Counting rational maps on ${\mathbb{P}}^1$ with prescribed local conditions

We explore distribution questions for rational maps on the projective line ${\mathbb{P}}^1$ over $\mathbb{Q}$ within the framework of arithmetic dynamics, drawing analogies to elliptic curves. Specifically, we investigate counting problems for rational maps $\phi$ of fixed degree $d\ge 2$ with prescribed reduction properties. Our main result establishes that the set of rational maps with minimal resultant has positive density. Additionally, for degree 2 rational maps, we perform explicit computations demonstrating that over $32.7\mathrm{\%}$ possess a squarefree, and hence minimal, resultant.