# Wydawnictwa / Czasopisma IMPAN / Annales Polonici Mathematici / Wszystkie zeszyty

## A Green's function for $\theta$-incomplete polynomials

### Tom 90 / 2007

Annales Polonici Mathematici 90 (2007), 21-35 MSC: Primary 32U35. DOI: 10.4064/ap90-1-2

#### Streszczenie

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.

#### Autorzy

• Joe CallaghanDepartment of Mathematics
University of Toronto
Toronto, Ontario