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❖♥ ❈♦♥str✉❝t✐♥❣ ❋❛♠✐❧✐❡s ♦❢ P❛✐r✐♥❣✲❋r✐❡♥❞❧② ❊❧❧✐♣t✐❝ ❈✉r✈❡s ✇✐t❤ ❱❛r✐❛❜❧❡ ❉✐s❝r✐♠✐♥❛♥t ❘♦❜❡rt ❉r②➟♦ ■♥❞♦❝r②♣t ✷✵✶✶✱ ✶✶✕✶✹ ❉❡❝❡♠❜❡r ✷✵✶✶✱ ❈❤❡♥♥❛✐ ❘✳ ❉r②➟♦ ✭✮ ❈♦♥str✉❝t✐♥❣ P❛✐r✐♥❣✲❢r✐❡♥❞❧② ❈✉r✈❡s ✶ ✴ ✷✼

P❛✐r✐♥❣s ▲❡t ❊ / F q ❜❡ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡✱ r ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r✱ r � | q , ❊ [ r ] = { P ∈ ❊ ( F q ) : [ r ] P = ✵ } ✱ µ r = { ζ ∈ F q : ζ r = ✶ } ✱ ❲❡ ❤❛✈❡ t✇♦ ♠❛✐♥ ♣❛✐r✐♥❣s✿ ✶ ❚❤❡ ❲❡✐❧ ♣❛✐r✐♥❣ ❡ r : ❊ [ r ] × ❊ [ r ] → µ r ⊂ F q ❦ , ✇❤❡r❡ F q ❦ = F q ( µ r ) ✳ ❚❤❡ ❡①♣♦♥❡♥t ❦ ✐s ❝❛❧❧❡❞ t❤❡ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ♦❢ ❊ ✇✐t❤ r❡s♣❡❝t t♦ r ✷ ❚❤❡ ❚❛t❡ ♣❛✐r✐♥❣ ❊ ( F q ❦ )[ r ] × ❊ ( F q ❦ ) / r❊ ( F q ❦ ) → µ r ⊂ F q ❦ ❘✳ ❉r②➟♦ ✭✮ ❈♦♥str✉❝t✐♥❣ P❛✐r✐♥❣✲❢r✐❡♥❞❧② ❈✉r✈❡s ✷ ✴ ✷✼

■❢ t❤❡ ❛r✐t❤♠❡t✐❝ ✐♥ t❤❡ ✜❡❧❞ F q ❦ ✐s ❢❡❛s✐❜❧❡✱ ♦♥❡ ❝❛♥ ❝♦♠♣✉t❡ ♣❛✐r✐♥❣s ✉s✐♥❣ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠✳ ❚❤❡ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ❦ ✐s ❡q✉❛❧ t♦ ❦ = ♠✐♥ { ❧ ∈ N : r | ( q ❧ − ✶ ) } = t❤❡ ♦r❞❡r ♦❢ q ♠♦❞ r ✐♥ F ∗ r ❚❤❡r❡❢♦r❡✱ ❦ ✐s ✉s✉❛❧❧② ♦❢ t❤❡ s✐♠✐❧❛r ❜✐t s✐③❡ ❛s r ✳ P❛✐r✐♥❣✲❢r✐❡♥❞❧② ❝✉r✈❡s r❡q✉✐r❡ s♣❡❝✐❛❧ ❝♦♥str✉❝t✐♦♥s✳ ❘✳ ❉r②➟♦ ✭✮ ❈♦♥str✉❝t✐♥❣ P❛✐r✐♥❣✲❢r✐❡♥❞❧② ❈✉r✈❡s ✸ ✴ ✷✼

