❡t✐♦♥ ♣r♦❜❧❡♠✳
✶✹ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❛r❳✐✈✿✶✺✶✵✳✵✺✻✷✷
✶✺ P❏ ❙♣❛❡♥❧❡❤❛✉❡r