Some generalizations of Chirka's extension theorem

In this paper, we generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of $S \cup (\partial D \times D)$ - where $D$ is the open unit disc in $\mathbb{C}$ and $S$ is the graph of a continuous $D-$valued function on $\overline{D}$ - to higher dimensions, for certain classes of graphs $S \subseteq \overline{D} \times {D}^{n}, n>1$. In particular, we show that Chirka's extension theorem generalizes to configurations in ${\mathbb{C} }^{n+1}, n>1$, involving graphs of (non-holomorphic) polynomial maps with small coefficients.