A new transcendence measure for the values of the exponential function at algebraic arguments
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909510414630912 |
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| author | Fischler, Stéphane Rivoal, Tanguy |
| author_facet | Fischler, Stéphane Rivoal, Tanguy |
| contents | Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $δ\ge 1$ and usual height $H\ge 1$, and let $α\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. Mahler proved in 1931 the following transcendence measure for $e^α$: for any $\varepsilon\>0$, there exists $c\>0$ such that $\vert P(e^α)\vert\>c/H^{μ(d,δ)+\varepsilon}$ where the exponent $μ(d,δ)=(4d^2-2d)δ+2d-1$. Zheng obtained a better result in 1991 with $μ(d,δ)=(4d^2-2d)δ-1$. In this paper, we provide a new explicit exponent $μ(d,δ)$ which improves on Zheng's transcendence measure for all $δ\ge 2$ and all $d\ge 2$. When $δ=1$, we recover his bound for all $d\ge 2$, which had in fact already been obtained by Kappe in 1966. Our improvement rests upon the optimization of an accessory parameter in Siegel's classical determinant method applied to Hermite-Pad{é} approximants to powers of the exponential function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17992 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new transcendence measure for the values of the exponential function at algebraic arguments Fischler, Stéphane Rivoal, Tanguy Number Theory Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $δ\ge 1$ and usual height $H\ge 1$, and let $α\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. Mahler proved in 1931 the following transcendence measure for $e^α$: for any $\varepsilon\>0$, there exists $c\>0$ such that $\vert P(e^α)\vert\>c/H^{μ(d,δ)+\varepsilon}$ where the exponent $μ(d,δ)=(4d^2-2d)δ+2d-1$. Zheng obtained a better result in 1991 with $μ(d,δ)=(4d^2-2d)δ-1$. In this paper, we provide a new explicit exponent $μ(d,δ)$ which improves on Zheng's transcendence measure for all $δ\ge 2$ and all $d\ge 2$. When $δ=1$, we recover his bound for all $d\ge 2$, which had in fact already been obtained by Kappe in 1966. Our improvement rests upon the optimization of an accessory parameter in Siegel's classical determinant method applied to Hermite-Pad{é} approximants to powers of the exponential function. |
| title | A new transcendence measure for the values of the exponential function at algebraic arguments |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.17992 |