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 ($m\geq 2$) with polynomial hulls containing no analytic discs constructed by Izzo, Samuelsson Kalm, and Wold and by Arosio and Wold.