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❖✉t❧✐♥❡ ▼♦t✐✈❛t✐♦♥s ❚❤❡ ❝❛s❡ ❢♦r ♠❛tr✐① ♠♦❞❡❧s ❚❤❡ ❝❛s❡ ❢♦r t❡♥s♦r ♠♦❞❡❧ ❊①t❡♥s✐♦♥❄ ❈♦♥❝❧✉s✐♦♥ ❇❧♦❜❜❡❞ ❚♦♣♦❧♦❣✐❝❛❧ ❘❡❝✉rs✐♦♥ ❢♦r ❚❡♥s♦r ▼♦❞❡❧s ◗✉❛♥t✉♠ ●r❛✈✐t② ✐♥ P❛r✐s ❙té♣❤❛♥❡ ❉❛rt♦✐s ■♥st✐t✉t ❍❡♥r✐ P♦✐♥❝❛ré✱ P❛r✐s ✷✵✶✼ ❙té♣❤❛♥❡ ❉❛rt♦✐s ❇❧♦❜❜❡❞ ❚♦♣♦❧♦❣✐❝❛❧ ❘❡❝✉rs✐♦♥ ❢♦r ❚❡♥s♦r ▼♦❞❡❧s

❖✉t❧✐♥❡ ▼♦t✐✈❛t✐♦♥s ❚❤❡ ❝❛s❡ ❢♦r ♠❛tr✐① ♠♦❞❡❧s ❚❤❡ ❝❛s❡ ❢♦r t❡♥s♦r ♠♦❞❡❧ ❊①t❡♥s✐♦♥❄ ❈♦♥❝❧✉s✐♦♥ ❖✉t❧✐♥❡ ✶ ▼♦t✐✈❛t✐♦♥s ✷ ❚❤❡ ❝❛s❡ ❢♦r ♠❛tr✐① ♠♦❞❡❧s ✸ ❚❤❡ ❝❛s❡ ❢♦r ❛ t❡♥s♦r ♠♦❞❡❧ ✹ ❊①t❡♥s✐♦♥❄ ✺ ❈♦♥❝❧✉s✐♦♥ ❙té♣❤❛♥❡ ❉❛rt♦✐s ❇❧♦❜❜❡❞ ❚♦♣♦❧♦❣✐❝❛❧ ❘❡❝✉rs✐♦♥ ❢♦r ❚❡♥s♦r ▼♦❞❡❧s

❖✉t❧✐♥❡ ▼♦t✐✈❛t✐♦♥s ❚❤❡ ❝❛s❡ ❢♦r ♠❛tr✐① ♠♦❞❡❧s ❚❤❡ ❝❛s❡ ❢♦r t❡♥s♦r ♠♦❞❡❧ ❊①t❡♥s✐♦♥❄ ❈♦♥❝❧✉s✐♦♥ ▼❛tr✐① ♠♦❞❡❧s ❢♦r ✷❞ q✉❛♥t✉♠ ❣r❛✈✐t② ■♥t❡❣r❛❧s ♦❢ ♠❛tr✐❝❡s ✇✐t❤ ❋❡②♥♠❛♥ ❣r❛♣❤s ❂ ♣♦❧②✲❛♥❣✉❧❛t✐♦♥s ♦❢ s✉r❢❛❝❡s✳ p = 4 p = 2 p = 3 p = 5 ❋✐❣✉r❡✿ ❊①❛♠♣❧❡ ♦❢ ❛ ♣♦❧②✲❛♥❣✉❧❛t✐♦♥ ❛♥❞ ✐ts ❞✉❛❧ ❋❡②♥♠❛♥ ●r❛♣❤✳ ❙té♣❤❛♥❡ ❉❛rt♦✐s ❇❧♦❜❜❡❞ ❚♦♣♦❧♦❣✐❝❛❧ ❘❡❝✉rs✐♦♥ ❢♦r ❚❡♥s♦r ▼♦❞❡❧s

