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| In the theory of functions of [[several complex variables]], a branch of [[mathematics]], a '''polydisc''' is a [[Cartesian product]] of [[disc (mathematics)|disc]]s.
| | Sail Maker Edgar Sanderford from Norwich, loves to spend time ghost hunting, diet and dominoes. Recalls what a marvelous place it have been having visited Abbey and Altenmünster of Lorsch. |
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| More specifically, if we denote by <math> D(z,r)</math> the [[open set|open]] disc of center ''z'' and radius ''r'' in the [[complex plane]], then an open polydisc is a set of the form
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| :<math>D(z_1,r_1) \times \dots \times D(z_n,r_n).</math>
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| It can be equivalently written as
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| :<math>\{ w=(w_1, w_2, \dots, w_n) \in {\mathbf{C}}^n \mid \vert z_k - w_k \vert < r_k, \mbox{ for all } k = 1,\dots,n \}.</math>
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| One should not confuse the polydisc with the [[open ball]] in '''C'''<sup>n</sup>, which is defined as
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| :<math>\{ w \in \mathbf{C}^n \mid \lVert z - w \rVert < r \}.</math>
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| Here, the [[norm (mathematics)|norm]] is the [[Euclidean distance]] in '''C'''<sup>n</sup>.
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| When <math>n > 1</math>, open balls and open polydiscs are ''not'' biholomorphically equivalent, that is, there is no [[biholomorphic mapping]] between the two. This was proven by [[Henri Poincaré|Poincaré]] in 1907 by showing that their [[automorphism group]]s have different dimensions as [[Lie group]]s.
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| When <math>n=2</math> the term ''bidisc'' is sometimes used.
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| A polydisc is an example of [[logarithmically convex set|logarithmically convex]] [[Reinhardt domain]].
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| ==References==
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| * {{cite book | author=Steven G Krantz | title=Function Theory of Several Complex Variables | publisher=American Mathematical Society | year=Jan 1, 2002 | isbn= 0-8218-2724-3}}
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| * {{cite book | author=John P D'Angelo, D'Angelo P D'Angelo | title=Several Complex Variables and the Geometry of Real Hypersurfaces | publisher=CRC Press | year=Jan 6, 1993 | isbn= 0-8493-8272-6}}
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| {{PlanetMath attribution|id=6030|title=polydisc}}
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| [[Category:Several complex variables]]
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Sail Maker Edgar Sanderford from Norwich, loves to spend time ghost hunting, diet and dominoes. Recalls what a marvelous place it have been having visited Abbey and Altenmünster of Lorsch.