True trees are dense

TL;DR

This paper demonstrates that any compact, connected planar set can be approximated by the critical points of a polynomial with two critical values, linking complex analysis and topological graph theory.

Contribution

It establishes a novel approximation method connecting critical points of polynomials to true trees in the sense of dessins d'enfants.

Findings

Any compact, connected set in the plane can be approximated by polynomial critical points.

The approximation uses polynomials with only two critical values.

This bridges complex polynomial critical points and topological structures in the plane.

Abstract

We show that any compact, connected set $K$ in the plane can be approximated by the critical points of a polynomial with two critical values. Equivalently, $K$ can be approximated in the Hausdorff metric by a true tree in the sense of Grothendieck's dessins d'enfants.