# Polynomial approximation avoiding values in countable sets

## TL;DR

This paper extends classical polynomial approximation theorems to ensure the approximating polynomials avoid specified countable sets of values, broadening the scope of uniform approximation on complex domains.

## Contribution

It generalizes Lavrenté's and Mergelyan's theorems to include polynomial approximations that avoid given countable sets of values on complex domains.

## Key findings

- Polynomial approximations can avoid countable sets on compact sets with connected complements.
- Extension of Lavrenté's theorem to avoid countable sets.
- Extension of Mergelyan's theorem for finite unions of Jordan domains.

## Abstract

We generalize a version of Lavrent\'ev's theorem which says that a function that is continuous on a compact set K with connected complement and without interior points can be uniformly approximated as closely as desired by a polynomial without zeros on the set K, so that the polynomial can avoid values from any given countable set. We also prove a corresponding version of Mergelyan's theorem when the interior of K is a finite union of Jordan domains, pairwise separated by a positive distance.