❖✉t❧✐♥❡ ▼♦t✐✈❛t✐♦♥s ❚❤❡ ❝❛s❡ ❢♦r ♠❛tr✐① ♠♦❞❡❧s ❚❤❡ ❝❛s❡ ❢♦r t❡♥s♦r ♠♦❞❡❧ ❊①t❡♥s✐♦♥❄ ❈♦♥❝❧✉s✐♦♥ ❖❜s❡r✈❛❜❧❡s ♦❢ t❤❡ ♠♦❞❡❧✿ { O p = tr ( M p ) | p ∈ N } ❘❡♣r❡s❡♥t ❜♦✉♥❞❛r② st❛t❡s ♦❢ t❤❡ tr✐❛♥❣✉❧❛t❡❞ s✉r❢❛❝❡s✳ ⇒ ♠✉❧t✐♣❧❡ ❜♦✉♥❞❛r② ❝♦♠♣♦♥❡♥ts✳ Pr♦❞✉❝t ♦❢ O p ✏❚r❛♥s✐t✐♦♥ ❛♠♣❧✐t✉❞❡s✑ ❂ ♥✉♠❜❡rs ♦❢ tr✐❛♥❣✉❧❛t✐♦♥s ♦❢ s✉r❢❛❝❡s ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ ❜♦✉♥❞❛r✐❡s✳ ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s❄ ❚❡♥s♦r ♠♦❞❡❧s ✳ ●❡♥❡r❛❧✐③❡ t❤❡ t❡❝❤♥✐q✉❡s ♦❢ ♠❛tr✐① ♠♦❞❡❧s t♦ t❡♥s♦r ♠♦❞❡❧s✳ ❙té♣❤❛♥❡ ❉❛rt♦✐s ❇❧♦❜❜❡❞ ❚♦♣♦❧♦❣✐❝❛❧ ❘❡❝✉rs✐♦♥ ❢♦r ❚❡♥s♦r ▼♦❞❡❧s

❖✉t❧✐♥❡ ▼♦t✐✈❛t✐♦♥s ❚❤❡ ❝❛s❡ ❢♦r ♠❛tr✐① ♠♦❞❡❧s ❚❤❡ ❝❛s❡ ❢♦r t❡♥s♦r ♠♦❞❡❧ ❊①t❡♥s✐♦♥❄ ❈♦♥❝❧✉s✐♦♥ ❙❝❤✇✐♥❣❡r✲❉②s♦♥ ❊q✉❛t✐♦♥s ❖♥❡ ▼❛tr✐① ♠♦❞❡❧✿ � Z = dM ❡①♣ ( − N ❚r V ( M )) , V ( M ) = ♣♦❧②♥♦♠✐❛❧ ♣♦t❡♥t✐❛❧ � p max p = ✷ t p x p ✳ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ♠❛tr✐① ✐♥t❡❣r❛❧ ✐♠♣❧✐❡s k ✶ − ✶ n � � ❚r ( M k ✶ − ✶ − m ) ❚r ( M m ) � ❚r ( M k i ) � m = ✵ i = ✷ n n � k j � ❚r ( M k j + k ✶ − ✶ ) � ❚r ( M k i ) � + j = ✷ i = ✷ , i � = j n − N � ❚r ( M k ✶ V ′ ( M )) � ❚r ( M k i ) � = ✵ i = ✷ ❙té♣❤❛♥❡ ❉❛rt♦✐s ❇❧♦❜❜❡❞ ❚♦♣♦❧♦❣✐❝❛❧ ❘❡❝✉rs✐♦♥ ❢♦r ❚❡♥s♦r ▼♦❞❡❧s

