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Determining sets for holomorphic functions on the symmetrized bidisk
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 66 / Issue 3 / September 2023
- Published online by Cambridge University Press:
- 31 January 2023, pp. 984-996
- Print publication:
- September 2023
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- Article
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A subset
${\mathcal D}$ of a domain 
$\Omega \subset {\mathbb C}^d$ is determining for an analytic function 
$f:\Omega \to \overline {{\mathbb D}}$ if whenever an analytic function 
$g:\Omega \rightarrow \overline {{\mathbb D}}$ coincides with f on 
${\mathcal D}$, equals to f on whole 
$\Omega $. This note finds several sufficient conditions for a subset of the symmetrized bidisk to be determining. For any 
$N\geq 1$, a set consisting of 
$N^2-N+1$ many points is constructed which is determining for any rational inner function with a degree constraint. We also investigate when the intersection of the symmetrized bidisk intersected with some special algebraic varieties can be determining for rational inner functions.
