Polynomial inscriptions

Given n points and a smooth Jordan curve in the complex plane, what is the minimum degree of a non-constant polynomial which maps all of the points to the curve?  It is easy to bound the degree above by n-1, while if the points are collinear and the curve is an ellipse, then the degree is bounded below by n/2.  In joint work with Andrew Lobb, we conjecture that the answer is bounded above by (n/2)-1 whenever the points are concyclic, and we prove it in certain cases.  The proofs in the cases n=4 and n=6 rely on properties of Lagrangian submanifolds of a symplectic vector space due to Polterovich, Viterbo, and Fukaya-Irie, while the proof of a special case when n \ge 8 relies on an argument in Floer theory.