❖✉t❧✐♥❡ ▼♦t✐✈❛t✐♦♥s ❚❤❡ ❝❛s❡ ❢♦r ♠❛tr✐① ♠♦❞❡❧s ❚❤❡ ❝❛s❡ ❢♦r t❡♥s♦r ♠♦❞❡❧ ❊①t❡♥s✐♦♥❄ ❈♦♥❝❧✉s✐♦♥ ▲♦♦♣ ❊q✉❛t✐♦♥s ■♥ t❤❡ ♣❡rt✉r❜❛t✐✈❡ ✭❢♦r♠❛❧✮ s❡tt✐♥❣✿ n n � � � ❚r ( M k i ) � c = N ✷ − ✷ g − n � ❚r ( M k i ) � g � c . i = ✶ g ≥ ✵ i = ✶ ❈✉♠✉❧❛♥ts ❤❛✈❡ ❛ ✶ / N ❡①♣❛♥s✐♦♥✳ ❋r♦♠ ❝✉♠✉❧❛♥ts ❞❡✜♥❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ♦❢ ♦❜s❡r✈❛❜❧❡s✿ i tr ( M p i ) � g � � c � W g ∀ ( g , n ) s✳t✳ ✷ g − ✷ + n ≥ − ✷ , n ( x ✶ , . . . , x n ) = i x p i + ✶ � p i ≥ ✵ i ❙té♣❤❛♥❡ ❉❛rt♦✐s ❇❧♦❜❜❡❞ ❚♦♣♦❧♦❣✐❝❛❧ ❘❡❝✉rs✐♦♥ ❢♦r ❚❡♥s♦r ▼♦❞❡❧s

❖✉t❧✐♥❡ ▼♦t✐✈❛t✐♦♥s ❚❤❡ ❝❛s❡ ❢♦r ♠❛tr✐① ♠♦❞❡❧s ❚❤❡ ❝❛s❡ ❢♦r t❡♥s♦r ♠♦❞❡❧ ❊①t❡♥s✐♦♥❄ ❈♦♥❝❧✉s✐♦♥ ❙✲❉ ❊q✉❛t✐♦♥s ❜❡❝♦♠❡ ▲♦♦♣ ❡q✉❛t✐♦♥s✿ W g − ✶ ✶ + | J | ( x , x J ) W g − h � W h n + ✶ ( x , x , x I ) + ✶ + | I − J | ( x , x | I − J | ) ✵ ≤ h ≤ g J ⊆ I W g x i , . . . , x n ) − W g n − ✶ ( x , x ✷ , . . . , ˆ n − ✶ ( x ✷ , . . . , x n ) ∂ � + ( x − x i ) ✷ ∂ x i i ∈ I − V ′ ( x ) W g n ( x , x I ) + P g n ( x ; x ✷ , . . . , x n ) = ✵ . P g n ( x ; x ✷ , . . . , x n ) ❛ ♣♦❧②♥♦♠✐❛❧ ✐♥ x ✳ ❙té♣❤❛♥❡ ❉❛rt♦✐s ❇❧♦❜❜❡❞ ❚♦♣♦❧♦❣✐❝❛❧ ❘❡❝✉rs✐♦♥ ❢♦r ❚❡♥s♦r ▼♦❞❡❧s

❖✉t❧✐♥❡ ▼♦t✐✈❛t✐♦♥s ❚❤❡ ❝❛s❡ ❢♦r ♠❛tr✐① ♠♦❞❡❧s ❚❤❡ ❝❛s❡ ❢♦r t❡♥s♦r ♠♦❞❡❧ ❊①t❡♥s✐♦♥❄ ❈♦♥❝❧✉s✐♦♥ ❙♦❧✉t✐♦♥ ♦❢ t❤❡ ❧♦♦♣ ❡q✉❛t✐♦♥s ❈♦♠♣✉t❡ W ✵ ✶ ❛♥❞ W ✵ ✷ ✳ ✶ ( x ) = ✶ � V ′ ( x ) ✷ − ✹ P ✵ ✷ ( V ′ ( x ) − W ✵ ✶ ( x )) ❚❤❡ ❢♦r♠ ♦❢ W ✵ ✶ t❡❧❧s ✉s ❛❜♦✉t ❛ ❢✉♥❝t✐♦♥ ✭❝♦✈❡r✐♥❣✮ x : Σ → C \ � i γ i ✳ ■♥ t❤❡ ✶✲❝✉t ❝❛s❡✿ � V ′ ( x ) ✷ − ✹ P ✵ � ✶ ( x ) = M ( x ) ( x − a )( x − b ) M ( x ) ❛ ♣♦❧②♥♦♠✐❛❧ ♥♦t ✈❛♥✐s❤✐♥❣ ❛t a , b ✳ a , b ❂ r❛♠✐✜❝❛t✐♦♥ ♣♦✐♥ts✳ Σ ✐s ❛ s♣❤❡r❡✳ x ( z ) = a + b + a − b ✹ ( z + ✶ / z ) ✳ ❉❡✜♥❡ ✷ ω g n = W g n ( x ( z ✶ ) , . . . , x ( z n )) dx ✶ . . . dx n ✳ ❖♥❡ s❤♦✇s dz ✶ dz ✷ ω ✵ ✷ ( z ✶ , z ✷ ) = ( z ✶ − z ✷ ) ✷ . ❙té♣❤❛♥❡ ❉❛rt♦✐s ❇❧♦❜❜❡❞ ❚♦♣♦❧♦❣✐❝❛❧ ❘❡❝✉rs✐♦♥ ❢♦r ❚❡♥s♦r ▼♦❞❡❧s

