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

Abstract

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 $\ddot{I}•$ 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 $32.7\%$ possess a squarefree, and hence minimal, resultant.

Version 3: 29 pages, accepted for publication in Research in Number theory