An estimate for polynomials on analytic sets

Let $A$ be a connected, analytic (in general, not closed) subset of the complex space $\mathbf C^n$ and let $K\subset A$ be a compact set which is not pluri-polar in $A$. In this article it is proved that the extremal function $V(z,K)$ is locally bounded on $A$ if and only if $A$ belongs to some algebraic set of the same dimension as $A$. Moreover, it is shown that for an algebraic set $A$ in a neighborhood of any ordinary point $z^0\in A_0$ the function $V(z,K)$ can be represented as the limit of an increasing sequence of maximal functions.
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