Siciak-Zahariuta extremal functions, analytic discs and polynomial hulls

TL;DR

This paper introduces two disc formulas for the Siciak-Zahariuta extremal function and uses them to characterize polynomial hulls of arbitrary compact sets in complex affine space through analytic discs.

Contribution

It extends previous results by removing the connectedness requirement, providing new formulas and characterizations for polynomial hulls in complex analysis.

Findings

Derived two disc formulas for the extremal function

Characterized polynomial hulls via analytic discs

Extended previous work to disconnected sets

Abstract

We prove two disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. We use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in terms of analytic discs. Similar results in previous work of ours required the subsets to be connected.