Solving the Gleason problem on linearly convex domains

Abstract.

Let $\Omega$ be a bounded, connected linearly convex set in ${\mathbf{C}} ^n$ with $C^{1+\epsilon}$ boundary. We show that the maximal ideal (both in $H^{\infty}(\Omega)$) and $A^{m}(\Omega), 0 \leq m \leq \infty$) consisting of all functions vanishing at $p \in \Omega$ is generated by the coordinate functions $z_1 - p_1, \ldots, z_n - p_n$.