Degree and holomorphic extensions

Let $D$ be a bounded convex domain in $\C^N,\ N\geq 2$. We prove that a continuous map $\Phi\colon\ bD\rightarrow\C^N$ extends holomorphically through $D$ if and only if for every polynomial map $P\colon\ \C^N\rightarrow\C^N$ such that $\Phi+P\not= 0$ on $bD$, the degree of $\Phi+P|bD$ is nonnegative. We also prove another such theorem for more general domains.

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Published 1 January 2007