Counting rational maps on ℙ¹ with prescribed local conditions

2024/08/28 by Nguyen, Khoa D., Ray, Anwesh
#11G50 #11R45 #37P35 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2408.15648

Abstract

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.

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