❖✉t❧✐♥❡ ▼♦t✐✈❛t✐♦♥s ❚❤❡ ❝❛s❡ ❢♦r ♠❛tr✐① ♠♦❞❡❧s ❚❤❡ ❝❛s❡ ❢♦r t❡♥s♦r ♠♦❞❡❧ ❊①t❡♥s✐♦♥❄ ❈♦♥❝❧✉s✐♦♥ ❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❧♦♦♣ ❡q✉❛t✐♦♥s ❙♦❧✈❡ t❤❡ ❧♦♦♣ ❡q✉❛t✐♦♥s ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rr❡♥❝❡ ❢♦r♠✉❧❛✿ ω g − ✶ ω g � � n ( z ✶ , . . . , z n ) = z → p i K ( z , z ✶ ) n + ✶ ( z , ι ( z ) , z ✷ , . . . , z n ) Res p i ′ ✶ + | J | ( z , z J ) · ω g − h � ω h � + ✶ + | I − J | ( ι ( z ) , z I − J ) . ✵ ≤ h ≤ g J ⊆ I ✇✐t❤ x ◦ ι = x ✱ ι ( z ) = ✶ / z ✭ ι ❂❞❡❝❦ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❝♦✈❡r✮ ❛♥❞ � z ι ( z ) ω ✵ ✷ ( · , z ✶ ) K ( z , z ✶ ) = ✶ ω ✵ ✶ ( z ) − ω ✵ ✷ ✶ ( ι ( z )) ❙té♣❤❛♥❡ ❉❛rt♦✐s ❇❧♦❜❜❡❞ ❚♦♣♦❧♦❣✐❝❛❧ ❘❡❝✉rs✐♦♥ ❢♦r ❚❡♥s♦r ▼♦❞❡❧s

❖✉t❧✐♥❡ ▼♦t✐✈❛t✐♦♥s ❚❤❡ ❝❛s❡ ❢♦r ♠❛tr✐① ♠♦❞❡❧s ❚❤❡ ❝❛s❡ ❢♦r t❡♥s♦r ♠♦❞❡❧ ❊①t❡♥s✐♦♥❄ ❈♦♥❝❧✉s✐♦♥ ■❞❡❛ ♦❢ ♣r♦♦❢ ▲♦♦♣ ❊q✉❛t✐♦♥s ⇔ ▲✐♥❡❛r ▲♦♦♣ ❡q✉❛t✐♦♥s ✭▲▲❊✮ ✰ ◗✉❛❞r❛t✐❝ ▲♦♦♣ ❊q✉❛t✐♦♥s ✭◗▲❊✮✳ ▲▲❊✿ ■♥tr♦❞✉❝❡ S ✱ ❧✐♥❡❛r ♦♣❡r❛t♦r ❛❝ts ♦♥ ❢♦r♠s f ( z ) ❜② Sf ( z ) = f ( z ) + f ( ✶ / z ) ✳ S z ✶ ω g n ( z ✶ , . . . , z n ) = ✵ , ( g , n ) � = ( ✵ , ✶ ) , ( ✵ , ✷ ) . ❖❜t❛✐♥❡❞ ❢r♦♠ ❧♦♦♣ ❡q✉❛t✐♦♥s✳ ❙té♣❤❛♥❡ ❉❛rt♦✐s ❇❧♦❜❜❡❞ ❚♦♣♦❧♦❣✐❝❛❧ ❘❡❝✉rs✐♦♥ ❢♦r ❚❡♥s♦r ▼♦❞❡❧s