Counting rational maps on P¹ with prescribed local conditions

We explore distribution questions for rational maps on the projective line P¹ over Q within the framework of arithmetic dynamics, drawing analogies to elliptic curves. Specifically, we investigate counting problems for rational maps φ of fixed degree d ≥ 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 $32.7%$ possess a squarefree, and hence minimal, resultant.