The Runge approximation theorem for generalized polynomial hulls

Abstract

It is known from the Runge approximation theorem that every function which is holomorphic in a neighborhood of a compact polynomially convex set $K\subset \complexes^{n}$ can be approximated uniformly on $K$ by analytic polynomials. We shall here prove the same result when the role of the polynomially convex hull $\hat {K}$ is played by the generalized polynomial hull $h_{q}(K)$ introduced by Basener and which can be defined, for each integer $q\in {0,...,n-1}$, by $h_{q}(K)=\displaystyle\bigcup_{P\in \complexes [z_{1},...,z_{n}]} A_{P}$ where $A_{P}=\{z\in \complexes^{n}: |P(z)|\leq \delta_{K}(P,z)\}$, and where $\delta_{K}(P,z)$ denotes the lowest value of $||P||_{K\cap f^{-1}(0)}$ when $f$ ranges in the set of holomorphic polynomial maps $\complexes^{n}\to \complexes^{q}$ vanishing at $z$.