Polynomial approximation, local polynomial convexity, and degenerate CR singularities -- II

TL;DR

This paper establishes new conditions for polynomial convexity of graphs of Hölder-continuous functions on a disk in the complex plane, using plurisubharmonic functions to address cases where traditional smoothness assumptions fail.

Contribution

It introduces a novel technique involving plurisubharmonic functions to analyze polynomial convexity for non-smooth functions, extending previous results beyond smooth cases.

Findings

Provided conditions for polynomial convexity of Hölder-continuous graphs

Developed a technique using plurisubharmonic functions to handle non-maximal rank cases

Made observations on polynomial hulls at isolated complex tangencies in C^2

Abstract

We provide some conditions for the graph of a Hoelder-continuous function on \bar{D}, where \bar{D} is a closed disc in the complex plane, 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 \bar{D}. These conditions often fail to yield any conclusion if rank_R(DF) is not maximal on a sufficiently large subset of \bar{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.