Polynomially convex embeddings of odd-dimensional closed manifolds

It is shown that any smooth closed orientable manifold of dimension 2 ⁢ k + 1 {2k+1} , k ≥ 2 {k\geq 2} , admits a smooth polynomially convex embedding into ℂ 3 ⁢ k {\mathbb{C}^{3k}} . This improves by 1 the previously known lower bound of 3 ⁢ k + 1 {3k+1} on the possible ambient complex dimension for such embeddings (which is sharp when k = 1 {k=1} ). It is further shown that the embeddings produced have the property that all continuous functions on the image can be uniformly approximated by holomorphic polynomials. Lastly, the same technique is modified to construct embeddings whose images have nontrivial hulls containing no nontrivial analytic disks. The distinguishing feature of this dimensional setting is the appearance of nonisolated CR-singularities, which cannot be tackled using only local analytic methods (as done in earlier results of this kind), and a topological approach is required.

Characterizations of the symmetrized polydisc via another family of domains

We find new characterizations for the points in the symmetrized polydisc [Formula: see text], a family of domains associated with the spectral interpolation, defined by [Formula: see text] We introduce a new family of domains which we call the extended symmetrized polydisc [Formula: see text], and define in the following way: [Formula: see text] [Formula: see text] We show that [Formula: see text] for [Formula: see text] and that [Formula: see text] for [Formula: see text]. We first obtain a variety of characterizations for the points in [Formula: see text] and we apply these necessary and sufficient conditions to produce an analogous set of characterizations for the points in [Formula: see text]. Also, we obtain similar characterizations for the points in [Formula: see text], where [Formula: see text]. A set of [Formula: see text] fractional linear transformations plays central role in the entire program. We also show that for [Formula: see text], [Formula: see text] is nonconvex but polynomially convex and is starlike about the origin but not circled.

POLYNOMIAL APPROXIMATION, LOCAL POLYNOMIAL CONVEXITY, AND DEGENERATE CR SINGULARITIES — II

We provide some conditions for the graph of a Hölder-continuous function on [Formula: see text], where [Formula: see text] is a closed disk in ℂ, to be polynomially convex. Almost all sufficient conditions known to date — provided the function (say F) is smooth — arise from versions of the Weierstrass Approximation Theorem on [Formula: see text]. These conditions often fail to yield any conclusion if rank ℝDF is not maximal on a sufficiently large subset of [Formula: see text]. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in ℂ2 at an isolated complex tangency.