Fields of definition of dynamical systems on ${\mathbb{P}}^1$. Improvements on a result of Silverman

J. Silverman proved that a dynamical system on ${\mathbb{P}}^1$ descends to the field of moduli if it is polynomial or it has even degree, but for non-polynomial ones of odd degree the picture is less clear. We give a complete characterization of which dynamical systems over ${\mathbb{P}}^1$ descend to the field of moduli.