Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

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.