❚♦ ❝♦♥str✉❝t ❛♥ ♦r❞✐♥❛r② ❡❧❧✐♣t✐❝ ❝✉r✈❡ ✇✐t❤ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ❦ ✇❡ ✜rst ✜♥❞ ♣❛r❛♠❡t❡rs ( r , t , q ) ✱ ✇❤❡r❡ r ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ t❤❡r❡ ❡①✐sts ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❊ / F q ✇✐t❤ tr❛❝❡ t ❛♥❞ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ❦ ✇✐t❤ r❡s♣❡❝t t♦ r ✳ ❚❤❡♥ ✉s✐♥❣ ❈▼ ♠❡t❤♦❞✱ ✇❡ ✜♥❞ ❛♥ ❡q✉❛t✐♦♥ ♦❢ ❊ ✳ ❚❤❡r❡❢♦r❡ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ❉ ♦❢ ❊ ♠✉st ❜❡ s✉✣❝✐❡♥t❧② s♠❛❧❧✳ ❘❡❝❛❧❧ t❤❛t t❤❡ tr❛❝❡ t ♦❢ ❊ / F q s❛t✐s✜❡s t = q + ✶ − # ❊ ( F q ) ❛♥❞ | t | ≤ ✷ √ q ✳ ❚❤❡♥ t❤❡ ❋r♦❜❡♥✐✉s ❡♥❞♦♠♦r♣❤✐s♠ π : ❊ → ❊ π ( ① , ② ) = ( ① q , ② q ) s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥ π ✷ − t π + q = ✵✳ ❚❤❡ ❞✐s❝r✐♠✐♥❛♥t ❉ ♦❢ ❊ ✐s t❤❡ sq✉❛r❡✲❢r❡❡ ♣❛rt ♦❢ ✹ q − t ✷ = ❉② ✷ > ✵✳ ■❢ ❊ ✐s ♦r❞✐♥❛r② ✭✐✳❡✳✱ ❣❝❞ ( t , q ) = ✶✮✱ t❤❡♥ ❊♥❞ ( ❊ ) ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛♥ ♦r❞❡r ✐♥ t❤❡ √ ✐♠❛❣✐♥❛r② q✉❛❞r❛t✐❝ ✜❡❧❞ ❑ = Q ( − ❉ ) ✳ ❘✳ ❉r②➟♦ ✭✮ ❈♦♥str✉❝t✐♥❣ P❛✐r✐♥❣✲❢r✐❡♥❞❧② ❈✉r✈❡s ✹ ✴ ✷✼

❚❤❡ ❈▼ ♠❡t❤♦❞ ❚❤❡ ❈▼ ♠❡t❤♦❞ ✐s ✉s❡❞ t♦ ❝♦♥str✉❝t ❛♥ ♦r❞✐♥❛r② ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❊ / F q ♦❢ ♦r❞❡r ♥ = # ❊ ( F q ) ✳ ❙✉❝❤ ❛ ❝✉r✈❡ ❊ ❡①✐sts ✐❢ ❛♥❞ ♦♥❧② ✐❢ | t | ≤ ✷ √ q ❛♥❞ ❣❝❞ ( t , q ) = ✶✱ ✇❤❡r❡ t = q + ✶ − ♥ ✳ ❚❤❡♥ ❊♥❞ ( ❊ ) ✐s ❛♥ ♦r❞❡r ✐♥ t❤❡ ✐♠❛❣✐♥❛r② q✉❛❞r❛t✐❝ ✜❡❧❞ √ ❑ = Q ( − ❉ ) ✱ ✇❤❡r❡ ❉ ✐s ❛ ❞✐s❝r✐♠✐♥❛♥t ♦❢ ❊ ✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ ❊♥❞ ( ❊ ) ✐s ❛♥ ♦r❞❡r ✐♥ ❑ ✱ t❤❡♥ s♦♠❡ t✇✐st ♦❢ ❊ ❤❛s ♦r❞❡r ♥ ✳ ❚❤❡r❡❢♦r❡✱ ✐t s✉✣❝❡s t♦ ❝♦♥str✉❝t ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❊ / F q s✉❝❤ t❤❛t ❊♥❞ ( ❊ ) ✐s t❤❡ ♠❛①✐♠❛❧ ♦r❞❡r O ❑ ✐♥ ❑ ✳ ❘✳ ❉r②➟♦ ✭✮ ❈♦♥str✉❝t✐♥❣ P❛✐r✐♥❣✲❢r✐❡♥❞❧② ❈✉r✈❡s ✺ ✴ ✷✼

