●r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳ ❆❧❢ ❖♥s❤✉✉s ▼❛r❝❤ ✷✻ ✷✵✶✽ ❆❧❢ ❖♥s❤✉✉s ●r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳ ▼❛r❝❤ ✷✻ ✷✵✶✽ ✶ ✴ ✷✻
❈♦♥t❡①t ●❡♦♠❡tr✐❝ ❋✐❡❧❞s • ❚❤r♦✉❣❤♦✉t t❤✐s t❛❧❦✱ ✇❡ ✇✐❧❧ ❜❡ ✇♦r❦✐♥❣ ✐♥ ❛ t❤❡♦r② T ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ❝♦♥t❛✐♥✐♥❣ t❤❡ t❤❡♦r② ♦❢ ✜❡❧❞s✳ • ❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t ♠♦❞❡❧s ♦❢ T ❛r❡ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳ ❚❤✐s ✐s ✜❡❧❞s ✇❤✐❝❤ ❛r❡✿ F ✐s ❛❧❣❡❜r❛✐❝❛❧❧② ❜♦✉♥❞❡❞✳ F ✐s ❞❡✜♥❛❜❧② ❝❧♦s❡❞ ✐♥ ✐ts ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡ ✭s♦ ♣❡r❢❡❝t✮✳ ❆❧❢ ❖♥s❤✉✉s ●r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳ ▼❛r❝❤ ✷✻ ✷✵✶✽ ✷ ✴ ✷✻
❈♦♥t❡①t ❆❧❣❡❜r❛✐❝❛❧❧② ❜♦✉♥❞❡❞ ❉❡✜♥✐t✐♦♥ ❆ t❤❡♦r② ✐s ❛❧❣❡❜r❛✐❝❛❧❧② ❜♦✉♥❞❡❞ ✐❢✱ ●✐✈❡♥ ❛♥② ❢♦r♠✉❧❛ φ (¯ x , y ) ✱ t❤❡r❡ ❛r❡ ♣♦❧②♥♦♠✐❛❧s f ✶ (¯ x , y ) , . . . , f n (¯ x , y ) ∈ Z [¯ x , y ] ❙✉❝❤ t❤❛t ✐♥ ❛♥② ♠♦❞❡❧ K ♦❢ T ❛♥❞ ❛♥② ¯ a ❛ t✉♣❧❡ ♦❢ ❡❧❡♠❡♥ts ♦❢ K s✉❝❤ t❤❛t φ (¯ a , K ) := { y ∈ K : φ (¯ a , y ) } ✐s ✜♥✐t❡✱ t❤❡♥ t❤❡r❡ ✐s ❛♥ ✐♥❞❡① i s✉❝❤ t❤❛t t❤❡ ♣♦❧②♥♦♠✐❛❧ f i (¯ a , y ) ✐s ♥♦t ✐❞❡♥t✐❝❛❧❧② ✵ ♦♥ K ❛♥❞ φ (¯ a , K ) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ s❡t ♦❢ r♦♦ts ♦❢ f i (¯ a , y ) = ✵ ✳ ❆❧❢ ❖♥s❤✉✉s ●r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳ ▼❛r❝❤ ✷✻ ✷✵✶✽ ✸ ✴ ✷✻
❈♦♥t❡①t ❊①❛♠♣❧❡s ♦❢ ❣❡♦♠❡tr✐❝ ✜❡❧❞s ✐♥❝❧✉❞❡✿ ❘❡❛❧ ❝❧♦s❡❞ ✜❡❧❞s✳ ♣✲❛❞✐❝❛❧❧② ❝❧♦s❡❞ ✜❡❧❞s✳ Ps❡✉❞♦✜♥✐t❡ ✜❡❧❞s✳ ❇♦✉♥❞❡❞ Ps❡✉❞♦✲❛❧❣❡❜r❛✐❝❛❧❧② ❝❧♦s❡❞ ✜❡❧❞s✳ ❇♦✉♥❞❡❞ Ps❡✉❞♦✲r❡❛❧ ❝❧♦s❡❞ ✜❡❧❞s✳ ❇♦✉♥❞❡❞ Ps❡✉❞♦✲♣✲❛❞✐❝❛❧❧② ❝❧♦s❡❞ ✜❡❧❞s✳ ❆❧❢ ❖♥s❤✉✉s ●r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳ ▼❛r❝❤ ✷✻ ✷✵✶✽ ✹ ✴ ✷✻
