Polynomial Inscriptions

Abstract

We prove that for every smooth Jordan curve ${\hat{I}}^3\subset \mathbb{C}$ 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 $p(Q)\subset {\hat{I}}^3$. 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 $p(Q)\subset {\hat{I}}^3$. 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 ${\hat{I}}^3$.

18 pages