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 uncountable, compact subset K K of a Euclidean space, there exists a set X X , in some C N \mathbb {C}^N , that is homeomorphic to K K and is rationally convex but not polynomially convex. In addition, it is shown that for the surfaces in C 3 \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 m -dimensional manifolds ( m ≥ 2 m\geq 2 ) with polynomial hulls containing no analytic discs constructed by Izzo, Samuelsson Kalm, and Wold and by Arosio and Wold.