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◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s ❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❆❧❡①❛♥❞❡r ❇♦r✐❝❤❡✈ ✭▼❛rs❡✐❧❧❡✮ ❛♥❞ ❑♦♥st❛♥t✐♥ ❋❡❞♦r♦✈s❦✐✐ ✭▼♦s❝♦✇✮ ❚♦♣✐❝s ✐♥ ●❡♦♠❡tr✐❝ ❋✉♥❝t✐♦♥ ❚❤❡♦r②✱ ✶✻ ❋❡❜r✉❛r② ✷✵✶✽ ❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ❉❡✜♥✐t✐♦♥ ❆ ❜♦✉♥❞❡❞ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥ G ⊂ C ✐s s❛✐❞ t♦ ❜❡ ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✱ ✐❢ t❤❡r❡ ❡①✐st ❢✉♥❝t✐♦♥s u , v ∈ H ∞ ( G ) s✉❝❤ t❤❛t t❤❡ ❡q✉❛❧✐t② z = u ( z ) v ( z ) , ❤♦❧❞s ♦♥ ∂ G ❛✳ ❡✳ ✐♥ s❡♥s❡ ♦❢ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣s✳ ❚❤❛t ♠❡❛♥s t❤❛t ϕ ( ζ ) = u ( ϕ ( ζ )) v ( ϕ ( ζ )) , ❛✳❡✳ ♦♥ T ❢♦r s♦♠❡✭❛♥②✮ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣ ϕ : D �→ G ✳ ◗✉❡st✐♦♥ ❍♦✇ ❧❛r❣❡ ❝❛♥ ❜❡ ✭❛❝❝❡ss✐❜❧❡✮ ❜♦✉♥❞❛r② ♦❢ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥❄ ❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ❉❡✜♥✐t✐♦♥ ❆ ❜♦✉♥❞❡❞ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥ G ⊂ C ✐s s❛✐❞ t♦ ❜❡ ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✱ ✐❢ t❤❡r❡ ❡①✐st ❢✉♥❝t✐♦♥s u , v ∈ H ∞ ( G ) s✉❝❤ t❤❛t t❤❡ ❡q✉❛❧✐t② z = u ( z ) v ( z ) , ❤♦❧❞s ♦♥ ∂ G ❛✳ ❡✳ ✐♥ s❡♥s❡ ♦❢ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣s✳ ❚❤❛t ♠❡❛♥s t❤❛t ϕ ( ζ ) = u ( ϕ ( ζ )) v ( ϕ ( ζ )) , ❛✳❡✳ ♦♥ T ❢♦r s♦♠❡✭❛♥②✮ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣ ϕ : D �→ G ✳ ◗✉❡st✐♦♥ ❍♦✇ ❧❛r❣❡ ❝❛♥ ❜❡ ✭❛❝❝❡ss✐❜❧❡✮ ❜♦✉♥❞❛r② ♦❢ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥❄ ❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

❊①❛♠♣❧❡s ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥s✿ ✉♥✐t ❞✐s❦ D ✱ ◆❡✉♠❛♥♥✬s ♦✈❛❧ ✭✐♠❛❣❡ ♦❢ ❡❧❧✐♣s❡ ✇✐t❤ ❝❡♥t❡r ❛t ♦r✐❣✐♥ ✉♥❞❡r z �→ 1 z ✮✳ ◆♦♥✲ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥s✿ ❡❧❧✐♣s❡✱ ♣♦❧②❣♦♥✳ z = x + iy : x 2 a 2 + y 2 � � G a , b = b 2 < 1 , b < a . ❇♦✉♥❞❛r② ♦❢ G a , b √ S a , b ( z ) = ( a 2 + b 2 ) z − 2 ab z 2 − c 2 z = S a , b ( z ) , , c 2 � a 2 − b 2 . c = ❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

▼♦t✐✈❛t✐♦♥s P♦❧②❛♥❛❧②t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ◗✉❛❞r❛t✉tr❡ ❞♦♠❛✐♥s✳ ❙❝❤✇❛r③ ❢✉♥❝t✐♦♥s❀ ❯♥✐✈❛❧❡♥t ❢✉♥❝t✐♦♥s ✐♥ ♠♦❞❡❧ s✉❜s♣❛❝❡s ♦❢ ❍❛r❞② s♣❛❝❡ ❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

P♦❧②❛♥❛❧②t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ❲❡ ✇✐❧❧ s❛② t❤❛t ❢✉♥❝t✐♦♥ f ✐s ♥✲❛♥❛❧②t✐❝ ✐❢ f ( z ) = z n − 1 f n − 1 ( z ) + ... + zf 1 ( z ) + f 0 ( z ) , ✇❤❡r❡ f k ❛r❡ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✳ X ✲ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ C A n ( X ) := { f ∈ C ( X ) : f − ♥✲❛♥❛❧②t✐❝ ✐♥ Int X } , P n ( X ) = Clos C ( X ) { P : P − ♥✲❛♥❛❧②t✐❝ ♣♦❧②♥♦♠✐❛❧ } . ◗✉❡st✐♦♥ ❋♦r ✇❤✐❝❤ X P n ( X ) = A n ( X )? ❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

