Polynomial inscriptions

We prove that, for every smooth Jordan curve γ⊂C and for every set Q⊂C of six concyclic points, there exists a nonconstant quadratic polynomial p∈C[z] such that p(Q)⊂γ. 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 nonconstant polynomial p∈C[z] of degree at most n−1 such that p(Q)⊂γ. The proof is based on a computation in Floer homology. These results support a conjecture about which point sets Q⊂C admit a polynomial inscription of a given degree into every smooth Jordan curve γ.