❈♦♥t❡①t ❖✉r st❛rt✐♥❣ ♣♦✐♥t ❋❛❝t ❬❍r✉s❤♦✈s❦✐✲P✐❧❧❛②❪ ▲❡t G ❜❡ ❛ ❣r♦✉♣ ❞❡✜♥❛❜❧❡ ✐♥ T ♦✈❡r ❛ s❡t A ✱ ❛♥❞ ❧❡t a , b , c ∈ G ( U ) ❜❡ ❞✐♠❡♥s✐♦♥ ❣❡♥❡r✐❝ ❡❧❡♠❡♥ts s✉❝❤ t❤❛t a · G b = c ✱ ✇✐t❤ dim ( atp ( a / A ) = dim ( b / A ) = dim ( G ) ❛♥❞ s✉❝❤ t❤❛t a ❛♥❞ b ❛r❡ ❛❧❣❡❜r❛✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t ♦✈❡r A ✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ s❡t B ❝♦♥t❛✐♥✐♥❣ A s✉❝❤ t❤❛t a ❛♥❞ b ❛r❡ st✐❧❧ ❛❧❣❡❜r❛✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t ♦✈❡r B ✱ ❛ B ✲❞❡✜♥❛❜❧❡ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ H ❛♥❞ ❞✐♠❡♥s✐♦♥✲❣❡♥❡r✐❝ ❡❧❡♠❡♥ts a ′ , b ′ , c ′ ∈ H ( U ) s✉❝❤ t❤❛t a ′ · b ′ = c ′ ❛♥❞ acl ( Ba ) = acl ( Ba ′ ) ✱ acl ( Bb ) = acl ( Bb ′ ) ❛♥❞ acl ( Bc ) = acl ( Bc ′ ) ✳ ❆❧❢ ❖♥s❤✉✉s ●r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳ ▼❛r❝❤ ✷✻ ✷✵✶✽ ✺ ✴ ✷✻
❈♦♥t❡①t ❍♦✇ ❝❧♦s❡ ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ G ❛♥❞ H ❄ ✭❍r✉s❤♦✈s❦✐✲P✐❧❧❛②✮ ✶ ❲❤❡♥ ♦♥❡ ❤❛s ❛ t♦♣♦❧♦❣②✿ ■❢ F ✐s ❛ r❡❛❧ ❝❧♦s❡❞ ♦r ♣✲❛❞✐❝❛❧❧② ❝❧♦s❡❞ ✜❡❧❞✱ ♦♥❡ ❝❛♥ ✉s❡ t❤❡ t♦♣♦❧♦❣② ♦♥ t❤❡ ✜❡❧❞ t♦ ✜♥❞ ❧♦❝❛❧ ✐s♦♠♦r♣❤✐s♠ f ❜❡t✇❡❡♥ ♥❡✐❣❤❜♦r❤♦♦❞s ♦❢ t❤❡ ✐❞❡♥t✐t② a − ✶ U ❛♥❞ a ′− ✶ U ′ ✇❤❡r❡ U ✐s ❛ ✭t✲t♦♣♦❧♦❣②✮ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ a ❛♥❞ U ′ ✐s ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ H ( F ) ✳ ✷ ■♥ t❤❡ ♣s❡✉❞♦ ✜♥✐t❡ ✜❡❧❞ ❝❛s❡✿ ❯s❡ st❛❜✐❧✐③❡r ♦❢ tp ( b , b ′ / B ) ✐♥ t❤❡ ❣r♦✉♣ G × H t♦ ✜♥❞ ❛ t②♣❡ ❞❡✜♥❛❜❧❡ s✉❜❣r♦✉♣ ♦❢ G × H ✇✐t❤ ❧❛r❣❡ ♣r♦❥❡❝t✐♦♥s ❛♥❞ ✜♥✐t❡ ✜❜❡rs ♦♥ ❜♦t❤ G ❛♥❞ H ✳ ❖❜s❡r✈❛t✐♦♥ ■❢ F ✐s R ♦r Q p ❛♥❞ G ✐s ❛ ◆❛s❤ ❣r♦✉♣ t❤❡♥ t❤❡ ❧♦❝❛❧ ✐s♦♠♦r♣❤✐s♠ ✐s ❛ ◆❛s❤ ✐s♦♠♦r♣❤✐s♠✳ ■❢ F = R ♦♥❡ ❝❛♥ t❤❡♥ t❛❦❡ t❤❡ ◆❛s❤ ❝❧♦s✉r❡ ♦❢ U × f ( U ) ✐♥ G × H ❛♥❞ ❣❡t ❛ ◆❛s❤ ✐s♦❣❡♥② ❜❡t✇❡❡♥ G ❛♥❞ H ✳ ❆❧❢ ❖♥s❤✉✉s ●r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳ ▼❛r❝❤ ✷✻ ✷✵✶✽ ✻ ✴ ✷✻
❈♦♥t❡①t ❇❛rr✐❣❛ ✉s❡❞ t❤❡ ✜rst str❛t❡❣② t♦ ❛❞❛♣t t❤❡ ♣r♦♦❢ ✐♥ t❤❡ r❡❛❧ ❝❛s❡ t♦ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✿ ❚❤❡♦r❡♠ ✭❇❛rr✐❣❛✮ ■❢ G ✐s ❜♦✉♥❞❡❞ ✭✐♥ t❤❡ ♦r❞❡r t♦♣♦❧♦❣②✮ ❣r♦✉♣ ❞❡✜♥❛❜❧❡ ✐♥ ❛ r❡❛❧ ❝❧♦s❡❞ ✜❡❧❞ F t❤❡♥ t❤❡r❡ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ H ❛♥❞ ❛ ❧♦❝❛❧ ❤♦♠♦♠♦r♣❤✐s♠ f ❜❡t✇❡❡♥ ❛ ❣❡♥❡r✐❝ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ t❤❡ ✐❞❡♥t✐t② ✐♥ G ❛♥❞ ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ H ( F ) ✳ ❙❤❡ t❤❡♥ ✉s❡❞ t❤✐s t♦ ❝❧❛ss✐❢② ❛❧❧ ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧ ❣r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ r❡❛❧ ❝❧♦s❡❞ ✜❡❧❞s ✐♥ t❡r♠s ♦❢ ❞❡✜♥❛❜❧❡ s✉❜s❡ts ♦❢ ✉♥✐✈❡rs❛❧ ❝♦✈❡rs ♦❢ ✭♣♦ss✐❜❧② ✐♥✜♥✐t❡s✐♠❛❧✮ ♥❡✐❣❤❜♦r❤♦♦❞s ♦❢ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣s✳ ❆❧❢ ❖♥s❤✉✉s ●r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳ ▼❛r❝❤ ✷✻ ✷✵✶✽ ✼ ✴ ✷✻
❚❤❡ ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠ ❲❤✐❧❡ ✇♦r❦✐♥❣ ✐♥ ❛♥❛❧♦❣✉❡s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❜♦✉♥❞❡❞ P❘❈ ✜❡❧❞s ✇❡ ✇❡♥t ❜❛❝❦ t♦ t❤❡ st❛❜✐❧✐③❡r str❛t❡❣②✳ ❉❡✜♥✐t✐♦♥ ✭❙t❛❜✐❧✐③❡r ♦❢ tp ( b , b ′ / B ) ✱ ✐♥ ❜♦✉♥❞❡❞ P❆❈✮ ▲❡t q ( x , x ′ ) ❜❡ ❛ t②♣❡ ✐♥ G × H ✱ ❞❡✜♥❡ St ( q ) : { ( x , x ′ ) | ( x , x ′ ) · q ∪ q } ✐s ❛ ♥♦♥ ❢♦r❦✐♥❣ ❡①t❡♥s✐♦♥ ♦❢ q ✳ Stab ( q ) ✇✐❧❧ ❜❡ t❤❡ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② St ( q ) ■❢ q := tp ( b , b ′ / B ) t❤❡♥ Stab ( q ) = St ( q ) St ( q ) ✐s ❛ t②♣❡ ❞❡✜♥❛❜❧❡ s✉❜❣r♦✉♣ ♦❢ G × H ✱ ✇❤✐❝❤ ♠❛♣s ✇✐t❤ ✜♥✐t❡ ✜❜❡rs ♦♥t♦ t②♣❡ ❞❡✜♥❛❜❧❡ s✉❜❣r♦✉♣s ♦❢ ❜♦✉♥❞❡❞ ✐♥❞❡① ♦❢ G ❛♥❞ H ✱ r❡s♣❡❝t✐✈❡❧②✳ ❆❧❢ ❖♥s❤✉✉s ●r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳ ▼❛r❝❤ ✷✻ ✷✵✶✽ ✽ ✴ ✷✻
❚❤❡ ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠ ❖♥❡ ♥❡❡❞s t❤❡ ❙✶ ♣r♦♣❡rt② ♦♥ t❤❡ ❢♦r❦✐♥❣ ✐❞❡❛❧✿ ❉❡✜♥✐t✐♦♥ ▲❡t µ ❜❡ ❛♥ A ✲✐♥✈❛r✐❛♥t ✐❞❡❛❧ ♦♥ ❞❡✜♥❛❜❧❡ s❡ts✳ ❚❤❡♥ µ ❤❛s t❤❡ ❙✶ ♣r♦♣❡rt② ✐❢ ❣✐✈❡♥ ❛♥② φ ( x , a ✵ ) ❛♥❞ φ ( x , a ✶ ) ✱ ❢♦r a ✵ ❛♥❞ a ✶ st❛rt✐♥❣ ❛♥ A ✲✐♥❞✐s❝❡r♥✐❜❧❡ s❡q✉❡♥❝❡✱ φ ( x , a ✵ ) ∧ φ ( x , a ✶ ) ∈ µ ⇒ φ ( x , a ✵ ) ∈ µ. ❆❧❢ ❖♥s❤✉✉s ●r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳ ▼❛r❝❤ ✷✻ ✷✵✶✽ ✾ ✴ ✷✻
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