Local polynomial convexity of certain graphs in C3

Dieu, Nguyen Quang; Phuong, Kieu

doi:10.1307/mmj/1250169073

Cited by 2 publications

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“…• 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.…”

Section: This Paper Has Evolved From the Following Two Considerationsmentioning

confidence: 99%

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

Bharali

2011

Int. J. Math.

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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.

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Section: This Paper Has Evolved From the Following Two Considerationsmentioning

confidence: 99%

“…A note on our notation: given a compact K ⋐ C n , K denotes the polynomially convex hull of K. The idea behind Proposition 1.7 is taken from a step in the proof of Theorem 2.1 in Dieu & Chi's paper [8]. However, their idea is not quite in a form that we can directly use.…”

Section: Then [Z F ] D = C(d) In Particular γ D (F ) Is Polynomiamentioning

confidence: 99%