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■✈❛♥ ❆r③❤❛♥ts❡✈ ✭❍❙❊ ❯♥✐✈❡rs✐t②✱ ▼♦s❝♦✇✮ ■♥✜♥✐t❡ tr❛♥s✐t✐✈✐t②✱ ✜♥✐t❡ ❣❡♥❡r❛t✐♦♥✱ ❛♥❞ ❉❡♠❛③✉r❡ r♦♦ts ✇✐t❤ ❑❛r✐♥❡ ❑✉②✉♠③❤✐②❛♥ ❛♥❞ ▼✐❦❤❛✐❧ ❩❛✐❞❡♥❜❡r❣ ❆❞✈❛♥❝❡s ✐♥ ▼❛t❤❡♠❛t✐❝s ✸✺✶ ✭✷✵✶✾✮ ✶✕✸✷ ✶✹✳✵✺✳✷✵✷✵

■♥✜♥✐t❡ ❚r❛♥s✐t✐✈✐t② ❉❡✜♥✐t✐♦♥ ▲❡t G ❜❡ ❛ ❣r♦✉♣✱ X ❛ s❡t✱ ❛♥❞ m ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❆♥ ❛❝t✐♦♥ G × X → X ✐s ❝❛❧❧❡❞ m ✲tr❛♥s✐t✐✈❡ ✐❢ ❢♦r ❛♥② t✇♦ t✉♣❧❡s ( a ✶ , . . . , a m ) ❛♥❞ ( b ✶ , . . . , b m ) ♦❢ ♣❛✐r✇✐s❡ ❞✐st✐♥❝t ♣♦✐♥ts ♦♥ X t❤❡r❡ ✐s ❛♥ ❡❧❡♠❡♥t g ∈ G s✉❝❤ t❤❛t ( ga ✶ , . . . , ga m ) = ( b ✶ , . . . , b m ) ✳ ❉❡✜♥✐t✐♦♥ ❆♥ ❛❝t✐♦♥ G × X → X ✐s ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡ ✐❢ ✐t ✐s m ✲tr❛♥s✐t✐✈❡ ❢♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r m ✳ ❊①❛♠♣❧❡ ✶✮ ▲❡t X ❜❡ ❛♥ ✐♥✜♥✐t❡ s❡t ❛♥❞ G t❤❡ ❣r♦✉♣ ♦❢ ❛❧❧ ♣❡r♠✉t❛t✐♦♥s ♦♥ X ✳ ✷✮ ▲❡t X ❜❡ ❛♥ ✐♥✜♥✐t❡ s❡t ❛♥❞ G t❤❡ ❣r♦✉♣ ♦❢ ❛❧❧ ♣❡r♠✉t❛t✐♦♥s ✇✐t❤ ✜♥✐t❡ s✉♣♣♦rt ♦♥ X ✳

❆✣♥❡ ❙♣❛❝❡s ❚❤❡♦r❡♠ ❚❤❡ ❣r♦✉♣ Aut( A n ) ✐s ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡ ♦♥ A n ❢♦r ❛♥② n � ✷ ✳ ■❞❡❛ ✭ n = ✷✮✿ ✉s❡ ♣❛r❛❧❧❡❧ tr❛♥s❧❛t✐♦♥s ( x ✶ + a , x ✷ ) ✱ ( x ✶ , x ✷ + b ) ❛♥❞ t❤❡✐r r❡♣❧✐❝❛s ( x ✶ + af ✶ ( x ✷ ) , x ✷ ) ✱ ( x ✶ , x ✷ + bf ✷ ( x ✶ )) ✱ ✇❤❡r❡ a , b ∈ K ✳ ❊①❛♠♣❧❡ ❚❤❡ ❣r♦✉♣ Aut( A ✶ ) ✐s ✐s♦♠♦r♣❤✐❝ t♦ K × ⋌ K ✳ ■t ✐s ✷✲tr❛♥s✐t✐✈❡✱ ❜✉t ♥♦t ✸✲tr❛♥s✐t✐✈❡ ♦♥ A ✶ ✳

