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❙♣❛rs❡ ♣♦❧②♥♦♠✐❛❧ s②st❡♠s ✇✐t❤ ♠❛♥② ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ❢r♦♠ ❜✐♣❛rt✐t❡ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①❡s ❋ré❞ér✐❝ ❇✐❤❛♥✱ P✐❡rr❡✲❏❡❛♥ ❙♣❛❡♥❧❡❤❛✉❡r ▲❆▼❆✱ ❯♥✐✈✳ ❙❛✈♦✐❡ ▼♦♥t ❇❧❛♥❝ ■♥r✐❛✴▲❖❘■❆✴❈◆❘❙✱ ♣r♦❥❡❝t ❈❆❘❆▼❊▲ ❏♦✉r♥é❡s ◆❛t✐♦♥❛❧❡s ❞✉ ❈❛❧❝✉❧ ❋♦r♠❡❧ ✷✵✶✺✱ ❈❧✉♥② ✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

✶ ✇✐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 ✭✶✻✸✼✮ ✶ ✐s ❜♦✉♥❞❡❞ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ r♦♦ts ♦❢ ❛ ▲❛✉r❡♥t ♣♦❧②♥♦♠✐❛❧ ❜② t❤❡ ♥✉♠❜❡r ♦❢ s✐❣♥ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ❝♦♥s❡❝✉t✐✈❡ ❝♦❡✣❝✐❡♥ts✳ ❛❧❧ ♥♦♥③❡r♦ ❝♦♠♣❧❡① r♦♦ts ♦❢ ✭sq✉❛r❡❢r❡❡✮ ❛r❡ ♣♦s✐t✐✈❡ ✶ ✶ ❢♦r s♦♠❡ ✳ Pr♦❜❧❡♠ st❛t❡♠❡♥t X α ✶ · · · X α d A ⊂ Z d ✿ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts✳ α ∈ A ↔ d ✳ ✶ ✷ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

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 ✭✶✻✸✼✮ ✶ ✐s ❜♦✉♥❞❡❞ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ r♦♦ts ♦❢ ❛ ▲❛✉r❡♥t ♣♦❧②♥♦♠✐❛❧ ❜② t❤❡ ♥✉♠❜❡r ♦❢ s✐❣♥ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ❝♦♥s❡❝✉t✐✈❡ ❝♦❡✣❝✐❡♥ts✳ ❛❧❧ ♥♦♥③❡r♦ ❝♦♠♣❧❡① r♦♦ts ♦❢ ✭sq✉❛r❡❢r❡❡✮ ❛r❡ ♣♦s✐t✐✈❡ ✶ ✶ ❢♦r s♦♠❡ ✳ Pr♦❜❧❡♠ st❛t❡♠❡♥t X α ✶ · · · X α d A ⊂ Z d ✿ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts✳ α ∈ A ↔ d ✳ ✶ f ✶ , . . . , f d ∈ R [ X ± ✶ , . . . , X ± ✶ ] ✇✐t❤ s✉♣♣♦rt A ✳ ✶ d ❆ s♦❧✉t✐♦♥ f ✶ ( ① ) = · · · = f d ( ① ) = ✵ ✐s ♣♦s✐t✐✈❡ ✐❢ ① ∈ R d ❛♥❞ x i > ✵ ❢♦r ❛❧❧ i ❀ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ t❤❡ ❥❛❝♦❜✐❛♥ ♠❛tr✐① ♦❢ ( f ✶ , . . . , f d ) ✐s ✐♥✈❡rt✐❜❧❡ ❛t ① ✷ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

