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Abstract

Let $K$ be any subset of $ \mathbb C^N $. We define a pluricomplex Green's function $V_{K,\theta} $ for $ \theta $-incomplete polynomials. We establish properties of $V_{K,\theta} $ analogous to those of the weighted pluricomplex Green's function. When $K$ is a regular compact subset of $ \mathbb R^N $, we show that every continuous function that can be approximated uniformly on $K$ by $\theta$-incomplete polynomials, must vanish on $ K \setminus {\rm supp}\,(dd^{c} V_{K,\theta})^N $. We prove a version of Siciak's theorem and a comparison theorem for $ \theta $-incomplete polynomials. We compute $ {\rm supp}\,(dd^{c} V_{K,\theta})^N $ when $K$ is a compact section.