### A Green's function for θ-incomplete polynomials

Let K be any subset of ${\u2102}^{N}$. We define a pluricomplex Green’s function ${V}_{K,\theta}$ for θ-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 θ-incomplete polynomials, must vanish on $K\setminus supp{\left(d{d}^{c}{V}_{K,\theta}\right)}^{N}$. We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute $supp{\left(d{d}^{c}{V}_{K,\theta}\right)}^{N}$ when K is a compact section.