▼♦t✐✈❛t✐♦♥ ✭ ✶✮✿ ❉❡s❝❛rt❡s✬ r✉❧❡ ♦❢ s✐❣♥s ✭✶✻✸✼✮ ✶ ✐s ❜♦✉♥❞❡❞ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ r♦♦ts ♦❢ ❛ ▲❛✉r❡♥t ♣♦❧②♥♦♠✐❛❧ ❜② t❤❡ ♥✉♠❜❡r ♦❢ s✐❣♥ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ❝♦♥s❡❝✉t✐✈❡ ❝♦❡✣❝✐❡♥ts✳ ❛❧❧ ♥♦♥③❡r♦ ❝♦♠♣❧❡① r♦♦ts ♦❢ ✭sq✉❛r❡❢r❡❡✮ ❛r❡ ♣♦s✐t✐✈❡ ✶ ✶ ❢♦r s♦♠❡ ✳ Pr♦❜❧❡♠ st❛t❡♠❡♥t X α ✶ · · · X α d A ⊂ Z d ✿ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts✳ α ∈ A ↔ d ✳ ✶ f ✶ , . . . , f d ∈ R [ X ± ✶ , . . . , X ± ✶ ] ✇✐t❤ s✉♣♣♦rt A ✳ ✶ d ❆ s♦❧✉t✐♦♥ f ✶ ( ① ) = · · · = f d ( ① ) = ✵ ✐s ♣♦s✐t✐✈❡ ✐❢ ① ∈ R d ❛♥❞ x i > ✵ ❢♦r ❛❧❧ i ❀ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ t❤❡ ❥❛❝♦❜✐❛♥ ♠❛tr✐① ♦❢ ( f ✶ , . . . , f d ) ✐s ✐♥✈❡rt✐❜❧❡ ❛t ① Pr♦❜❧❡♠ st❛t❡♠❡♥ts ●✐✈❡♥ A ✱ ❝♦♥str✉❝t f ✶ , . . . , f d s✉❝❤ t❤❛t f ✶ ( ❳ ) = · · · = f d ( ❳ ) = ✵ ❤❛s ✶ ♠❛♥② ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s❀ ●✐✈❡♥ s ∈ N ❝♦♥str✉❝t A ❛♥❞ f ✶ , . . . , f d s✉❝❤ t❤❛t |A| = s ❛♥❞ ✷ f ✶ ( ❳ ) = · · · = f d ( ❳ ) = ✵ ❤❛s ♠❛♥② ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s✳ ✷ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

Pr♦❜❧❡♠ st❛t❡♠❡♥t X α ✶ · · · X α d A ⊂ Z d ✿ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts✳ α ∈ A ↔ d ✳ ✶ f ✶ , . . . , f d ∈ R [ X ± ✶ , . . . , X ± ✶ ] ✇✐t❤ s✉♣♣♦rt A ✳ ✶ d ❆ s♦❧✉t✐♦♥ f ✶ ( ① ) = · · · = f d ( ① ) = ✵ ✐s ♣♦s✐t✐✈❡ ✐❢ ① ∈ R d ❛♥❞ x i > ✵ ❢♦r ❛❧❧ i ❀ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ t❤❡ ❥❛❝♦❜✐❛♥ ♠❛tr✐① ♦❢ ( f ✶ , . . . , f d ) ✐s ✐♥✈❡rt✐❜❧❡ ❛t ① Pr♦❜❧❡♠ st❛t❡♠❡♥ts ●✐✈❡♥ A ✱ ❝♦♥str✉❝t f ✶ , . . . , f d s✉❝❤ t❤❛t f ✶ ( ❳ ) = · · · = f d ( ❳ ) = ✵ ❤❛s ✶ ♠❛♥② ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s❀ ●✐✈❡♥ s ∈ N ❝♦♥str✉❝t A ❛♥❞ f ✶ , . . . , f d s✉❝❤ t❤❛t |A| = s ❛♥❞ ✷ f ✶ ( ❳ ) = · · · = f d ( ❳ ) = ✵ ❤❛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❧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✳ ✹ ✷ ✸ ✸ ✸ ✹ ❊①❛♠♣❧❡ ✿ ✱ ✜♥❞ ✷ s✳t✳ ✵ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝✐r❝❧❡ ✳ ✵ ❈♦♥t❡①t ❱✐r♦✬s ♠❡t❤♦❞ ✭✼✵s✮✿ ❡✛❡❝t✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ♣r❡s❝r✐❜❡❞ t♦♣♦❧♦❣② ❛♥❞ s✉♣♣♦rt✳ ✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡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✳ ❈♦♥t❡①t ❱✐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❝❧❡ ✳ ✸ P❏ ❙♣❛❡♥❧❡❤❛✉❡r