❚❤❡ ❈▼ ♠❡t❤♦❞ ❚❤❡r❡ ❡①✐sts t❤❡ ❍✐❧❜❡rt ❝❧❛ss ♣♦❧②♥♦♠✐❛❧ ❍ ❑ ( ① ) ∈ Z [ ① ] s✉❝❤ t❤❛t ❥ ∈ F q ✐s ❛ ❥✲✐♥✈❛r✐❛♥t ♦❢ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❊ / F q ✇✐t❤ ❊♥❞ ( ❊ ) = O ❑ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❍ ❑ ( ❥ ) = ✵ . ❆❧❣♦r✐t❤♠s ❢♦r ❝♦♠♣✉t✐♥❣ ❍ ❑ ( ① ) ❤❛✈❡ ❝♦♠♣❧❡①✐t② ❛t ❧❡❛st ❖ ( ❉ ) ✱ t❤❡r❡❢♦r❡ ❉ ♠✉st ❜❡ s✉✣❝✐❡♥t❧② s♠❛❧❧ ✭❝✉rr❡♥t❧②✱ ❉ ≤ ✶✵ ✶✸ ✮✳ ❘✳ ❉r②➟♦ ✭✮ ❈♦♥str✉❝t✐♥❣ P❛✐r✐♥❣✲❢r✐❡♥❞❧② ❈✉r✈❡s ✻ ✴ ✷✼

P❛r❛♠❡t❡rs ( r , t , q ) ♦❢ ❛♥ ♦r❞✐♥❛r② ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❊ ✇✐t❤ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ❦ ❛♥❞ ❞✐s❝r✐♠✐♥❛♥t ❉ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ q ♠♦❞ r ≡ ( t − ✶ ) ♠♦❞ r ✐s ❛ ❦ t❤ ♣r✐♠✐t✐✈❡ r♦♦t ♦❢ ✉♥✐t② ζ ❦ ∈ F r ❀ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❦ | ( r − ✶ ) ✳ − ❉ ♠♦❞ r ✐s ❛ sq✉❛r❡ ✐♥ F r ✳ √ − ❉ , ✇❤❡r❡ ✹ q − t ✷ = ❉② ✷ . ② ♠♦❞ r = ( ζ ❦ − ✶ ) / ❘✳ ❉r②➟♦ ✭✮ ❈♦♥str✉❝t✐♥❣ P❛✐r✐♥❣✲❢r✐❡♥❞❧② ❈✉r✈❡s ✼ ✴ ✷✼

❚❤❡ ❈♦❝❦s✲P✐♥❝❤ ▼❡t❤♦❞ ■♥♣✉t✿ ❦ ✱ ❛ ♣r✐♠❡ ♥✉♠❜❡r r s✉❝❤ t❤❛t ❦ | ( r − ✶ ) ✱ ❛♥❞ ❛ ❞✐s❝r✐♠✐♥❛♥t ❉ > ✵ s✉❝❤ t❤❛t − ❉ ♠♦❞ r ✐s ❛ sq✉❛r❡ ✐♥ F r ✳ ❖✉t♣✉t✿ P❛r❛♠❡t❡rs ( r , t , q ) ♦❢ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ✇✐t❤ ❞✐s❝r✐♠✐♥❛♥t ❉ ❛♥❞ ❡♠❜❡❞❞✐♥❣ ❦ ✇✐t❤ r❡s♣❡❝t t♦ r ✳ ❚❛❦❡ ❦ t❤ ♣r✐♠✐t✐✈❡ r♦♦t ♦❢ ✉♥✐t② ζ ❦ ∈ F r ✳ √ ▲❡t t , ② ∈ Z ❜❡ ❧✐❢ts ♦❢ ζ ❦ + ✶ ❛♥❞ ( ζ ❦ − ✶ ) / − ❉ ✱ r❡s♣❡❝t✐✈❡❧②✳ ▲❡t q = ( t ✷ + ❉② ✷ ) / ✹✳ ■❢ q ✐s ♣r✐♠❡✱ r❡t✉r♥ ( r , t , q ) ✳ ❉❡❢✳ ❋♦r ♣❛r❛♠❡t❡rs ( r , t , q ) ♦❢ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❊ ✇❡ ❞❡✜♥❡ ♣❛r❛♠❡t❡r ρ := ❧♦❣ q ❧♦❣ r ≈ ❧♦❣ # ❊ ( F q ) . ❧♦❣ r ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♦❜t❛✐♥ ρ ❝❧♦s❡ t♦ ✶✳ ❚❤❡ ♠❛✐♥ ❞r❛✇❜❛❝❦ ♦❢ t❤❡ ❈♦❝❦s✲P✐♥❝❤ ♠❡t❤♦❞ ✐s t❤❛t ❣❡♥❡r✐❝❛❧❧② ✇❡ ❤❛✈❡ ρ ≈ ✷✱ s✐♥❝❡ ✉s✉❛❧❧② t , ② ❛r❡ ♦❢ t❤❡ s✐♠✐❧❛r s✐③❡ ❛s r ✳ ❘✳ ❉r②➟♦ ✭✮ ❈♦♥str✉❝t✐♥❣ P❛✐r✐♥❣✲❢r✐❡♥❞❧② ❈✉r✈❡s ✽ ✴ ✷✼

