The moduli space of rational maps and surjectivity of multiplier representation (Q925155)

Let \(\text{Rat}_n\) consist of functions \(f=p/q\) where \(p\) and \(q\) are coprime polynomials of the maximal degree \(n\). The moduli space of \(\text{Rat}_n\) denoted my \(\mathcal M_n\) is the set of all Möbius conjugacy classes of maps in \(\text{Rat}_n\). Let \(\text{Poly}_n\) be the space of all polynomials of degree \(n\). The moduli space of \(\text{Poly}_n\) denoted by \(M_n\) is the set of all affine conjugacy classes of maps in \(\text{Poly}_n\). The map \(\Psi_{\text{Rat}_n}: \mathcal M_n\to\mathbb C^n\) is a natural map to an \(n\)-tuple of multipliers at fixed points. The map \(\Psi_{\text{Poly}_n}\) is defined similarly. The author proves Theorem 1: For every \(n\geq2\), the map \(\Psi_{\text{Rat}_n}: \mathcal M_n\to\mathbb C^n\) is surjective. Theorem 2: The exceptional set \(\mathcal E(n)=\mathbb C^{n-1}\setminus\Psi_{\text{Poly}_n}(M_n)\) is non-empty for every \(n\geq4\). In the case of Theorem 2, the author gives a sufficient condition and a necessary one for points to be not in the image of \(\Psi_{\text{Poly}_n}\), and presents an explicit parametrization for all such points if \(n=4\) or 5. Also, it is shown that the preimage of a generic point by \(\Psi_{\text{Poly}_n}\) consists of \((n-2)!\) points.