On the zero-free polynomial approximation problem

TL;DR

This paper investigates whether functions that are zero-free inside a compact set with connected complement can be uniformly approximated by polynomials that are also zero-free on the entire set, extending previous specific cases.

Contribution

It identifies classes of functions for which zero-free polynomial approximation is possible on any compact set with connected complement.

Findings

Zero-free approximation is achievable for certain classes of functions.

The paper extends known results to arbitrary compact sets with connected complement.

Provides conditions under which zero-free polynomial approximation can be realized.

Abstract

Let $E$ be a compact set in $\mathbb C$ with connected complement, and let $A(E)$ be the class of all complex continuous function on $E$ that are analytic in the interior $E^0$ of $E$. Let $f \in A(E)$ be zero free on $E^0$. By Mergelyan's theorem $f$ can be uniformly approximated on $E$ by polynomials, but is it possible to realize such approximation by polynomials that are zero-free on $E$? This natural question has been proposed by J. Andersson and P. Gauthier. So far it has been settled for some particular sets $E$. The present paper describes classes of functions for which zero free approximation is possible on an arbitrary $E$.