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.

Given a real analytic set $X$ in a complex manifold and a positive integer $d$, denote by ${{\mathcal{A}}^{d}}$ the set of points $p$ in $X$ at which there exists a germ of a complex analytic set of dimension $d$ contained in $X$. It is proved that ${{\mathcal{A}}^{d}}$ is a closed semianalytic subset of $X$.

Let $V$ be an analytic variety in some open set in ${{\mathbb{C}}^{n}}$. For a real analytic curve $\gamma $ with $\gamma (0)\,=\,0$ and $d\,\ge \,1$, define ${{V}_{t}}\,=\,{{t}^{-d}}(V\,-\,\gamma (t))$. It was shown in a previous paper that the currents of integration over ${{V}_{t}}$ converge to a limit current whose support ${{T}_{\gamma ,d}}V$ is an algebraic variety as $t$ tends to zero. Here, it is shown that the canonical defining function of the limit current is the suitably normalized limit of the canonical defining functions of the ${{V}_{t}}$. As a corollary, it is shown that ${{T}_{\gamma ,d}}V$ is either inhomogeneous or coincides with ${{T}_{\gamma ,\,\delta }}V$ for all $\delta $ in some neighborhood of $d$. As another application it is shown that for surfaces only a finite number of curves lead to limit varieties that are interesting for the investigation of Phragmén-Lindelöf conditions. Corresponding results for limit varieties ${{T}_{\sigma ,\delta }}W$ of algebraic varieties $W$ along real analytic curves tending to infinity are derived by a reduction to the local case.

In this paper, we study a problem of extension of holomorphic functions given on a complex hypersurface with singularities on the boundary of a strictly pseudoconvex domain.

We consider bihomogeneous polynomials on complex Euclidean spaces that are positive outside the origin and obtain effective estimates on certain modifications needed to turn them into squares of norms of vector-valued polynomials on complex Euclidean space. The corresponding results for hypersurfaces in complex Euclidean spaces are also proved. The results can be considered as Hermitian analogues of Hilbert's seventeenth problem on representing a positive definite quadratic form on $\mathbb{R}^n$ as a sum of squares of rational functions. They can also be regarded as effective estimates on the power of a Hermitian line bundle required for isometric projective embedding. Further applications are discussed.

Let $V$ be an analytic variety in some open set in ${{\mathbb{C}}^{n}}$ which contains the origin and which is purely $k$-dimensional. For a curve $\gamma $ in ${{\mathbb{C}}^{n}}$, defined by a convergent Puiseux series and satisfying $\gamma (0)\,=\,0$, and $d\,\ge \,1$, define ${{V}_{t}}\,:=\,{{t}^{-d}}\,\left( V\,-\,\gamma \left( t \right) \right)$. Then the currents defined by ${{V}_{t}}$ converge to a limit current ${{T}_{\gamma ,d}}\left[ V \right]$ as $t$ tends to zero. ${{T}_{\gamma ,d}}\left[ V \right]$ is either zero or its support is an algebraic variety of pure dimension $k$ in ${{\mathbb{C}}^{n}}$. Properties of such limit currents and examples are presented. These results will be applied in a forthcoming paper to derive necessary conditions for varieties satisfying the local Phragmén-Lindelöf condition that was used by Hörmander to characterize the constant coefficient partial differential operators which act surjectively on the space of all real analytic functions on ${{\mathbb{R}}^{n}}$.