Polynomial Approximation, Local Polynomial Convexity, and Degenerate Cr Singularities — Ii

Abstract

Abstract. We provide some conditions for the graph of a Hölder-continuous function on D, where D is a closed disc 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 D. These conditions often fail to yield any conclusion if rank R DF is not maximal on a sufficiently large subset of D. 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 C 2 at an isolated complex tangency. Introduction and statement of results This paper has evolved from the following two considerations:• Let D be a closed disc in C and let F ∈ C(D). There are numerous results that provide sufficient conditions for the uniform algebra on D generated by z and F to equal C(D); see [21,19,16,9,17,3]. These conditions are sufficient, naturally, for the graph of F to be polynomially convex. The aforementioned results require -either by explicit fiat or through some a priori condition on F -that F −1 {F (ζ)} be at most countable for a.e. ζ ∈ D. This is troublesome because it excludes, for instance, C( ∼ = R 2 )-valued functions in C 1 (D) having rank R DF < 2 on a non-empty open subset of D. It would thus be useful to devise techniques that allow us to detect polynomial convexity without imposing such restrictions.• In a recent work, Dieu & Chi [8] employed a technique that can be applied to situations very different from the one that they study. Their idea, suitably adapted to the given context, might serve as quite a general tool in the study of polynomial convexity of graphs. We wish to argue this case by presenting a couple of adaptations of their idea. Given F as above and a set S ⊆ Dom(F ), we shall write Γ S (F ) := Graph(F ) ∩ (S × C). Let us understand some of the known sufficient conditions on F , for Γ D (F ) to be polynomially convex, by examining critically a representative result selected from the aforementioned papers. Hence consider: 2000 Mathematics Subject Classification. Primary 30E10, 32E20, 32F05.

Normal Forms and Degenerate CR Singularities

“…Problems regarding the local polynomial convexity when 2λ = 1 have been studied by Jöricke [17]. Surfaces in C 2 with degenerate CR singularities and related local polynomial convexity problems near degenerate CR singularities have been studied by Bharali [1], [2], [3], [4]. In codimensions different from 2, we mention the work of Coffman [7], [8] about CR singularities.…”

The local polynomial hull near a degenerate CR singularity: Bishop discs revisited

“…Taking X := (F m + R; D(0; δ)) and f (z, w) := w, and observing that each of the sets in (5.1) is polynomially convex, we conclude from Result 5.3 that (F m + R) is locally polynomially convex at (0, 0)-or, equivalently, that S is locally polynomially convex at p. We now consider Part (2). As before, we shall work in the coordinate system (z, w) with respect to which (S, p) has the representation (3.1).…”

“…Hence, our assumption about the existence of g : (0, 1) −→ A α (D; C 2 ) must be wrong, which establishes Part (2).…”

“…Wiegerinck's examples suggest that very different considerations must apply when (S, p) is not thin. These considerations have been examined-although more from the viewpoint of detecting polynomial convexity than of polynomial hulls-in [1] and in a recent article [2]. As for the assumptions in Wiegerinck's work: we refer the reader to [14,Theorems 3.3,3.4].…”