Polynomial Inscriptions

TL;DR

This paper proves the existence of polynomial mappings inscribed in any smooth Jordan curve for specific point configurations, using advanced geometric and Floer homology techniques, advancing the understanding of polynomial inscribability.

Contribution

It establishes new existence results for polynomial inscribed configurations on Jordan curves, connecting complex analysis, topology, and Floer homology.

Findings

Existence of quadratic polynomials mapping six concyclic points into any smooth Jordan curve.

Construction of degree at most n-1 polynomials for two regular n-gons' vertices.

Supports a conjecture on polynomial inscribability of point sets in Jordan curves.

Abstract

We prove that for every smooth Jordan curve $\gamma \subset \mathbb{C}$ and for every set $Q \subset \mathbb{C}$ of six concyclic points, there exists a non-constant quadratic polynomial $p \in \mathbb{C}[z]$ such that $p(Q) \subset \gamma$. The proof relies on a theorem of Fukaya and Irie. We also prove that if $Q$ is the union of the vertex sets of two concyclic regular $n$-gons, there exists a non-constant polynomial $p \in \mathbb{C}[z]$ of degree at most $n-1$ such that $p(Q) \subset \gamma$. The proof is based on a computation in Floer homology. These results support a conjecture about which point sets $Q \subset \mathbb{C}$ admit a polynomial inscription of a given degree into every smooth Jordan curve $\gamma$.