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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2404.06968 |
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| _version_ | 1866909165553713152 |
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| author | Kramer-Miller, Joe |
| author_facet | Kramer-Miller, Joe |
| contents | Let $E_p(x)$ denote the Artin-Hasse exponential and let $\overline{E}_p(x)$ denote its reduction modulo $p$ in $\mathbb{F}_p[[x]]$. In this article we study transcendence properties of $\overline{E}_p(x)$ over $\mathbb{F}_p[x]$. We give two proofs that $\overline{E}_p(x)$ is transcendental, affirmatively answering a question of Thakur. We also prove algebraic independence results: i) for $f_1,\dots,f_r \in x\mathbb{F}_p[x]$ satisfying certain linear independence properties, we show that the $\overline{E}_p(f_1), \dots, \overline{E}_p(f_r)$ are algebraically independent over $\mathbb{F}_p[x]$ and ii) we determine the algebraic relations between $\overline{E}_p(cx)$, where $c \in \mathbb{F}_p^\times$. Our proof studies the higher derivatives of $\overline{E}_p(x)$ and makes use of iterative differential Galois theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06968 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transcendence properties of the Artin-Hasse exponential modulo $p$ Kramer-Miller, Joe Number Theory Combinatorics 11J91, 11B85 Let $E_p(x)$ denote the Artin-Hasse exponential and let $\overline{E}_p(x)$ denote its reduction modulo $p$ in $\mathbb{F}_p[[x]]$. In this article we study transcendence properties of $\overline{E}_p(x)$ over $\mathbb{F}_p[x]$. We give two proofs that $\overline{E}_p(x)$ is transcendental, affirmatively answering a question of Thakur. We also prove algebraic independence results: i) for $f_1,\dots,f_r \in x\mathbb{F}_p[x]$ satisfying certain linear independence properties, we show that the $\overline{E}_p(f_1), \dots, \overline{E}_p(f_r)$ are algebraically independent over $\mathbb{F}_p[x]$ and ii) we determine the algebraic relations between $\overline{E}_p(cx)$, where $c \in \mathbb{F}_p^\times$. Our proof studies the higher derivatives of $\overline{E}_p(x)$ and makes use of iterative differential Galois theory. |
| title | Transcendence properties of the Artin-Hasse exponential modulo $p$ |
| topic | Number Theory Combinatorics 11J91, 11B85 |
| url | https://arxiv.org/abs/2404.06968 |