Polynomials on algebraic manifolds

In this paper, we study polynomials on algebraic subvarieties $A\subset{\mathbb{C}}^{N},$ $\dim A=n,$ $n\le N,$ in the point of view their polynomial extensions into the enclosing space ${\mathbb{C}}^{N}.$ It is important in observing extremal Green function $V(z,K)$ for compact subset $K\subset A.$

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