●❡♥❡r❛❧ Pr♦❜❧❡♠s ▲❡t X ❜❡ ❛♥ ❛✣♥❡ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t② ♦✈❡r t❤❡ ✜❡❧❞ C ✳ ❲❤❡♥ t❤❡ ❣r♦✉♣ Aut( X ) ♦❢ ♣♦❧②♥♦♠✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ X ✐s ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡ ♦♥ X ❄ ■❢ X ✐s s✐♥❣✉❧❛r✱ ✇❡ ❛s❦ t❤✐s q✉❡st✐♦♥ ❢♦r t❤❡ s♠♦♦t❤ ❧♦❝✉s X r❡❣ ✳ ■❞❡❛✿ t♦ ✉s❡ G a ✲s✉❜❣r♦✉♣s ✐♥ t❤❡ ❣r♦✉♣ Aut( X ) ❛♥❞ t❤❡✐r r❡♣❧✐❝❛s✳ ❍❡r❡ G a = ( C , +) ✳ ◆♦t❛t✐♦♥✿ SAut( X ) ✐s t❤❡ s✉❜❣r♦✉♣ ♦❢ Aut( X ) ❣❡♥❡r❛t❡❞ ❜② ❛❧❧ G a ✲s✉❜❣r♦✉♣s✳

▲♦❝❛❧❧② ◆✐❧♣♦t❡♥t ❉❡r✐✈❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ❆ ❞❡r✐✈❛t✐♦♥ D : A → A ♦❢ ❛♥ ❛❧❣❡❜r❛ A ✐s ❧♦❝❛❧❧② ♥✐❧♣♦t❡♥t ✐❢ ❢♦r ❛♥② a ∈ A t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r k s✉❝❤ t❤❛t D k ( a ) = ✵✳ ▲♦❝❛❧❧② ♥✐❧♣♦t❡♥t ❞❡r✐✈❛t✐♦♥s ♦♥ C [ X ] ⇔ G a ✲s✉❜❣r♦✉♣s ✐♥ Aut( X ) D ∈ LND( C [ X ]) ⇐ ⇒ exp( C D ) ⊆ Aut( X ) ■❢ D ∈ LND( A ) ❛♥❞ f ∈ Ker( D ) ✱ t❤❡♥ fD ∈ LND( A ) ✳ G a ✲s✉❜❣r♦✉♣s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ▲◆❉s ♦❢ t❤❡ ❢♦r♠ fD ❛r❡ r❡♣❧✐❝❛s ♦❢ t❤❡ G a ✲s✉❜❣r♦✉♣ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ D ✳

❋❧❡①✐❜✐❧✐t② ✈s ■♥✜♥✐t❡ ❚r❛♥s✐t✐✈✐t② ❉❡✜♥✐t✐♦♥ ❆♥ ❛✣♥❡ ✈❛r✐❡t② X ✐s ✢❡①✐❜❧❡ ✐❢ t❤❡ t❛♥❣❡♥t s♣❛❝❡ T x ( X ) ❛t ❛♥② s♠♦♦t❤ ♣♦✐♥t x ∈ X r❡❣ ✐s ❣❡♥❡r❛t❡❞ ❜② ✈❡❧♦❝✐t② ✈❡❝t♦rs t♦ ♦r❜✐ts ♦❢ G a ✲s✉❜❣r♦✉♣s ♣❛ss✐♥❣ t❤r♦✉❣❤ x ✳ ❚❤❡♦r❡♠ ✭❆✳✲❋❧❡♥♥❡r✲❑❛❧✐♠❛♥✲❑✉t③s❝❤❡❜❛✉❝❤✲❩❛✐❞❡♥❜❡r❣✬✷✵✶✸✮ ▲❡t X ❜❡ ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❛✣♥❡ ✈❛r✐❡t② ♦❢ ❞✐♠❡♥s✐♦♥ � ✷ ✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✭❛✮ t❤❡ ❣r♦✉♣ SAut( X ) ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ X r❡❣ ❀ ✭❜✮ t❤❡ ❣r♦✉♣ SAut( X ) ❛❝ts ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ♦♥ X r❡❣ ❀ ✭❝✮ t❤❡ ✈❛r✐❡t② X ✐s ✢❡①✐❜❧❡✳

