No topological condition implies equality of polynomial and rational hulls

Abstract

It is shown that no purely topological condition implies the equality of the polynomial and rational hulls of a set: For any compact subset $K$ of a Euclidean space, there exists a set $X$, in some ${\mathbb{C}}^N$, that is homeomorphic to $K$ and is rationally convex but not polynomially convex. In addition, it is shown that for the surfaces in ${\mathbb{C}}^3$ constructed by Izzo and Stout, whose polynomial hulls are nontrivial but contain no analytic discs, the polynomial and rational hulls coincide, thereby answering a question of Gupta. Equality of polynomial and rational hulls is shown also for $m$-dimensional manifolds ($m\ge 2$) with polynomial hulls containing no analytic discs constructed by Izzo, Samuelsson Kalm, and Wold and by Arosio and Wold.

arXiv commentSome results have been strengthened by decreasing the dimension of the ambient space