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arXiv · 2408.15648

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

Khoa D. Nguyen·Anwesh Ray
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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.

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Khoa D. Nguyen, Anwesh Ray. 2024-08-28. Counting rational maps on $\mathbb{P}^1$ with prescribed local conditions. https://doi.org/10.1007/s40993-026-00703-8

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