❊①❛♠♣❧❡s ♦❢ ❋❧❡①✐❜❧❡ ❱❛r✐❡t✐❡s • ❙✉s♣❡♥s✐♦♥s ❙✉s♣ ( X , f ) ❣✐✈❡♥ ❜② uv = f ( x ) ✐♥ A ✷ × X ♦✈❡r ❛ ✢❡①✐❜❧❡ ✈❛r✐❡t② X ❀ • ◆♦♥✲❞❡❣❡♥❡r❛t❡ ✭ C [ X ] × = C × ✮ ❛✣♥❡ t♦r✐❝ ✈❛r✐❡t✐❡s❀ • ◆♦♥✲❞❡❣❡♥❡r❛t❡ ❤♦r♦s♣❤❡r✐❝❛❧ ✈❛r✐❡t✐❡s ♦❢ r❡❞✉❝t✐✈❡ ❣r♦✉♣s❀ • ❍♦♠♦❣❡♥❡♦✉s s♣❛❝❡s G / F ✱ ✇❤❡r❡ G ✐s s❡♠✐s✐♠♣❧❡ ❛♥❞ F ✐s r❡❞✉❝t✐✈❡❀ • ◆♦r♠❛❧ ❛✣♥❡ ❙▲ ( ✷ ) ✲❡♠❜❡❞❞✐♥❣s❀ • ❆✣♥❡ ❝♦♥❡s ♦✈❡r ✢❛❣ ✈❛r✐❡t✐❡s ❛♥❞ ❞❡❧ P❡③③♦ s✉r❢❛❝❡s✳

❘♦♦t ❙✉❜❣r♦✉♣s ❛♥❞ ❉❡♠❛③✉r❡ ❘♦♦ts ▲❡t X ❜❡ ❛ ✈❛r✐❡t② ✇✐t❤ ❛♥ ❛❝t✐♦♥ ♦❢ ❛ t♦r✉s T ✳ ❆ G a ✲s✉❜❣r♦✉♣ H ✐♥ Aut( X ) ✐s ❝❛❧❧❡❞ ❛ r♦♦t s✉❜❣r♦✉♣ ✐❢ H ✐s ♥♦r♠❛❧✐③❡❞ ✐♥ Aut( X ) ❜② t❤❡ t♦r✉s T ✳ ■♥ t❤✐s ❝❛s❡ T ❛❝ts ♦♥ H ❜② s♦♠❡ ❝❤❛r❛❝t❡r e ✳ ❙✉❝❤ ❛ ❝❤❛r❛❝t❡r ✐s ❝❛❧❧❡❞ ❛ r♦♦t ♦❢ t❤❡ T ✲✈❛r✐❡t② X ✳ ❆ss✉♠❡ X ✐s t♦r✐❝ ✇✐t❤ ❛❝t✐♥❣ t♦r✉s T ✳ ❲❤❛t ❛r❡ t❤❡ r♦♦ts ♦❢ X ❄ ▲❡t p ✶ , . . . , p s ❜❡ t❤❡ ♣r✐♠✐t✐✈❡ ❧❛tt✐❝❡ ✈❡❝t♦rs ♦♥ r❛②s ♦❢ t❤❡ ❢❛♥ Σ X ✳ ❉❡✜♥✐t✐♦♥ ❆ ❉❡♠❛③✉r❡ r♦♦t ♦❢ t❤❡ ❢❛♥ Σ X ✐♥ ❛ ❝❤❛r❛❝t❡r e ∈ M s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ✶ � i � s ✇✐t❤ � e , p i � = − ✶ ❛♥❞ � e , p j � � ✵ ❢♦r j � = i ✳ ❚❤❡♦r❡♠ ✭❉❡♠❛③✉r❡✬ ✶✾✼✵✮ ▲❡t X ❜❡ ❛ ❝♦♠♣❧❡t❡ ♦r ❛♥ ❛✣♥❡ t♦r✐❝ ✈❛r✐❡t✐❡s✳ ❚❤❡♥ r♦♦t s✉❜❣r♦✉♣s ♦♥ X ❛r❡ ✐♥ ❜✐❥❡❝t✐♦♥ ✇✐t❤ ❉❡♠❛③✉r❡ r♦♦ts ♦❢ t❤❡ ❢❛♥ Σ X ✳