P♦❧②❛♥❛❧②t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ❲❡ ✇✐❧❧ s❛② t❤❛t ❢✉♥❝t✐♦♥ f ✐s ♥✲❛♥❛❧②t✐❝ ✐❢ f ( z ) = z n − 1 f n − 1 ( z ) + ... + zf 1 ( z ) + f 0 ( z ) , ✇❤❡r❡ f k ❛r❡ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✳ X ✲ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ C A n ( X ) := { f ∈ C ( X ) : f − ♥✲❛♥❛❧②t✐❝ ✐♥ Int X } , P n ( X ) = Clos C ( X ) { P : P − ♥✲❛♥❛❧②t✐❝ ♣♦❧②♥♦♠✐❛❧ } . ◗✉❡st✐♦♥ ❋♦r ✇❤✐❝❤ X P n ( X ) = A n ( X )? ❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

P♦❧②❛♥❛❧②t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ❚❤❡♦r❡♠ ✭▼❡r❣❡❧②❛♥ ✶✾✺✷✮ P 1 ( X ) = A 1 ( X ) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s❡t C \ X ✐s ❝♦♥♥❡❝t❡❞✳ ❚❤❡♦r❡♠ ✭❈❛r♠♦♥❛ ✶✾✽✺✮ ■❢ C \ X ✐s ❝♦♥♥❡❝t❡❞✱ t❤❡♥ P m ( X ) = A m ( X ) ❢♦r ❛♥② m ≥ 2 ✳ ❚❤❡♦r❡♠ ✭❈❛r♠♦♥❛✱ P❛r❛♠♦♥♦✈✱ ❋❡❞♦r♦✈s❦✐② ✷✵✵✷✮ ▲❡t X ❜❡ ❈❛r❛t❤❡♦❞♦r② ❝♦♠♣❛❝t s❡t✱ m ≥ 2 ✳ ❲❡ ❤❛✈❡ P m ( X ) = A m ( X ) ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② ❜♦✉♥❞❡❞ ❝♦♠♣♦♥❡♥t ♦❢ C \ X ✐s ♥♦t ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✳ ❋♦r ❛r❜✐tr❛r② ❝♦♠♣❛❝t X ❛♥s✇❡r ♠❛② ❞❡♣❡♥❞ ♦♥ m ✳ ❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

P♦❧②❛♥❛❧②t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ❚❤❡♦r❡♠ ✭▼❡r❣❡❧②❛♥ ✶✾✺✷✮ P 1 ( X ) = A 1 ( X ) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s❡t C \ X ✐s ❝♦♥♥❡❝t❡❞✳ ❚❤❡♦r❡♠ ✭❈❛r♠♦♥❛ ✶✾✽✺✮ ■❢ C \ X ✐s ❝♦♥♥❡❝t❡❞✱ t❤❡♥ P m ( X ) = A m ( X ) ❢♦r ❛♥② m ≥ 2 ✳ ❚❤❡♦r❡♠ ✭❈❛r♠♦♥❛✱ P❛r❛♠♦♥♦✈✱ ❋❡❞♦r♦✈s❦✐② ✷✵✵✷✮ ▲❡t X ❜❡ ❈❛r❛t❤❡♦❞♦r② ❝♦♠♣❛❝t s❡t✱ m ≥ 2 ✳ ❲❡ ❤❛✈❡ P m ( X ) = A m ( X ) ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② ❜♦✉♥❞❡❞ ❝♦♠♣♦♥❡♥t ♦❢ C \ X ✐s ♥♦t ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✳ ❋♦r ❛r❜✐tr❛r② ❝♦♠♣❛❝t X ❛♥s✇❡r ♠❛② ❞❡♣❡♥❞ ♦♥ m ✳ ❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

P♦❧②❛♥❛❧②t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ❚❤❡♦r❡♠ ✭▼❡r❣❡❧②❛♥ ✶✾✺✷✮ P 1 ( X ) = A 1 ( X ) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s❡t C \ X ✐s ❝♦♥♥❡❝t❡❞✳ ❚❤❡♦r❡♠ ✭❈❛r♠♦♥❛ ✶✾✽✺✮ ■❢ C \ X ✐s ❝♦♥♥❡❝t❡❞✱ t❤❡♥ P m ( X ) = A m ( X ) ❢♦r ❛♥② m ≥ 2 ✳ ❚❤❡♦r❡♠ ✭❈❛r♠♦♥❛✱ P❛r❛♠♦♥♦✈✱ ❋❡❞♦r♦✈s❦✐② ✷✵✵✷✮ ▲❡t X ❜❡ ❈❛r❛t❤❡♦❞♦r② ❝♦♠♣❛❝t s❡t✱ m ≥ 2 ✳ ❲❡ ❤❛✈❡ P m ( X ) = A m ( X ) ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② ❜♦✉♥❞❡❞ ❝♦♠♣♦♥❡♥t ♦❢ C \ X ✐s ♥♦t ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✳ ❋♦r ❛r❜✐tr❛r② ❝♦♠♣❛❝t X ❛♥s✇❡r ♠❛② ❞❡♣❡♥❞ ♦♥ m ✳ ❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s