❚♦ ❝♦♥str✉❝t ❡❧❧✐♣t✐❝ ❝✉r✈❡s ✇✐t❤ ρ < ✷✱ ♦♥❡ ♦❜t❛✐♥s ♣❛r❛♠❡t❡rs ( r , t , q ) ❛s ✈❛❧✉❡s ♦❢ ❝❡rt❛✐♥ ♣♦❧②♥♦♠✐❛❧s r ( ① ) , t ( ① ) , q ( ① ) ∈ Q [ ① ] ✳ ❚❤❡ ♠❛✐♥ t❤❡♦r❡t✐❝❛❧ ♣r♦❜❧❡♠ ✐s ✇❤❡♥ ❛ ♣♦❧②♥♦♠✐❛❧ q ( ① ) ∈ Q [ ① ] t❛❦❡s ✐♥✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s ✈❛❧✉❡s ❢♦r ① ∈ Z ✳ ❚❤❡ ❇✉♥✐❛❦♦✇s❦✐✲❙❝❤✐♥③❡❧ ❈♦♥❥❡❝t✉r❡✳ ❆ ♣♦❧②♥♦♠✐❛❧ q ( ① ) ∈ Q [ ① ] t❛❦❡s ✐♥✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡ ✈❛❧✉❡s ❢♦r ① ∈ Z ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✭✐✮ q ( ① ) ✐s ✐rr❡❞✉❝✐❜❧❡ ❛♥❞ ❤❛s ♣♦s✐t✐✈❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥t✱ ✭✐✐✮ t❤❡ s❡t ❙ = { ❢ ( ① ) | ① , ❢ ( ① ) ∈ Z } ✐s ♥♦♥✲❡♠♣t② ❛♥❞ ❣❝❞ ( ❙ ) = ✶✳ ❲❡ s❛② t❤❛t q ( ① ) r❡♣r❡s❡♥ts ♣r✐♠❡s ✐❢ ✐t s❛t✐s✜❡s t❤❡ ❛❜♦✈❡ t✇♦ ❝♦♥❞✐t✐♦♥s✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ s❛② t❤❛t ❛ ♣♦❧②♥♦♠✐❛❧ r ( ① ) ∈ Q [ ① ] ✐s ✐♥t❡❣❡r ✈❛❧✉❡❞ ✐❢ r ( ① ) ∈ Z ❢♦r ❛❧❧ ① ∈ Z ✳ ❘✳ ❉r②➟♦ ✭✮ ❈♦♥str✉❝t✐♥❣ P❛✐r✐♥❣✲❢r✐❡♥❞❧② ❈✉r✈❡s ✾ ✴ ✷✼