❋✐♥✐t❡ ❣❡♥❡r❛t✐♦♥ ❈♦♥❥❡❝t✉r❡ ❆✳ ❆♥② ❣❡♥❡r✐❝❛❧❧② ✢❡①✐❜❧❡ ❛✣♥❡ ✈❛r✐❡t② X ❛❞♠✐ts ❛ ✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥ { H ✶ , . . . , H k } ♦❢ G a ✲s✉❜❣r♦✉♣s ♦❢ Aut( X ) s✉❝❤ t❤❛t t❤❡ ❣r♦✉♣ G = � H ✶ , . . . , H k � ❛❝ts ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ♦♥ ✐ts ♦♣❡♥ ♦r❜✐t✳ ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢✿ ❙t❡♣ ✶✳ ❋✐♥❞ G = � H ✶ , . . . , H s � t❤❛t ❛❝ts ♦♥ X ✇✐t❤ ❛♥ ♦♣❡♥ ♦r❜✐t❀ ❙t❡♣ ✷✳ Pr♦✈❡ t❤❛t t❤❡ ❝❧♦s✉r❡ G ♦❢ t❤❡ s✉❜❣r♦✉♣ G ✐♥ Aut( X ) ✐♥ ✐♥❞✲t♦♣♦❧♦❣② ❝♦♥t❛✐♥s ❵♠❛♥② ♦t❤❡r✬ G a ✲s✉❜❣r♦✉♣s❀ ❙t❡♣ ✸✳ Pr♦✈❡ t❤❛t G ❛❝ts ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ♦♣❡♥ ♦r❜✐t❀ ❙t❡♣ ✹✳ Pr♦✈❡ t❤❛t G ❛❝ts ✐♥✜♥✐t❡❧② tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ♦♣❡♥ ♦r❜✐t✳ ❙t❡♣ ✸ ⇒ ❙t❡♣ ✹ t✉r♥s ♦✉t t♦ ❜❡ tr✉❡ ✐♥ ❣❡♥❡r❛❧✳

❆ ❈♦♥❥❡❝t✉r❡ ♦♥ ▲♦❝❛❧❧② ◆✐❧♣♦t❡♥t ❉❡r✐✈❛t✐♦♥s ❚♦ ❙t❡♣ ✷✿ ❈♦♥❥❡❝t✉r❡ ❇✳ ▲❡t X ❜❡ ❛♥ ❛✣♥❡ ✈❛r✐❡t②✱ ❛♥❞ A = C [ X ] ❜❡ ✐ts str✉❝t✉r❡ ❛❧❣❡❜r❛✳ ❈♦♥s✐❞❡r t❤❡ ❣r♦✉♣ G = � H ✶ , . . . , H k � ❣❡♥❡r❛t❡❞ ❜② ❛ ✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ G a ✲s✉❜❣r♦✉♣s H i = exp( C D i ) ⊂ SAut( X ) ✱ ✇❤❡r❡ D i ∈ LND( A ) ✱ i = ✶ , . . . , k ✳ ❚❤❡♥ t❤❡ G a ✲s✉❜❣r♦✉♣ H = exp( C D ) ⊂ SAut( X ) ✱ ✇❤❡r❡ D ∈ LND( A ) ✱ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ G ✐❢ ❛♥❞ ♦♥❧② ✐❢ D ∈ ▲✐❡ � D ✶ , . . . , D k � ✳