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 of a Euclidean space, there exists a set X, in some 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 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 ≥ 2) with polynomial hulls containing no analytic discs constructed by Izzo, Samuelsson Kalm, and Wold and by Arosio and Wold. Dedicated to Hari Bercovici2000 Mathematics Subject Classification. 32E20, 32A65 46J10, 46J15.

Polynomially convex embeddings of odd-dimensional closed manifolds

“…To show that b † D h r . †/, we use an argument due to Izzo, who showed in [22,Section 3] that E satisfies the generalized argument principle, i.e., if p is a polynomial that has a continuous logarithm on E, then 0 … p.E/. We now apply the following result due to Stolzenberg ([29]) to X D E and Y D B 0 .…”

Polynomially convex embeddings of even-dimensional compact manifolds

“…In [2], it is shown that if the embedding is also required to be totally real, then the optimal value of n is ⌊3m/2⌋, for any m ≥ 2. In [17], it is shown that the constructions in [18] and [2] can be done so that the rational and polynomial hulls of the embeddings coincide. In our next result, we show that the answer to the original question is strictly less than ⌊3m/2⌋ for even-dimensional manifolds.…”

“…To show that Σ = h r (Σ), we use Izzo's argument from [17,Section 3]. He shows that the set E satisfies the generalized argument principle, i.e., if p is a polynomial that has a continuous logarithm on E, then 0 / ∈ p(E).…”

“…To show that Σ = h r (Σ), we use an argument due to Izzo, who showed in [21,Section 3] that E satisfies the generalized argument principle, i.e., if p is a polynomial that has a continuous logarithm on E, then 0 / ∈ p(E). We now apply the following result due to Stolzenberg ([28]) to X = E and Y = B .…”

A Cantor set whose polynomial hull contains no analytic discs

“…This implies that the polynomial and rational hulls of X coincide. In contrast, the construction given here does not yield density of invertible elements, and as shown in the first author's paper [8], it can be used to obtain rationally convex examples.…”