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

TL;DR

This paper studies the distribution of rational maps on the projective line over the rationals, focusing on counting maps with specific reduction properties and showing that maps with minimal resultants are positively dense, with explicit degree 2 computations.

Contribution

It establishes positive density results for rational maps with minimal resultants and provides explicit computations for degree 2 maps, linking arithmetic dynamics to distribution questions.

Findings

Maps with minimal resultant have positive density.

Approximately 32.7% of degree 2 maps have squarefree, minimal resultants.

Explicit degree 2 computations demonstrate the distribution of minimal resultants.

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 $\phi$ of fixed degree $d \geq 2$ 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.