Counting rational maps on $\mathbb{P}^1$ with prescribed local conditions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918312033648640 |
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| author | Nguyen, Khoa D. Ray, Anwesh |
| author_facet | Nguyen, Khoa D. Ray, Anwesh |
| contents | 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 $ϕ$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_15648 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting rational maps on $\mathbb{P}^1$ with prescribed local conditions Nguyen, Khoa D. Ray, Anwesh Number Theory Dynamical Systems 11R45, 11G50, 37P35 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 $ϕ$ 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. |
| title | Counting rational maps on $\mathbb{P}^1$ with prescribed local conditions |
| topic | Number Theory Dynamical Systems 11R45, 11G50, 37P35 |
| url | https://arxiv.org/abs/2408.15648 |