Determining sets for holomorphic functions on the symmetrized bidisk

Abstract

Abstract 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.