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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2504.14041 |
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| _version_ | 1866916696636260352 |
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| author | Waldschmidt, Michel |
| author_facet | Waldschmidt, Michel |
| contents | For almost all tuples $(x_1,\dots,x_n)$ of complex numbers, a strong version of Schanuel's Conjecture is true: the $2n$ numbers $x_1,\dots,x_n, {\mathrm e}^{x_1},\dots, {\mathrm e}^{x_n}$ are algebraically independent. Similar statements hold when one replaces the exponential function ${\mathrm e}^z$ with algebraically independent functions. We give examples involving elliptic and quasi--elliptic functions, that we prove to be algebraically independent: $z$, $\wp(z)$, $ζ(z)$, $σ(z)$, exponential functions, and Serre functions related with integrals of the third kind. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14041 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schanuel Property for Elliptic and Quasi--Elliptic Functions Waldschmidt, Michel Number Theory 11J81 11J89 14K25 For almost all tuples $(x_1,\dots,x_n)$ of complex numbers, a strong version of Schanuel's Conjecture is true: the $2n$ numbers $x_1,\dots,x_n, {\mathrm e}^{x_1},\dots, {\mathrm e}^{x_n}$ are algebraically independent. Similar statements hold when one replaces the exponential function ${\mathrm e}^z$ with algebraically independent functions. We give examples involving elliptic and quasi--elliptic functions, that we prove to be algebraically independent: $z$, $\wp(z)$, $ζ(z)$, $σ(z)$, exponential functions, and Serre functions related with integrals of the third kind. |
| title | Schanuel Property for Elliptic and Quasi--Elliptic Functions |
| topic | Number Theory 11J81 11J89 14K25 |
| url | https://arxiv.org/abs/2504.14041 |