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Difference between revisions of "Thin set"

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A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092620/t0926201.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092620/t0926202.png" /> such that, for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092620/t0926203.png" />, there exists an open polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092620/t0926204.png" /> and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092620/t0926205.png" /> which is holomorphic, not identically equal to zero, but which vanishes on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092620/t0926206.png" />.
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A subset  $  A $
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of a domain  $  D \subset  \mathbf C  ^ {k} $
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such that, for each point  $  z \in D $,
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there exists an open polydisc  $  \Delta ( z, r) \subset  D $
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and a function  $  f $
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which is holomorphic, not identically equal to zero, but which vanishes on  $  A \cap \Delta ( z, r) $.
  
 
====Comments====
 
====Comments====

Latest revision as of 08:25, 6 June 2020


A subset $ A $ of a domain $ D \subset \mathbf C ^ {k} $ such that, for each point $ z \in D $, there exists an open polydisc $ \Delta ( z, r) \subset D $ and a function $ f $ which is holomorphic, not identically equal to zero, but which vanishes on $ A \cap \Delta ( z, r) $.

Comments

Usually, being thin means being a subset of an analytic set. Cf. also Thinness of a set.

References

[a1] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) pp. Chapt. 1, Sect. C
[a2] R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 1, Sect. 3
How to Cite This Entry:
Thin set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thin_set&oldid=48965
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article