On the Exceptional Sets of $p$-adic Transcendental Analytic Functions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914875328954368 |
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| author | Miranda, Bruno De Paula Lelis, Jean |
| author_facet | Miranda, Bruno De Paula Lelis, Jean |
| contents | In this paper, we study the exceptional sets $S_f$ of $p$-adic transcendental analytic functions $f$ with rational and algebraic coefficients. We establish a necessary condition for a subset $S \subseteq \overline{\mathbb{Q}} \cap B(0, ρ)$ to be the exceptional set of a $p$-adic transcendental analytic function with rational coefficients, demonstrating that, in general, the answer to Mahler's Problem C over $\mathbb{C}_p$ is negative. However, we prove that if $S$ is closed under algebraic conjugation and contains 0, there exist uncountably many transcendental analytic functions $f \in \mathbb{Q}_ρ[[z]]$ such that $S_f = S$. Furthermore, if $ρ\geq 1$, $f$ can be taken in $\mathbb{Z}_ρ[[z]]$. Additionally, we demonstrate that any $S \subseteq \overline{\mathbb{Q}} \cap B(0, ρ)$ containing 0 can be the exceptional set of uncountably many transcendental analytic functions $f \in \overline{\mathbb{Q}}_ρ[[z]]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13015 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Exceptional Sets of $p$-adic Transcendental Analytic Functions Miranda, Bruno De Paula Lelis, Jean Number Theory 11S80, 11J61 In this paper, we study the exceptional sets $S_f$ of $p$-adic transcendental analytic functions $f$ with rational and algebraic coefficients. We establish a necessary condition for a subset $S \subseteq \overline{\mathbb{Q}} \cap B(0, ρ)$ to be the exceptional set of a $p$-adic transcendental analytic function with rational coefficients, demonstrating that, in general, the answer to Mahler's Problem C over $\mathbb{C}_p$ is negative. However, we prove that if $S$ is closed under algebraic conjugation and contains 0, there exist uncountably many transcendental analytic functions $f \in \mathbb{Q}_ρ[[z]]$ such that $S_f = S$. Furthermore, if $ρ\geq 1$, $f$ can be taken in $\mathbb{Z}_ρ[[z]]$. Additionally, we demonstrate that any $S \subseteq \overline{\mathbb{Q}} \cap B(0, ρ)$ containing 0 can be the exceptional set of uncountably many transcendental analytic functions $f \in \overline{\mathbb{Q}}_ρ[[z]]$. |
| title | On the Exceptional Sets of $p$-adic Transcendental Analytic Functions |
| topic | Number Theory 11S80, 11J61 |
| url | https://arxiv.org/abs/2407.13015 |